\(\int \frac {x^2 (A+B x+C x^2)}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx\) [193]

Optimal result

Integrand size = 35, antiderivative size = 507 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 c^2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a-b x^2}}{d^3 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {2 (12 c C-5 B d) \sqrt {c+d x} \sqrt {a-b x^2}}{15 b d^3}-\frac {2 C (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b d^3}+\frac {2 \sqrt {a} \left (9 a^2 C d^4+a b d^2 \left (24 c^2 C-25 B c d+15 A d^2\right )-2 b^2 c^2 \left (24 c^2 C-20 B c d+15 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d^4 \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \left (a d^2 (12 c C-5 B d)+2 b c \left (24 c^2 C-20 B c d+15 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

2*c^2*(A*d^2-B*c*d+C*c^2)*(-b*x^2+a)^(1/2)/d^3/(-a*d^2+b*c^2)/(d*x+c)^(1/2 
)+2/15*(-5*B*d+12*C*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^3-2/5*C*(d*x+c)^ 
(3/2)*(-b*x^2+a)^(1/2)/b/d^3+2/15*a^(1/2)*(9*a^2*C*d^4+a*b*d^2*(15*A*d^2-2 
5*B*c*d+24*C*c^2)-2*b^2*c^2*(15*A*d^2-20*B*c*d+24*C*c^2))*(d*x+c)^(1/2)*(( 
-b*x^2+a)/a)^(1/2)*EllipticE(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^4/(-a*d^2+b*c^2)/((d 
*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)+2/15*a^(1/2)*(a*d^2*(- 
5*B*d+12*C*c)+2*b*c*(15*A*d^2-20*B*c*d+24*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^4/(d 
*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.24 (sec) , antiderivative size = 728, normalized size of antiderivative = 1.44 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (9 a^2 C d^4+a b d^2 \left (24 c^2 C-25 B c d+15 A d^2\right )-2 b^2 c^2 \left (24 c^2 C-20 B c d+15 A d^2\right )\right ) \left (a-b x^2\right )+b d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right ) \left (a d^2 (c+d x) (-9 c C+5 B d+3 C d x)+b c^2 \left (24 c^2 C+c (-20 B d+6 C d x)+d^2 \left (15 A-5 B x-3 C x^2\right )\right )\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (9 a^2 C d^4+a b d^2 \left (24 c^2 C-25 B c d+15 A d^2\right )-2 b^2 c^2 \left (24 c^2 C-20 B c d+15 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (9 a^{3/2} C d^3+a \sqrt {b} d^2 (12 c C-5 B d)+3 \sqrt {a} b d \left (12 c^2 C-10 B c d+5 A d^2\right )+2 b^{3/2} c \left (24 c^2 C-20 B c d+15 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{15 b^2 d^5 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:

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

Output:

(2*(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(9*a^2*C*d^4 + a*b*d^2*(24*c^2*C - 
25*B*c*d + 15*A*d^2) - 2*b^2*c^2*(24*c^2*C - 20*B*c*d + 15*A*d^2))*(a - b* 
x^2) + b*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2)*(a*d^2*(c + d*x)*( 
-9*c*C + 5*B*d + 3*C*d*x) + b*c^2*(24*c^2*C + c*(-20*B*d + 6*C*d*x) + d^2* 
(15*A - 5*B*x - 3*C*x^2))) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(9*a^2*C*d^ 
4 + a*b*d^2*(24*c^2*C - 25*B*c*d + 15*A*d^2) - 2*b^2*c^2*(24*c^2*C - 20*B* 
c*d + 15*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]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]* 
c - Sqrt[a]*d)] - I*Sqrt[a]*Sqrt[b]*d*(Sqrt[b]*c - Sqrt[a]*d)*(9*a^(3/2)*C 
*d^3 + a*Sqrt[b]*d^2*(12*c*C - 5*B*d) + 3*Sqrt[a]*b*d*(12*c^2*C - 10*B*c*d 
 + 5*A*d^2) + 2*b^(3/2)*c*(24*c^2*C - 20*B*c*d + 15*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)]))/(15*b^2*d^5*S 
qrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2 
])
 

Rubi [A] (verified)

