\(\int \sqrt {c+d x} (a-b x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [141]

Optimal result

Integrand size = 37, antiderivative size = 1077 \[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Output:

4/45045*(195*a^3*d^6*D-9*a^2*b*d^4*(-65*B*d^2+6*C*c*d+3*D*c^2)+3*a*b^2*c*d 
^2*(572*A*d^3-299*B*c*d^2+186*C*c^2*d-128*D*c^3)-4*b^3*c^3*(143*A*d^3-104* 
B*c*d^2+80*C*c^2*d-64*D*c^3))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^6+4/150 
15*(3*a^2*d^4*(77*C*d+6*D*c)-a*b*d^2*(-1001*A*d^3-208*B*c*d^2+127*C*c^2*d- 
86*D*c^3)+b^2*c^2*(143*A*d^3-104*B*c*d^2+80*C*c^2*d-64*D*c^3))*x*(d*x+c)^( 
1/2)*(-b*x^2+a)^(1/2)/b/d^5+2/9009*(39*a^2*d^4*D-3*a*b*d^2*(-39*B*d^2+19*C 
*c*d-10*D*c^2)+b^2*c*(143*A*d^3-104*B*c*d^2+80*C*c^2*d-64*D*c^3))*(d*x+c)^ 
(1/2)*(-b*x^2+a)^(3/2)/b^2/d^4+2/1287*(3*a*d^2*(11*C*d-D*c)-b*(-143*A*d^3- 
13*B*c*d^2+10*C*c^2*d-8*D*c^3))*x*(d*x+c)^(1/2)*(-b*x^2+a)^(3/2)/b/d^3-2/4 
29*(13*D*a-3*b*(-13*B*d^2+10*C*c*d-8*D*c^2)/d^2)*(d*x+c)^(1/2)*(-b*x^2+a)^ 
(5/2)/b^2-2/39*(3*C*d-5*D*c)*(d*x+c)^(3/2)*(-b*x^2+a)^(5/2)/b/d^2-2/15*D*( 
d*x+c)^(5/2)*(-b*x^2+a)^(5/2)/b/d^2-8/45045*a^(1/2)*(3*a^3*d^6*(231*C*d+83 
*D*c)+3*a*b^2*c^2*d^2*(715*A*d^3-403*B*c*d^2+266*C*c^2*d-192*D*c^3)-3*a^2* 
b*d^4*(-1001*A*d^3-403*B*c*d^2+145*C*c^2*d-77*D*c^3)-4*b^3*c^4*(143*A*d^3- 
104*B*c*d^2+80*C*c^2*d-64*D*c^3))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellip 
ticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c 
+a^(1/2)*d))^(1/2))/b^(3/2)/d^7/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b* 
x^2+a)^(1/2)+8/45045*a^(1/2)*(-a*d^2+b*c^2)*(195*a^3*d^6*D-9*a^2*b*d^4*(-6 
5*B*d^2+6*C*c*d+3*D*c^2)+3*a*b^2*c*d^2*(572*A*d^3-299*B*c*d^2+186*C*c^2*d- 
128*D*c^3)-4*b^3*c^3*(143*A*d^3-104*B*c*d^2+80*C*c^2*d-64*D*c^3))*((d*x...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.94 (sec) , antiderivative size = 9837, normalized size of antiderivative = 9.13 \[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Result too large to show} \] Input:

