\(\int \frac {x^4 (A+B x+C x^2)}{\sqrt {c+d x} (a-b x^2)^{5/2}} \, dx\) [232]

Optimal result

Integrand size = 35, antiderivative size = 681 \[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {a \sqrt {c+d x} (a (b B c-A b d-a C d)+b (A b c+a c C-a B d) x)}{3 b^3 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {\sqrt {c+d x} \left (a \left (13 a^2 C d^3+b^2 c^2 (12 B c-11 A d)-a b d \left (17 c^2 C+8 B c d-7 A d^2\right )\right )+b \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right ) x\right )}{6 b^3 \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}-\frac {2 C \sqrt {c+d x} \sqrt {a-b x^2}}{3 b^3 d}+\frac {\sqrt {a} \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (30 c^2 C-37 B c d-4 A d^2\right )+4 b^2 c^3 \left (2 c^2 C-3 B c d-2 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 b^{5/2} d^2 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \left (15 a^2 C d^4-a b d^2 \left (6 c^2 C+13 B c d-5 A d^2\right )-4 b^2 c^2 \left (2 c^2 C-3 B c d+A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 b^{7/2} d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*a*(d*x+c)^(1/2)*(a*(-A*b*d+B*b*c-C*a*d)+b*(A*b*c-B*a*d+C*a*c)*x)/b^3/( 
-a*d^2+b*c^2)/(-b*x^2+a)^(3/2)-1/6*(d*x+c)^(1/2)*(a*(13*a^2*C*d^3+b^2*c^2* 
(-11*A*d+12*B*c)-a*b*d*(-7*A*d^2+8*B*c*d+17*C*c^2))+b*(4*A*b*c*(-a*d^2+2*b 
*c^2)+a*(b*c^2*(-13*B*d+14*C*c)-a*d^2*(-9*B*d+10*C*c)))*x)/b^3/(-a*d^2+b*c 
^2)^2/(-b*x^2+a)^(1/2)-2/3*C*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^3/d+1/6*a^(1 
/2)*(3*a^2*d^4*(-7*B*d+6*C*c)-a*b*c*d^2*(-4*A*d^2-37*B*c*d+30*C*c^2)+4*b^2 
*c^3*(-2*A*d^2-3*B*c*d+2*C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellipt 
icE(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^(5/2)/d^2/(-a*d^2+b*c^2)^2/((d*x+c)/(c+a^(1/2)*d/b^(1 
/2)))^(1/2)/(-b*x^2+a)^(1/2)+1/6*a^(1/2)*(15*a^2*C*d^4-a*b*d^2*(-5*A*d^2+1 
3*B*c*d+6*C*c^2)-4*b^2*c^2*(A*d^2-3*B*c*d+2*C*c^2))*((d*x+c)/(c+a^(1/2)*d/ 
b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(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^(7/2)/d^2/ 
(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 32.21 (sec) , antiderivative size = 1061, normalized size of antiderivative = 1.56 \[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(x^4*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*C)/(3*b^3*d) + (a^2*b*B*c - a^2*A*b*d - 
 a^3*C*d + a*A*b^2*c*x + a^2*b*c*C*x - a^2*b*B*d*x)/(3*b^3*(b*c^2 - a*d^2) 
*(-a + b*x^2)^2) + (12*a*b^2*B*c^3 - 11*a*A*b^2*c^2*d - 17*a^2*b*c^2*C*d - 
 8*a^2*b*B*c*d^2 + 7*a^2*A*b*d^3 + 13*a^3*C*d^3 + 8*A*b^3*c^3*x + 14*a*b^2 
*c^3*C*x - 13*a*b^2*B*c^2*d*x - 4*a*A*b^2*c*d^2*x - 10*a^2*b*c*C*d^2*x + 9 
*a^2*b*B*d^3*x)/(6*b^3*(b*c^2 - a*d^2)^2*(-a + b*x^2))) + (Sqrt[a - (b*(c 
+ d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*a^2 
*d^4*(6*c*C - 7*B*d) + 4*b^2*c^3*(2*c^2*C - 3*B*c*d - 2*A*d^2) + a*b*c*d^2 
*(-30*c^2*C + 37*B*c*d + 4*A*d^2))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c 
+ d*x))^2) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3*a^2*d^4*(6*c*C - 7*B*d) 
 + 4*b^2*c^3*(2*c^2*C - 3*B*c*d - 2*A*d^2) + a*b*c*d^2*(-30*c^2*C + 37*B*c 
*d + 4*A*d^2))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqr 
t[1 - c/(c + d*x) + (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*EllipticE[I*ArcSinh[S 
qrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqr 
t[b]*c - Sqrt[a]*d)])/Sqrt[c + d*x] - (I*d*(Sqrt[b]*c - Sqrt[a]*d)*(12*A*b 
^(5/2)*c^3*d - 15*a^(5/2)*C*d^4 + 3*a^2*Sqrt[b]*d^3*(-11*c*C + 7*B*d) + a^ 
(3/2)*b*d^2*(6*c^2*C + 13*B*c*d - 5*A*d^2) + 3*a*b^(3/2)*c*d*(12*c^2*C - 8 
*B*c*d - 3*A*d^2) + 4*Sqrt[a]*b^2*c^2*(2*c^2*C - 3*B*c*d + A*d^2))*Sqrt[1 
- c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (S 
qrt[a]*d)/(Sqrt[b]*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d...
 