Time = 2.60 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2182, 27, 2185, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {2 \int \frac {b d (12 c C-5 B d) \left (b c^2-a d^2\right ) x^2+\left (3 a^2 C d^4-5 a b (B c-A d) d^3-2 b^2 c^2 \left (4 C c^2-5 B d c+5 A d^2\right )\right ) x+a c d \left (3 a C d^2-b \left (8 C c^2-5 B d c+5 A d^2\right )\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {b d (12 c C-5 B d) \left (b c^2-a d^2\right ) x^2+\left (3 a^2 C d^4-5 a b (B c-A d) d^3-2 b^2 c^2 \left (4 C c^2-5 B d c+5 A d^2\right )\right ) x+a c d \left (3 a C d^2-b \left (8 C c^2-5 B d c+5 A d^2\right )\right )}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {-\frac {2 \int \frac {b d^2 \left (a d \left (a (3 c C-5 B d) d^2+b c \left (12 C c^2-10 B d c+15 A d^2\right )\right )-\left (9 a^2 C d^4+a b \left (24 C c^2-25 B d c+15 A d^2\right ) d^2-2 b^2 c^2 \left (24 C c^2-20 B d c+15 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {1}{3} \int \frac {a d \left (a (3 c C-5 B d) d^2+b c \left (12 C c^2-10 B d c+15 A d^2\right )\right )-\left (9 a^2 C d^4+a b \left (24 C c^2-25 B d c+15 A d^2\right ) d^2-2 b^2 c^2 \left (24 C c^2-20 B d c+15 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (9 a^2 C d^4+a b d^2 \left (15 A d^2-25 B c d+24 c^2 C\right )-2 b^2 c^2 \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}+\frac {\left (b c^2-a d^2\right ) \left (a d^2 (12 c C-5 B d)+2 b c \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (9 a^2 C d^4+a b d^2 \left (15 A d^2-25 B c d+24 c^2 C\right )-2 b^2 c^2 \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {\left (b c^2-a d^2\right ) \left (a d^2 (12 c C-5 B d)+2 b c \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (b c^2-a d^2\right ) \left (a d^2 (12 c C-5 B d)+2 b c \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^2 C d^4+a b d^2 \left (15 A d^2-25 B c d+24 c^2 C\right )-2 b^2 c^2 \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (b c^2-a d^2\right ) \left (a d^2 (12 c C-5 B d)+2 b c \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^2 C d^4+a b d^2 \left (15 A d^2-25 B c d+24 c^2 C\right )-2 b^2 c^2 \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a d^2 (12 c C-5 B d)+2 b c \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^2 C d^4+a b d^2 \left (15 A d^2-25 B c d+24 c^2 C\right )-2 b^2 c^2 \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (12 c C-5 B d)+2 b c \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^2 C d^4+a b d^2 \left (15 A d^2-25 B c d+24 c^2 C\right )-2 b^2 c^2 \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a^2 C d^4+a b d^2 \left (15 A d^2-25 B c d+24 c^2 C\right )-2 b^2 c^2 \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (12 c C-5 B d)+2 b c \left (15 A d^2-20 B c d+24 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) (12 c C-5 B d)}{5 b d^3}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b d^3}}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (A d^2-B c d+c^2 C\right )}{d^3 \sqrt {c+d x} \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(2*c^2*(c^2*C - B*c*d + A*d^2)*Sqrt[a - b*x^2])/(d^3*(b*c^2 - a*d^2)*Sqrt[ 
c + d*x]) + ((-2*C*(b*c^2 - a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b*d 
^3) - ((-2*(12*c*C - 5*B*d)*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]) 
/3 + ((-2*Sqrt[a]*(9*a^2*C*d^4 + a*b*d^2*(24*c^2*C - 25*B*c*d + 15*A*d^2) 
- 2*b^2*c^2*(24*c^2*C - 20*B*c*d + 15*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)/((Sqr 
t[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sq 
rt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*(b*c^2 - a*d^2)*(a*d^2*(12*c*C - 5 
*B*d) + 2*b*c*(24*c^2*C - 20*B*c*d + 15*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 - (Sqr 
t[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sq 
rt[c + d*x]*Sqrt[a - b*x^2]))/3)/(5*b*d^3))/(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 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

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. \(887\) vs. \(2(437)=874\).

Time = 7.65 (sec) , antiderivative size = 888, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 \left (-b d \,x^{2}+d a \right ) c^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) d^{4} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+d a \right )}}-\frac {2 C x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 d^{2} b}-\frac {2 \left (\frac {B d -C c}{d^{2}}-\frac {4 C c}{5 d^{2}}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (-\frac {c \left (A \,d^{2}-B c d +C \,c^{2}\right )}{d^{4}}-\frac {b \,c^{3} \left (A \,d^{2}-B c d +C \,c^{2}\right )}{d^{4} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {2 C a c}{5 d^{2} b}+\frac {\left (\frac {B d -C c}{d^{2}}-\frac {4 C c}{5 d^{2}}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {A \,d^{2}-B c d +C \,c^{2}}{d^{3}}-\frac {b \,c^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right )}{d^{3} \left (a \,d^{2}-b \,c^{2}\right )}+\frac {3 C a}{5 d b}-\frac {2 \left (\frac {B d -C c}{d^{2}}-\frac {4 C c}{5 d^{2}}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\)	\(888\)
risch	\(\text {Expression too large to display}\)	\(1062\)
default	\(\text {Expression too large to display}\)	\(3971\)