Integrate[Sqrt[c + d*x]*(a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 1046, normalized size of antiderivative = 0.97, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.459, Rules used = {2185, 27, 2185, 27, 687, 27, 682, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {2 \int -\frac {5}{2} \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (b (3 C d-5 c D) x^2 d^2+(3 A b d+a c D) d^2+\left (-2 b D c^2+3 b B d^2+a d^2 D\right ) x d\right )dx}{15 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (b (3 C d-5 c D) x^2 d^2+(3 A b d+a c D) d^2+\left (-2 b D c^2+3 b B d^2+a d^2 D\right ) x d\right )dx}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {2 \int -\frac {1}{2} b d^3 \sqrt {c+d x} \left (d (39 A b d+9 a C d-2 a c D)-\left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) x\right ) \left (a-b x^2\right )^{3/2}dx}{13 b d^2}-\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{13} d \int \sqrt {c+d x} \left (d (39 A b d+9 a C d-2 a c D)-\left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) x\right ) \left (a-b x^2\right )^{3/2}dx-\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (-13 a d^2 D-39 b B d^2-24 b c^2 D+30 b c C d\right )}{11 b}-\frac {2 \int -\frac {\left (d \left (429 A c d b^2+a \left (13 a D d^2+b \left (2 D c^2+69 C d c+39 B d^2\right )\right )\right )+3 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{2 \sqrt {c+d x}}dx}{11 b}\right )-\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {\int \frac {\left (d \left (429 A c d b^2+a \left (13 a D d^2+b \left (2 D c^2+69 C d c+39 B d^2\right )\right )\right )+3 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}}dx}{11 b}+\frac {2 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (-13 a d^2 D-39 b B d^2-24 b c^2 D+30 b c C d\right )}{11 b}\right )-\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (39 a^2 d^4 D+7 b d x \left (3 a d^2 (11 C d-c D)-b \left (-143 A d^3-13 B c d^2-8 c^3 D+10 c^2 C d\right )\right )-3 a b d^2 \left (-39 B d^2-10 c^2 D+19 c C d\right )+b^2 c \left (143 A d^3-104 B c d^2-64 c^3 D+80 c^2 C d\right )\right )}{21 d^2}-\frac {4 \int -\frac {3 b \left (a d \left (39 a^2 D d^4+3 a b \left (3 D c^2+58 C d c+39 B d^2\right ) d^2+b^2 c \left (-8 D c^3+10 C d c^2-13 B d^2 c+1144 A d^3\right )\right )+b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{21 b d^2}}{11 b}+\frac {2 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (-13 a d^2 D-39 b B d^2-24 b c^2 D+30 b c C d\right )}{11 b}\right )-\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {\frac {2 \int \frac {\left (a d \left (39 a^2 D d^4+3 a b \left (3 D c^2+58 C d c+39 B d^2\right ) d^2+b^2 c \left (-8 D c^3+10 C d c^2-13 B d^2 c+1144 A d^3\right )\right )+b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{7 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (39 a^2 d^4 D+7 b d x \left (3 a d^2 (11 C d-c D)-b \left (-143 A d^3-13 B c d^2-8 c^3 D+10 c^2 C d\right )\right )-3 a b d^2 \left (-39 B d^2-10 c^2 D+19 c C d\right )+b^2 c \left (143 A d^3-104 B c d^2-64 c^3 D+80 c^2 C d\right )\right )}{21 d^2}}{11 b}+\frac {2 \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (-13 a d^2 D-39 b B d^2-24 b c^2 D+30 b c C d\right )}{11 b}\right )-\frac {2}{13} d \left (a-b x^2\right )^{5/2} (c+d x)^{3/2} (3 C d-5 c D)}{3 b d^3}-\frac {2 D \left (a-b x^2\right )^{5/2} (c+d x)^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {a-b x^2}}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (195 a^3 D d^6+9 a^2 b \left (3 D c^2+71 C d c+65 B d^2\right ) d^4+3 a b^2 c \left (-42 D c^3+59 C d c^2-91 B d^2 c+1573 A d^3\right ) d^2-b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )+b \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \int \frac {a d \left (195 a^3 D d^6+9 a^2 b \left (3 D c^2+71 C d c+65 B d^2\right ) d^4+3 a b^2 c \left (-42 D c^3+59 C d c^2-91 B d^2 c+1573 A d^3\right ) d^2-b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )+b \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {b \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {b \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \left (-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \left (-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {1}{13} d \left (\frac {2 \left (-24 b D c^2+30 b C d c-39 b B d^2-13 a d^2 D\right ) \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b}+\frac {\frac {2 \sqrt {c+d x} \left (39 a^2 D d^4-3 a b \left (-10 D c^2+19 C d c-39 B d^2\right ) d^2+7 b \left (3 a d^2 (11 C d-c D)-b \left (-8 D c^3+10 C d c^2-13 B d^2 c-143 A d^3\right )\right ) x d+b^2 c \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{21 d^2}+\frac {2 \left (\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2+3 b \left (3 a^2 (77 C d+6 c D) d^4-a b \left (-86 D c^3+127 C d c^2-208 B d^2 c-1001 A d^3\right ) d^2+b^2 c^2 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) x d-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right )}{15 d^2}+\frac {2 \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (195 a^3 D d^6-9 a^2 b \left (3 D c^2+6 C d c-65 B d^2\right ) d^4+3 a b^2 c \left (-128 D c^3+186 C d c^2-299 B d^2 c+572 A d^3\right ) d^2-4 b^3 c^3 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^3 (231 C d+83 c D) d^6-3 a^2 b \left (-77 D c^3+145 C d c^2-403 B d^2 c-1001 A d^3\right ) d^4+3 a b^2 c^2 \left (-192 D c^3+266 C d c^2-403 B d^2 c+715 A d^3\right ) d^2-4 b^3 c^4 \left (-64 D c^3+80 C d c^2-104 B d^2 c+143 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{7 d^2}}{11 b}\right )-\frac {2}{13} d (3 C d-5 c D) (c+d x)^{3/2} \left (a-b x^2\right )^{5/2}}{3 b d^3}-\frac {2 D (c+d x)^{5/2} \left (a-b x^2\right )^{5/2}}{15 b d^2}\)