Rubi [A] (verified)

Time = 3.72 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2180, 27, 2180, 27, 2185, 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.

\(\displaystyle \int \frac {x^4 \left (A+B x+C x^2\right )}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2180

\(\displaystyle \frac {\int -\frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^4+6 a B \left (c^2-\frac {a d^2}{b}\right ) x^3+\frac {6 a (A b+a C) \left (b c^2-a d^2\right ) x^2}{b^2}+\frac {3 a^2 (b c (2 B c-A d)-a d (c C+B d)) x}{b^2}+\frac {a^2 \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right )}{b^3}}{2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{3 a \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\int \frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^4+6 a B \left (c^2-\frac {a d^2}{b}\right ) x^3+\frac {6 a (A b+a C) \left (b c^2-a d^2\right ) x^2}{b^2}+\frac {3 a^2 (b c (2 B c-A d)-a d (c C+B d)) x}{b^2}+\frac {a^2 \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right )}{b^3}}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 2180

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\int -\frac {\frac {12 C \left (b c^2-a d^2\right )^2 x^2 a^2}{b^2}+\frac {\left (A b \left (12 b^2 c^4-13 a b d^2 c^2+5 a^2 d^4\right )+a \left (11 a^2 C d^4-a b c (31 c C-8 B d) d^2+12 b^2 c^3 (2 c C-B d)\right )\right ) a^2}{b^3}+\frac {\left (4 b^2 (3 B c+2 A d) c^3+a b d \left (14 C c^2-37 B d c-4 A d^2\right ) c-a^2 d^3 (10 c C-21 B d)\right ) x a^2}{b^2}}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\int \frac {\frac {12 C \left (b c^2-a d^2\right )^2 x^2 a^2}{b^2}+\frac {\left (A b \left (12 b^2 c^4-13 a b d^2 c^2+5 a^2 d^4\right )+a \left (11 a^2 C d^4-a b c (31 c C-8 B d) d^2+12 b^2 c^3 (2 c C-B d)\right )\right ) a^2}{b^3}+\frac {\left (4 b^2 (3 B c+2 A d) c^3+a b d \left (14 C c^2-37 B d c-4 A d^2\right ) c-a^2 d^3 (10 c C-21 B d)\right ) x a^2}{b^2}}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\frac {2 \int -\frac {3 a^2 d \left (d \left (A b \left (12 b^2 c^4-13 a b d^2 c^2+5 a^2 d^4\right )+a \left (15 a^2 C d^4-a b c (39 c C-8 B d) d^2+4 b^2 c^3 (7 c C-3 B d)\right )\right )-b \left (3 a^2 (6 c C-7 B d) d^4-a b c \left (30 C c^2-37 B d c-4 A d^2\right ) d^2+4 b^2 c^3 \left (2 C c^2-3 B d c-2 A d^2\right )\right ) x\right )}{2 b^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \int \frac {d \left (A b \left (12 b^2 c^4-13 a b d^2 c^2+5 a^2 d^4\right )+a \left (15 a^2 C d^4-a b c (39 c C-8 B d) d^2+4 b^2 c^3 (7 c C-3 B d)\right )\right )-b \left (3 a^2 (6 c C-7 B d) d^4-a b c \left (30 C c^2-37 B d c-4 A d^2\right ) d^2+4 b^2 c^3 \left (2 C c^2-3 B d c-2 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \left (-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 C d^4-a b d^2 \left (-5 A d^2+13 B c d+6 c^2 C\right )-4 b^2 c^2 \left (A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (-4 A d^2-37 B c d+30 c^2 C\right )+4 b^2 c^3 \left (-2 A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \left (-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 C d^4-a b d^2 \left (-5 A d^2+13 B c d+6 c^2 C\right )-4 b^2 c^2 \left (A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (-4 A d^2-37 B c d+30 c^2 C\right )+4 b^2 c^3 \left (-2 A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (-4 A d^2-37 B c d+30 c^2 C\right )+4 b^2 c^3 \left (-2 A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 C d^4-a b d^2 \left (-5 A d^2+13 B c d+6 c^2 C\right )-4 b^2 c^2 \left (A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (-4 A d^2-37 B c d+30 c^2 C\right )+4 b^2 c^3 \left (-2 A d^2-3 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (15 a^2 C d^4-a b d^2 \left (-5 A d^2+13 B c d+6 c^2 C\right )-4 b^2 c^2 \left (A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (-4 A d^2-37 B c d+30 c^2 C\right )+4 b^2 c^3 \left (-2 A d^2-3 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (15 a^2 C d^4-a b d^2 \left (-5 A d^2+13 B c d+6 c^2 C\right )-4 b^2 c^2 \left (A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (15 a^2 C d^4-a b d^2 \left (-5 A d^2+13 B c d+6 c^2 C\right )-4 b^2 c^2 \left (A d^2-3 B c d+2 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (-4 A d^2-37 B c d+30 c^2 C\right )+4 b^2 c^3 \left (-2 A d^2-3 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {a \sqrt {c+d x} (b x (-a B d+a c C+A b c)+a (-a C d-A b d+b B c))}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (a \left (13 a^2 C d^3-a b d \left (-7 A d^2+8 B c d+17 c^2 C\right )+b^2 c^2 (12 B c-11 A d)\right )+b x \left (4 A b c \left (2 b c^2-a d^2\right )+a \left (b c^2 (14 c C-13 B d)-a d^2 (10 c C-9 B d)\right )\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (15 a^2 C d^4-a b d^2 \left (-5 A d^2+13 B c d+6 c^2 C\right )-4 b^2 c^2 \left (A d^2-3 B c d+2 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (6 c C-7 B d)-a b c d^2 \left (-4 A d^2-37 B c d+30 c^2 C\right )+4 b^2 c^3 \left (-2 A d^2-3 B c d+2 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b^3 d}-\frac {8 a^2 C \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2}{b^3 d}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

Input:

Int[(x^4*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

(a*Sqrt[c + d*x]*(a*(b*B*c - A*b*d - a*C*d) + b*(A*b*c + a*c*C - a*B*d)*x) 
)/(3*b^3*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) - ((a*Sqrt[c + d*x]*(a*(13*a^2 
*C*d^3 + b^2*c^2*(12*B*c - 11*A*d) - a*b*d*(17*c^2*C + 8*B*c*d - 7*A*d^2)) 
 + b*(4*A*b*c*(2*b*c^2 - a*d^2) + a*(b*c^2*(14*c*C - 13*B*d) - a*d^2*(10*c 
*C - 9*B*d)))*x))/(b^3*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) - ((-8*a^2*C*(b*c^ 
2 - a*d^2)^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(b^3*d) + (a^2*((2*Sqrt[a]*Sqr 
t[b]*(3*a^2*d^4*(6*c*C - 7*B*d) - a*b*c*d^2*(30*c^2*C - 37*B*c*d - 4*A*d^2 
) + 4*b^2*c^3*(2*c^2*C - 3*B*c*d - 2*A*d^2))*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)*(15*a^2*C*d^4 - a*b*d^2*(6* 
c^2*C + 13*B*c*d - 5*A*d^2) - 4*b^2*c^2*(2*c^2*C - 3*B*c*d + A*d^2))*Sqrt[ 
(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF 
[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] 
 + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(b^3*d))/(2*a*(b*c^2 - 
 a*d^2)))/(6*a*(b*c^2 - a*d^2))
 

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 2180

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema 
inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 
, x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((a*(e*R - d*S) 
 + (b*d*R + a*e*S)*x)/(2*a*(p + 1)*(b*d^2 + a*e^2))), x] + Simp[1/(2*a*(p + 
 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a* 
(p + 1)*(b*d^2 + a*e^2)*Qx + b*d^2*R*(2*p + 3) - a*e*(d*S*m - e*R*(m + 2*p 
+ 3)) + e*(b*d*R + a*e*S)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, d, e 
, m}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] &&  !(IGtQ[ 
m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 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 [A] (verified)

Time = 14.95 (sec) , antiderivative size = 1187, normalized size of antiderivative = 1.74

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1187\)
risch	\(\text {Expression too large to display}\)	\(2483\)
default	\(\text {Expression too large to display}\)	\(9545\)

Input:

int(x^4*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x,method=_RETURNVERBO 
SE)
                                                                                    
                                                                                    
 

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*((-1/3*a*(A*b*c- 
B*a*d+C*a*c)/b^4/(a*d^2-b*c^2)*x+1/3*a^2*(A*b*d-B*b*c+C*a*d)/(a*d^2-b*c^2) 
/b^5)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x^2-a/b)^2-2*(-b*d*x-b*c)*(1/12/ 
b^3*(4*A*a*b*c*d^2-8*A*b^2*c^3-9*B*a^2*d^3+13*B*a*b*c^2*d+10*C*a^2*c*d^2-1 
4*C*a*b*c^3)/(a*d^2-b*c^2)^2*x-1/12*a*(7*A*a*b*d^3-11*A*b^2*c^2*d-8*B*a*b* 
c*d^2+12*B*b^2*c^3+13*C*a^2*d^3-17*C*a*b*c^2*d)/(a*d^2-b*c^2)^2/b^4)/((x^2 
-a/b)*(-b*d*x-b*c))^(1/2)-2/3*C/b^3/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2 
*((A*b+2*C*a)/b^3-1/6/b^3/(a*d^2-b*c^2)*(7*A*a*b*d^2-8*A*b^2*c^2+B*a*b*c*d 
+13*C*a^2*d^2-14*C*a*b*c^2)+1/12/b^3*d*a*(7*A*a*b*d^3-11*A*b^2*c^2*d-8*B*a 
*b*c*d^2+12*B*b^2*c^3+13*C*a^2*d^3-17*C*a*b*c^2*d)/(a*d^2-b*c^2)^2-1/6/b^2 
*c*(4*A*a*b*c*d^2-8*A*b^2*c^3-9*B*a^2*d^3+13*B*a*b*c^2*d+10*C*a^2*c*d^2-14 
*C*a*b*c^3)/(a*d^2-b*c^2)^2+1/3*C/b^3*a)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c 
/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1 
/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a 
*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b 
*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(B/b^2-1/12*d*(4*A*a*b*c*d^ 
2-8*A*b^2*c^3-9*B*a^2*d^3+13*B*a*b*c^2*d+10*C*a^2*c*d^2-14*C*a*b*c^3)/b^2/ 
(a*d^2-b*c^2)^2-2/3*C/b^2/d*c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a* 
b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/ 
b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1357 vs. \(2 (613) = 1226\).

