\(\int \frac {(a-b x^2)^{3/2} (A+B x+C x^2)}{\sqrt {c+d x}} \, dx\) [178]

Optimal result

Integrand size = 32, antiderivative size = 692 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\frac {4 \left (45 a^2 C d^4-4 b^2 c^2 \left (80 c^2 C-88 B c d+99 A d^2\right )+3 a b d^2 \left (98 c^2 C-121 B c d+165 A d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{3465 b d^5}-\frac {4 \left (a d^2 (61 c C-77 B d)-b c \left (80 c^2 C-88 B c d+99 A d^2\right )\right ) x \sqrt {c+d x} \sqrt {a-b x^2}}{1155 d^4}+\frac {2 \left (99 A b+9 a C+\frac {8 b c (10 c C-11 B d)}{d^2}\right ) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{693 b d}-\frac {2 (10 c C-11 B d) x \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{99 d^2}-\frac {2 C \sqrt {c+d x} \left (a-b x^2\right )^{5/2}}{11 b d}+\frac {8 \sqrt {a} \left (3 a^2 d^4 (46 c C-77 B d)+4 b^2 c^3 \left (80 c^2 C-88 B c d+99 A d^2\right )-3 a b c d^2 \left (178 c^2 C-209 B c d+264 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 \sqrt {b} d^6 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {8 \sqrt {a} \left (b c^2-a d^2\right ) \left (45 a^2 C d^4-4 b^2 c^2 \left (80 c^2 C-88 B c d+99 A d^2\right )+3 a b d^2 \left (98 c^2 C-121 B c d+165 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 b^{3/2} d^6 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

4/3465*(45*a^2*C*d^4-4*b^2*c^2*(99*A*d^2-88*B*c*d+80*C*c^2)+3*a*b*d^2*(165 
*A*d^2-121*B*c*d+98*C*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^5-4/1155*(a 
*d^2*(-77*B*d+61*C*c)-b*c*(99*A*d^2-88*B*c*d+80*C*c^2))*x*(d*x+c)^(1/2)*(- 
b*x^2+a)^(1/2)/d^4+2/693*(99*A*b+9*a*C+8*b*c*(-11*B*d+10*C*c)/d^2)*(d*x+c) 
^(1/2)*(-b*x^2+a)^(3/2)/b/d-2/99*(-11*B*d+10*C*c)*x*(d*x+c)^(1/2)*(-b*x^2+ 
a)^(3/2)/d^2-2/11*C*(d*x+c)^(1/2)*(-b*x^2+a)^(5/2)/b/d+8/3465*a^(1/2)*(3*a 
^2*d^4*(-77*B*d+46*C*c)+4*b^2*c^3*(99*A*d^2-88*B*c*d+80*C*c^2)-3*a*b*c*d^2 
*(264*A*d^2-209*B*c*d+178*C*c^2))*(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^(1/2)/d^6/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b* 
x^2+a)^(1/2)+8/3465*a^(1/2)*(-a*d^2+b*c^2)*(45*a^2*C*d^4-4*b^2*c^2*(99*A*d 
^2-88*B*c*d+80*C*c^2)+3*a*b*d^2*(165*A*d^2-121*B*c*d+98*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^(3/2)/d^6/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 30.36 (sec) , antiderivative size = 831, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-\left ((c+d x) \left (180 a^2 C d^4-a b d^2 \left (988 c^2 C-2 c d (583 B+358 C x)+d^2 \left (1485 A+847 B x+585 C x^2\right )\right )+b^2 \left (640 c^4 C-32 c^3 d (22 B+15 C x)+8 c^2 d^2 \left (99 A+66 B x+50 C x^2\right )+5 d^4 x^2 (99 A+7 x (11 B+9 C x))-2 c d^3 x (297 A+5 x (44 B+35 C x))\right )\right )\right )+\frac {4 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (3 a^2 d^4 (46 c C-77 B d)+4 b^2 c^3 \left (80 c^2 C-88 B c d+99 A d^2\right )-3 a b c d^2 \left (178 c^2 C-209 B c d+264 A d^2\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 a^2 d^4 (46 c C-77 B d)+4 b^2 c^3 \left (80 c^2 C-88 B c d+99 A d^2\right )-3 a b c d^2 \left (178 c^2 C-209 B c d+264 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (45 a^2 C d^4+3 a^{3/2} \sqrt {b} d^3 (61 c C-77 B d)-4 b^2 c^2 \left (80 c^2 C-88 B c d+99 A d^2\right )-3 \sqrt {a} b^{3/2} c d \left (80 c^2 C-88 B c d+99 A d^2\right )+3 a b d^2 \left (98 c^2 C-121 B c d+165 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{3465 b d^5 \sqrt {c+d x}} \] Input:

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

Output:

(2*Sqrt[a - b*x^2]*(-((c + d*x)*(180*a^2*C*d^4 - a*b*d^2*(988*c^2*C - 2*c* 
d*(583*B + 358*C*x) + d^2*(1485*A + 847*B*x + 585*C*x^2)) + b^2*(640*c^4*C 
 - 32*c^3*d*(22*B + 15*C*x) + 8*c^2*d^2*(99*A + 66*B*x + 50*C*x^2) + 5*d^4 
*x^2*(99*A + 7*x*(11*B + 9*C*x)) - 2*c*d^3*x*(297*A + 5*x*(44*B + 35*C*x)) 
))) + (4*(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(3*a^2*d^4*(46*c*C - 77*B*d) 
+ 4*b^2*c^3*(80*c^2*C - 88*B*c*d + 99*A*d^2) - 3*a*b*c*d^2*(178*c^2*C - 20 
9*B*c*d + 264*A*d^2))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3*a 
^2*d^4*(46*c*C - 77*B*d) + 4*b^2*c^3*(80*c^2*C - 88*B*c*d + 99*A*d^2) - 3* 
a*b*c*d^2*(178*c^2*C - 209*B*c*d + 264*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + 
x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3 
/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sq 
rt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*d*(Sqrt[b]*c - S 
qrt[a]*d)*(45*a^2*C*d^4 + 3*a^(3/2)*Sqrt[b]*d^3*(61*c*C - 77*B*d) - 4*b^2* 
c^2*(80*c^2*C - 88*B*c*d + 99*A*d^2) - 3*Sqrt[a]*b^(3/2)*c*d*(80*c^2*C - 8 
8*B*c*d + 99*A*d^2) + 3*a*b*d^2*(98*c^2*C - 121*B*c*d + 165*A*d^2))*Sqrt[( 
d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c 
+ d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] 
]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(d^2* 
Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2))))/(3465*b*d^5*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.94 (sec) , antiderivative size = 662, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2185, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {2 \int -\frac {d ((11 A b+a C) d-b (10 c C-11 B d) x) \left (a-b x^2\right )^{3/2}}{2 \sqrt {c+d x}}dx}{11 b d^2}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {((11 A b+a C) d-b (10 c C-11 B d) x) \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}}dx}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}-\frac {4 \int -\frac {b \left (a d \left (9 a C d^2+b \left (10 C c^2-11 B d c+99 A d^2\right )\right )-b \left (a d^2 (61 c C-77 B d)-b c \left (80 C c^2-88 B d c+99 A d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{21 b d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \int \frac {\left (a d \left (9 a C d^2+b \left (10 C c^2-11 B d c+99 A d^2\right )\right )-b \left (a d^2 (61 c C-77 B d)-b c \left (80 C c^2-88 B d c+99 A d^2\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (45 a^2 C d^4+3 a b \left (37 C c^2-44 B d c+165 A d^2\right ) d^2-b^2 c^2 \left (80 C c^2-88 B d c+99 A d^2\right )\right )-b \left (3 a^2 (46 c C-77 B d) d^4-3 a b c \left (178 C c^2-209 B d c+264 A d^2\right ) d^2+4 b^2 c^3 \left (80 C c^2-88 B d c+99 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (\frac {2 \int \frac {a d \left (45 a^2 C d^4+3 a b \left (37 C c^2-44 B d c+165 A d^2\right ) d^2-b^2 c^2 \left (80 C c^2-88 B d c+99 A d^2\right )\right )-b \left (3 a^2 (46 c C-77 B d) d^4-3 a b c \left (178 C c^2-209 B d c+264 A d^2\right ) d^2+4 b^2 c^3 \left (80 C c^2-88 B d c+99 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 C d^4+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (3 a^2 d^4 (46 c C-77 B d)-3 a b c d^2 \left (264 A d^2-209 B c d+178 c^2 C\right )+4 b^2 c^3 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 C d^4+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (46 c C-77 B d)-3 a b c d^2 \left (264 A d^2-209 B c d+178 c^2 C\right )+4 b^2 c^3 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (46 c C-77 B d)-3 a b c d^2 \left (264 A d^2-209 B c d+178 c^2 C\right )+4 b^2 c^3 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 C d^4+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (46 c C-77 B d)-3 a b c d^2 \left (264 A d^2-209 B c d+178 c^2 C\right )+4 b^2 c^3 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 C d^4+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (46 c C-77 B d)-3 a b c d^2 \left (264 A d^2-209 B c d+178 c^2 C\right )+4 b^2 c^3 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (45 a^2 C d^4+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (45 a^2 C d^4+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (46 c C-77 B d)-3 a b c d^2 \left (264 A d^2-209 B c d+178 c^2 C\right )+4 b^2 c^3 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (45 a^2 C d^4+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4 (46 c C-77 B d)-3 a b c d^2 \left (264 A d^2-209 B c d+178 c^2 C\right )+4 b^2 c^3 \left (99 A d^2-88 B c d+80 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (45 a^2 C d^4-3 b d x \left (a d^2 (61 c C-77 B d)-b c \left (99 A d^2-88 B c d+80 c^2 C\right )\right )+3 a b d^2 \left (165 A d^2-121 B c d+98 c^2 C\right )-4 b^2 c^2 \left (99 A d^2-88 B c d+80 c^2 C\right )\right )}{15 d^2}\right )}{21 d^2}+\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a C d^2+b \left (99 A d^2-88 B c d+80 c^2 C\right )-7 b d x (10 c C-11 B d)\right )}{63 d^2}}{11 b d}-\frac {2 C \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}{11 b d}\)