Input:

Int[Sqrt[c + d*x]*(a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

(-2*D*(c + d*x)^(5/2)*(a - b*x^2)^(5/2))/(15*b*d^2) + ((-2*d*(3*C*d - 5*c* 
D)*(c + d*x)^(3/2)*(a - b*x^2)^(5/2))/13 + (d*((2*(30*b*c*C*d - 39*b*B*d^2 
 - 24*b*c^2*D - 13*a*d^2*D)*Sqrt[c + d*x]*(a - b*x^2)^(5/2))/(11*b) + ((2* 
Sqrt[c + d*x]*(39*a^2*d^4*D - 3*a*b*d^2*(19*c*C*d - 39*B*d^2 - 10*c^2*D) + 
 b^2*c*(80*c^2*C*d - 104*B*c*d^2 + 143*A*d^3 - 64*c^3*D) + 7*b*d*(3*a*d^2* 
(11*C*d - c*D) - b*(10*c^2*C*d - 13*B*c*d^2 - 143*A*d^3 - 8*c^3*D))*x)*(a 
- b*x^2)^(3/2))/(21*d^2) + (2*((2*Sqrt[c + d*x]*(195*a^3*d^6*D - 9*a^2*b*d 
^4*(6*c*C*d - 65*B*d^2 + 3*c^2*D) + 3*a*b^2*c*d^2*(186*c^2*C*d - 299*B*c*d 
^2 + 572*A*d^3 - 128*c^3*D) - 4*b^3*c^3*(80*c^2*C*d - 104*B*c*d^2 + 143*A* 
d^3 - 64*c^3*D) + 3*b*d*(3*a^2*d^4*(77*C*d + 6*c*D) - a*b*d^2*(127*c^2*C*d 
 - 208*B*c*d^2 - 1001*A*d^3 - 86*c^3*D) + b^2*c^2*(80*c^2*C*d - 104*B*c*d^ 
2 + 143*A*d^3 - 64*c^3*D))*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((-2*Sqrt[a]* 
Sqrt[b]*(3*a^3*d^6*(231*C*d + 83*c*D) + 3*a*b^2*c^2*d^2*(266*c^2*C*d - 403 
*B*c*d^2 + 715*A*d^3 - 192*c^3*D) - 3*a^2*b*d^4*(145*c^2*C*d - 403*B*c*d^2 
 - 1001*A*d^3 - 77*c^3*D) - 4*b^3*c^4*(80*c^2*C*d - 104*B*c*d^2 + 143*A*d^ 
3 - 64*c^3*D))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - 
 (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[ 
(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a] 
*(b*c^2 - a*d^2)*(195*a^3*d^6*D - 9*a^2*b*d^4*(6*c*C*d - 65*B*d^2 + 3*c^2* 
D) + 3*a*b^2*c*d^2*(186*c^2*C*d - 299*B*c*d^2 + 572*A*d^3 - 128*c^3*D) ...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4512\) vs. \(2(981)=1962\).