Time = 0.13 (sec) , antiderivative size = 1357, normalized size of antiderivative = 1.99 \[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="f 
ricas")
 

Output:

-1/18*((8*C*a^2*b^3*c^6 - 12*B*a^2*b^3*c^5*d + B*a^3*b^2*c^3*d^3 + 3*B*a^4 
*b*c*d^5 + 2*(27*C*a^3*b^2 + 14*A*a^2*b^3)*c^4*d^2 - (99*C*a^4*b + 35*A*a^ 
3*b^2)*c^2*d^4 + 15*(3*C*a^5 + A*a^4*b)*d^6 + (8*C*b^5*c^6 - 12*B*b^5*c^5* 
d + B*a*b^4*c^3*d^3 + 3*B*a^2*b^3*c*d^5 + 2*(27*C*a*b^4 + 14*A*b^5)*c^4*d^ 
2 - (99*C*a^2*b^3 + 35*A*a*b^4)*c^2*d^4 + 15*(3*C*a^3*b^2 + A*a^2*b^3)*d^6 
)*x^4 - 2*(8*C*a*b^4*c^6 - 12*B*a*b^4*c^5*d + B*a^2*b^3*c^3*d^3 + 3*B*a^3* 
b^2*c*d^5 + 2*(27*C*a^2*b^3 + 14*A*a*b^4)*c^4*d^2 - (99*C*a^3*b^2 + 35*A*a 
^2*b^3)*c^2*d^4 + 15*(3*C*a^4*b + A*a^3*b^2)*d^6)*x^2)*sqrt(-b*d)*weierstr 
assPInverse(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*(8*C*a^2*b^3*c^5*d - 12*B*a^2*b^3*c^4*d^2 + 37* 
B*a^3*b^2*c^2*d^4 - 21*B*a^4*b*d^6 - 2*(15*C*a^3*b^2 + 4*A*a^2*b^3)*c^3*d^ 
3 + 2*(9*C*a^4*b + 2*A*a^3*b^2)*c*d^5 + (8*C*b^5*c^5*d - 12*B*b^5*c^4*d^2 
+ 37*B*a*b^4*c^2*d^4 - 21*B*a^2*b^3*d^6 - 2*(15*C*a*b^4 + 4*A*b^5)*c^3*d^3 
 + 2*(9*C*a^2*b^3 + 2*A*a*b^4)*c*d^5)*x^4 - 2*(8*C*a*b^4*c^5*d - 12*B*a*b^ 
4*c^4*d^2 + 37*B*a^2*b^3*c^2*d^4 - 21*B*a^3*b^2*d^6 - 2*(15*C*a^2*b^3 + 4* 
A*a*b^4)*c^3*d^3 + 2*(9*C*a^3*b^2 + 2*A*a^2*b^3)*c*d^5)*x^2)*sqrt(-b*d)*we 
ierstrassZeta(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), 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)) + 3*(4*C*a^2*b^3*c^4*d^2 + 10*B*a^2* 
b^3*c^3*d^3 - 6*B*a^3*b^2*c*d^5 - (23*C*a^3*b^2 + 9*A*a^2*b^3)*c^2*d^4 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**4*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^4*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="m 
axima")
 

Output:

integrate((C*x^2 + B*x + A)*x^4/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
 

Giac [F]

\[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^4*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="g 
iac")
 

Output:

integrate((C*x^2 + B*x + A)*x^4/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (C\,x^2+B\,x+A\right )}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int((x^4*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
 

Output:

int((x^4*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

int(x^4*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
 

Output:

int(x^4*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)