Input:

int(x^2*(C*x^2+B*x+A)/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2),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)*(-2*(-b*d*x^2+a* 
d)/(a*d^2-b*c^2)/d^4*c^2*(A*d^2-B*c*d+C*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2 
)-2/5*C/d^2/b*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(1/d^2*(B*d-C*c)-4/ 
5*C/d^2*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-c/d^4*(A*d^2-B*c*d+C 
*c^2)-b/d^4*c^3*(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c^2)+2/5*C/d^2/b*a*c+1/3*(1/d 
^2*(B*d-C*c)-4/5*C/d^2*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*(1/d^3*(A*d^2-B*c*d+C*c^2)-b/d^3*c^2* 
(A*d^2-B*c*d+C*c^2)/(a*d^2-b*c^2)+3/5*C/d/b*a-2/3*(1/d^2*(B*d-C*c)-4/5*C/d 
^2*c)/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*c)^(1/2)*((-c/d-1/b*(a* 
b)^(1/2))*EllipticE(((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))+1/b*(a*b)^(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))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.31 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^{3/2} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (48 \, C b^{2} c^{6} - 40 \, B b^{2} c^{5} d + 55 \, B a b c^{3} d^{3} + 15 \, B a^{2} c d^{5} - 30 \, {\left (2 \, C a b - A b^{2}\right )} c^{4} d^{2} - 6 \, {\left (3 \, C a^{2} + 10 \, A a b\right )} c^{2} d^{4} + {\left (48 \, C b^{2} c^{5} d - 40 \, B b^{2} c^{4} d^{2} + 55 \, B a b c^{2} d^{4} + 15 \, B a^{2} d^{6} - 30 \, {\left (2 \, C a b - A b^{2}\right )} c^{3} d^{3} - 6 \, {\left (3 \, C a^{2} + 10 \, A a b\right )} c d^{5}\right )} x\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 ) + 3 \, {\left (48 \, C b^{2} c^{5} d - 40 \, B b^{2} c^{4} d^{2} + 25 \, B a b c^{2} d^{4} - 6 \, {\left (4 \, C a b - 5 \, A b^{2}\right )} c^{3} d^{3} - 3 \, {\left (3 \, C a^{2} + 5 \, A a b\right )} c d^{5} + {\left (48 \, C b^{2} c^{4} d^{2} - 40 \, B b^{2} c^{3} d^{3} + 25 \, B a b c d^{5} - 6 \, {\left (4 \, C a b - 5 \, A b^{2}\right )} c^{2} d^{4} - 3 \, {\left (3 \, C a^{2} + 5 \, A a b\right )} d^{6}\right )} x\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 (24 \, C b^{2} c^{4} d^{2} - 20 \, B b^{2} c^{3} d^{3} + 5 \, B a b c d^{5} - 3 \, {\left (3 \, C a b - 5 \, A b^{2}\right )} c^{2} d^{4} - 3 \, {\left (C b^{2} c^{2} d^{4} - C a b d^{6}\right )} x^{2} + {\left (6 \, C b^{2} c^{3} d^{3} - 5 \, B b^{2} c^{2} d^{4} - 6 \, C a b c d^{5} + 5 \, B a b d^{6}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{45 \, {\left (b^{3} c^{3} d^{5} - a b^{2} c d^{7} + {\left (b^{3} c^{2} d^{6} - a b^{2} d^{8}\right )} x\right )}} \] Input:

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

Output:

2/45*((48*C*b^2*c^6 - 40*B*b^2*c^5*d + 55*B*a*b*c^3*d^3 + 15*B*a^2*c*d^5 - 
 30*(2*C*a*b - A*b^2)*c^4*d^2 - 6*(3*C*a^2 + 10*A*a*b)*c^2*d^4 + (48*C*b^2 
*c^5*d - 40*B*b^2*c^4*d^2 + 55*B*a*b*c^2*d^4 + 15*B*a^2*d^6 - 30*(2*C*a*b 
- A*b^2)*c^3*d^3 - 6*(3*C*a^2 + 10*A*a*b)*c*d^5)*x)*sqrt(-b*d)*weierstrass 
PInverse(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*(48*C*b^2*c^5*d - 40*B*b^2*c^4*d^2 + 25*B*a*b*c^2* 
d^4 - 6*(4*C*a*b - 5*A*b^2)*c^3*d^3 - 3*(3*C*a^2 + 5*A*a*b)*c*d^5 + (48*C* 
b^2*c^4*d^2 - 40*B*b^2*c^3*d^3 + 25*B*a*b*c*d^5 - 6*(4*C*a*b - 5*A*b^2)*c^ 
2*d^4 - 3*(3*C*a^2 + 5*A*a*b)*d^6)*x)*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), weierstrassPInver 
se(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*(24*C*b^2*c^4*d^2 - 20*B*b^2*c^3*d^3 + 5*B*a*b*c*d^5 - 
3*(3*C*a*b - 5*A*b^2)*c^2*d^4 - 3*(C*b^2*c^2*d^4 - C*a*b*d^6)*x^2 + (6*C*b 
^2*c^3*d^3 - 5*B*b^2*c^2*d^4 - 6*C*a*b*c*d^5 + 5*B*a*b*d^6)*x)*sqrt(-b*x^2 
 + a)*sqrt(d*x + c))/(b^3*c^3*d^5 - a*b^2*c*d^7 + (b^3*c^2*d^6 - a*b^2*d^8 
)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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