Time = 5.34 (sec) , antiderivative size = 4513, normalized size of antiderivative = 4.19

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(4513\)
default	\(\text {Expression too large to display}\)	\(8269\)

Input:

int((d*x+c)^(1/2)*(-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVER 
BOSE)
 

Output:

1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(-2/15*D*b*x^6 
*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/13*(b^2*d*C+1/15*b^2*c*D)/b/d*x^5*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/11*(B*b^2*d+C*b^2*c-17/15*d*a*b*D-12/13 
*(b^2*d*C+1/15*b^2*c*D)/d*c)/b/d*x^4*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/ 
9*(A*b^2*d+B*b^2*c-2*d*a*b*C-6/5*a*b*c*D+11/13*(b^2*d*C+1/15*b^2*c*D)/b*a- 
10/11*(B*b^2*d+C*b^2*c-17/15*d*a*b*D-12/13*(b^2*d*C+1/15*b^2*c*D)/d*c)/d*c 
)/b/d*x^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(A*b^2*c-2*B*a*b*d-2*C*a* 
b*c+a^2*d*D+10/13*(b^2*d*C+1/15*b^2*c*D)/b/d*a*c+9/11*(B*b^2*d+C*b^2*c-17/ 
15*d*a*b*D-12/13*(b^2*d*C+1/15*b^2*c*D)/d*c)/b*a-8/9*(A*b^2*d+B*b^2*c-2*d* 
a*b*C-6/5*a*b*c*D+11/13*(b^2*d*C+1/15*b^2*c*D)/b*a-10/11*(B*b^2*d+C*b^2*c- 
17/15*d*a*b*D-12/13*(b^2*d*C+1/15*b^2*c*D)/d*c)/d*c)/d*c)/b/d*x^2*(-b*d*x^ 
3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(-2*A*a*b*d-2*B*a*b*c+a^2*C*d+a^2*c*D+8/11* 
(B*b^2*d+C*b^2*c-17/15*d*a*b*D-12/13*(b^2*d*C+1/15*b^2*c*D)/d*c)/b/d*a*c+7 
/9*(A*b^2*d+B*b^2*c-2*d*a*b*C-6/5*a*b*c*D+11/13*(b^2*d*C+1/15*b^2*c*D)/b*a 
-10/11*(B*b^2*d+C*b^2*c-17/15*d*a*b*D-12/13*(b^2*d*C+1/15*b^2*c*D)/d*c)/d* 
c)/b*a-6/7*(A*b^2*c-2*B*a*b*d-2*C*a*b*c+a^2*d*D+10/13*(b^2*d*C+1/15*b^2*c* 
D)/b/d*a*c+9/11*(B*b^2*d+C*b^2*c-17/15*d*a*b*D-12/13*(b^2*d*C+1/15*b^2*c*D 
)/d*c)/b*a-8/9*(A*b^2*d+B*b^2*c-2*d*a*b*C-6/5*a*b*c*D+11/13*(b^2*d*C+1/15* 
b^2*c*D)/b*a-10/11*(B*b^2*d+C*b^2*c-17/15*d*a*b*D-12/13*(b^2*d*C+1/15*b^2* 
c*D)/d*c)/d*c)/d*c)/d*c)/b/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 1032, normalized size of antiderivative = 0.96 \[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)*(-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= 
"fricas")
 