Input:

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

Output:

(-2*C*Sqrt[c + d*x]*(a - b*x^2)^(5/2))/(11*b*d) + ((2*Sqrt[c + d*x]*(9*a*C 
*d^2 + b*(80*c^2*C - 88*B*c*d + 99*A*d^2) - 7*b*d*(10*c*C - 11*B*d)*x)*(a 
- b*x^2)^(3/2))/(63*d^2) + (2*((2*Sqrt[c + d*x]*(45*a^2*C*d^4 - 4*b^2*c^2* 
(80*c^2*C - 88*B*c*d + 99*A*d^2) + 3*a*b*d^2*(98*c^2*C - 121*B*c*d + 165*A 
*d^2) - 3*b*d*(a*d^2*(61*c*C - 77*B*d) - b*c*(80*c^2*C - 88*B*c*d + 99*A*d 
^2))*x)*Sqrt[a - b*x^2])/(15*d^2) + (2*((2*Sqrt[a]*Sqrt[b]*(3*a^2*d^4*(46* 
c*C - 77*B*d) + 4*b^2*c^3*(80*c^2*C - 88*B*c*d + 99*A*d^2) - 3*a*b*c*d^2*( 
178*c^2*C - 209*B*c*d + 264*A*d^2))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*Elli 
pticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sq 
rt[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)*(45*a^2*C*d^4 - 4*b^2*c^2*(80*c^2*C 
- 88*B*c*d + 99*A*d^2) + 3*a*b*d^2*(98*c^2*C - 121*B*c*d + 165*A*d^2))*Sqr 
t[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ellipti 
cF[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])))/(15*d^2)))/(21*d^2)) 
/(11*b*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 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. \(1539\) vs. \(2(608)=1216\).

Time = 6.66 (sec) , antiderivative size = 1540, normalized size of antiderivative = 2.23

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1540\)
risch	\(\text {Expression too large to display}\)	\(1781\)
default	\(\text {Expression too large to display}\)	\(4856\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 608, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{\sqrt {c+d x}} \, dx=-\frac {2 \, {\left (4 \, {\left (320 \, C b^{3} c^{6} - 352 \, B b^{3} c^{5} d + 891 \, B a b^{2} c^{3} d^{3} - 627 \, B a^{2} b c d^{5} - 18 \, {\left (43 \, C a b^{2} - 22 \, A b^{3}\right )} c^{4} d^{2} + 3 \, {\left (157 \, C a^{2} b - 363 \, A a b^{2}\right )} c^{2} d^{4} + 135 \, {\left (C a^{3} + 11 \, A a^{2} b\right )} d^{6}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 12 \, {\left (320 \, C b^{3} c^{5} d - 352 \, B b^{3} c^{4} d^{2} + 627 \, B a b^{2} c^{2} d^{4} - 231 \, B a^{2} b d^{6} - 6 \, {\left (89 \, C a b^{2} - 66 \, A b^{3}\right )} c^{3} d^{3} + 6 \, {\left (23 \, C a^{2} b - 132 \, A a b^{2}\right )} c d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (315 \, C b^{3} d^{6} x^{4} + 640 \, C b^{3} c^{4} d^{2} - 704 \, B b^{3} c^{3} d^{3} + 1166 \, B a b^{2} c d^{5} - 4 \, {\left (247 \, C a b^{2} - 198 \, A b^{3}\right )} c^{2} d^{4} + 45 \, {\left (4 \, C a^{2} b - 33 \, A a b^{2}\right )} d^{6} - 35 \, {\left (10 \, C b^{3} c d^{5} - 11 \, B b^{3} d^{6}\right )} x^{3} + 5 \, {\left (80 \, C b^{3} c^{2} d^{4} - 88 \, B b^{3} c d^{5} - 9 \, {\left (13 \, C a b^{2} - 11 \, A b^{3}\right )} d^{6}\right )} x^{2} - {\left (480 \, C b^{3} c^{3} d^{3} - 528 \, B b^{3} c^{2} d^{4} + 847 \, B a b^{2} d^{6} - 2 \, {\left (358 \, C a b^{2} - 297 \, A b^{3}\right )} c d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{10395 \, b^{2} d^{7}} \] Input:

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

Output:

-2/10395*(4*(320*C*b^3*c^6 - 352*B*b^3*c^5*d + 891*B*a*b^2*c^3*d^3 - 627*B 
*a^2*b*c*d^5 - 18*(43*C*a*b^2 - 22*A*b^3)*c^4*d^2 + 3*(157*C*a^2*b - 363*A 
*a*b^2)*c^2*d^4 + 135*(C*a^3 + 11*A*a^2*b)*d^6)*sqrt(-b*d)*weierstrassPInv 
erse(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*(320*C*b^3*c^5*d - 352*B*b^3*c^4*d^2 + 627*B*a*b^2*c^ 
2*d^4 - 231*B*a^2*b*d^6 - 6*(89*C*a*b^2 - 66*A*b^3)*c^3*d^3 + 6*(23*C*a^2* 
b - 132*A*a*b^2)*c*d^5)*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), 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*(315*C*b^3*d^6*x^4 + 640*C*b^3*c^4*d^2 - 704*B*b^3*c^3*d^3 + 1166*B*a 
*b^2*c*d^5 - 4*(247*C*a*b^2 - 198*A*b^3)*c^2*d^4 + 45*(4*C*a^2*b - 33*A*a* 
b^2)*d^6 - 35*(10*C*b^3*c*d^5 - 11*B*b^3*d^6)*x^3 + 5*(80*C*b^3*c^2*d^4 - 
88*B*b^3*c*d^5 - 9*(13*C*a*b^2 - 11*A*b^3)*d^6)*x^2 - (480*C*b^3*c^3*d^3 - 
 528*B*b^3*c^2*d^4 + 847*B*a*b^2*d^6 - 2*(358*C*a*b^2 - 297*A*b^3)*c*d^5)* 
x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^2*d^7)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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