Output:

2/135135*(4*(256*D*b^4*c^8 - 320*C*b^4*c^7*d - 32*(24*D*a*b^3 - 13*B*b^4)* 
c^6*d^2 + 2*(519*C*a*b^3 - 286*A*b^4)*c^5*d^3 + 3*(203*D*a^2*b^2 - 507*B*a 
*b^3)*c^4*d^4 - 6*(161*C*a^2*b^2 - 429*A*a*b^3)*c^3*d^5 + 12*(14*D*a^3*b + 
 169*B*a^2*b^2)*c^2*d^6 - 6*(204*C*a^3*b + 1859*A*a^2*b^2)*c*d^7 - 585*(D* 
a^4 + 3*B*a^3*b)*d^8)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2) 
/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 12*(256* 
D*b^4*c^7*d - 320*C*b^4*c^6*d^2 - 32*(18*D*a*b^3 - 13*B*b^4)*c^5*d^3 + 2*( 
399*C*a*b^3 - 286*A*b^4)*c^4*d^4 + 3*(77*D*a^2*b^2 - 403*B*a*b^3)*c^3*d^5 
- 15*(29*C*a^2*b^2 - 143*A*a*b^3)*c^2*d^6 + 3*(83*D*a^3*b + 403*B*a^2*b^2) 
*c*d^7 + 231*(3*C*a^3*b + 13*A*a^2*b^2)*d^8)*sqrt(-b*d)*weierstrassZeta(4/ 
3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstras 
sPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3) 
, 1/3*(3*d*x + c)/d)) - 3*(3003*D*b^4*d^8*x^6 - 512*D*b^4*c^6*d^2 + 640*C* 
b^4*c^5*d^3 + 64*(17*D*a*b^3 - 13*B*b^4)*c^4*d^4 - 4*(379*C*a*b^3 - 286*A* 
b^4)*c^3*d^5 - 2*(174*D*a^2*b^2 - 1157*B*a*b^3)*c^2*d^6 + (708*C*a^2*b^2 - 
 4147*A*a*b^3)*c*d^7 + 780*(D*a^3*b + 3*B*a^2*b^2)*d^8 + 231*(D*b^4*c*d^7 
+ 15*C*b^4*d^8)*x^5 - 21*(12*D*b^4*c^2*d^6 - 15*C*b^4*c*d^7 + 13*(17*D*a*b 
^3 - 15*B*b^4)*d^8)*x^4 + 7*(40*D*b^4*c^3*d^5 - 50*C*b^4*c^2*d^6 - (81*D*a 
*b^3 - 65*B*b^4)*c*d^7 - 55*(15*C*a*b^3 - 13*A*b^4)*d^8)*x^3 - (320*D*b^4* 
c^4*d^4 - 400*C*b^4*c^3*d^5 - 2*(327*D*a*b^3 - 260*B*b^4)*c^2*d^6 + 5*(...
 

Sympy [F]

\[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int \left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x} \left (A + B x + C x^{2} + D x^{3}\right )\, dx \] Input:

integrate((d*x+c)**(1/2)*(-b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Integral((a - b*x**2)**(3/2)*sqrt(c + d*x)*(A + B*x + C*x**2 + D*x**3), x)
 

Maxima [F]

\[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= 
"maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)*sqrt(d*x + c), x)
 

Giac [F]

\[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \,d x } \] Input:

integrate((d*x+c)^(1/2)*(-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= 
"giac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)*sqrt(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (a-b\,x^2\right )}^{3/2}\,\sqrt {c+d\,x}\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:

int((a - b*x^2)^(3/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

int((a - b*x^2)^(3/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D), x)
 

Reduce [F]

\[ \int \sqrt {c+d x} \left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int \sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )d x \] Input:

int((d*x+c)^(1/2)*(-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x)
                                                                                    
                                                                                    
 

Output:

int((d*x+c)^(1/2)*(-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x)