\(\int \frac {(a-b x^2)^{3/2} (A+B x+C x^2)}{x^5 \sqrt {c+d x}} \, dx\) [183]

Optimal result

Integrand size = 35, antiderivative size = 757 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=-\frac {a A \sqrt {c+d x} \sqrt {a-b x^2}}{4 c x^4}-\frac {a (8 B c-7 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{24 c^2 x^3}+\frac {\left (60 A b c^2-48 a c^2 C+40 a B c d-35 a A d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{96 c^3 x^2}+\frac {\left (4 b c^2 (64 B c-47 A d)+3 a d \left (48 c^2 C-40 B c d+35 A d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{192 c^4 x}-\frac {\sqrt {a} \sqrt {b} \left (4 b c^2 \left (96 c^2 C+64 B c d-47 A d^2\right )+3 a d^2 \left (48 c^2 C-40 B c d+35 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{192 c^4 d \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \sqrt {b} \left (4 b c^2 \left (96 c^2 C-32 B c d-17 A d^2\right )+a d^2 \left (48 c^2 C-40 B c d+35 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{192 c^3 d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (A \left (48 b^2 c^4-72 a b c^2 d^2+35 a^2 d^4\right )+8 a c \left (a d^2 (6 c C-5 B d)-12 b c^2 (2 c C-B d)\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{64 c^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-1/4*a*A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c/x^4-1/24*a*(-7*A*d+8*B*c)*(d*x+c 
)^(1/2)*(-b*x^2+a)^(1/2)/c^2/x^3+1/96*(-35*A*a*d^2+60*A*b*c^2+40*B*a*c*d-4 
8*C*a*c^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c^3/x^2+1/192*(4*b*c^2*(-47*A*d+ 
64*B*c)+3*a*d*(35*A*d^2-40*B*c*d+48*C*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2) 
/c^4/x-1/192*a^(1/2)*b^(1/2)*(4*b*c^2*(-47*A*d^2+64*B*c*d+96*C*c^2)+3*a*d^ 
2*(35*A*d^2-40*B*c*d+48*C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellipti 
cE(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))/c^4/d/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^( 
1/2)+1/192*a^(1/2)*b^(1/2)*(4*b*c^2*(-17*A*d^2-32*B*c*d+96*C*c^2)+a*d^2*(3 
5*A*d^2-40*B*c*d+48*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))/c^3/d/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2) 
-1/64*(A*(35*a^2*d^4-72*a*b*c^2*d^2+48*b^2*c^4)+8*a*c*(a*d^2*(-5*B*d+6*C*c 
)-12*b*c^2*(-B*d+2*C*c)))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a 
)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a 
^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/c^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.41 (sec) , antiderivative size = 2808, normalized size of antiderivative = 3.71 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\text {Result too large to show} \] Input:

Integrate[((a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^5*Sqrt[c + d*x]),x]
 

Output:

(-1/4*(a*A)/(c*x^4) - (a*(8*B*c - 7*A*d))/(24*c^2*x^3) + (60*A*b*c^2 - 48* 
a*c^2*C + 40*a*B*c*d - 35*a*A*d^2)/(96*c^3*x^2) + (256*b*B*c^3 - 188*A*b*c 
^2*d + 144*a*c^2*C*d - 120*a*B*c*d^2 + 105*a*A*d^3)/(192*c^4*x))*Sqrt[c + 
d*x]*Sqrt[a - b*x^2] + (Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2] 
*(-384*b^2*c^5*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 256*b^2*B*c^4*d*Sqrt[-c 
+ (Sqrt[a]*d)/Sqrt[b]] + 188*A*b^2*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] 
- 144*a*b*c^3*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 120*a*b*B*c^2*d^3*Sqr 
t[-c + (Sqrt[a]*d)/Sqrt[b]] - 105*a*A*b*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b 
]] - (384*b^2*c^7*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 - (256*b^2 
*B*c^6*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (188*A*b^2*c^5*d^2* 
Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (240*a*b*c^5*C*d^2*Sqrt[-c + 
 (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (376*a*b*B*c^4*d^3*Sqrt[-c + (Sqrt[a] 
*d)/Sqrt[b]])/(c + d*x)^2 - (293*a*A*b*c^3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[ 
b]])/(c + d*x)^2 + (144*a^2*c^3*C*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + 
 d*x)^2 - (120*a^2*B*c^2*d^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + 
 (105*a^2*A*c*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x)^2 + (768*b^2*c 
^6*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) + (512*b^2*B*c^5*d*Sqrt[-c 
+ (Sqrt[a]*d)/Sqrt[b]])/(c + d*x) - (376*A*b^2*c^4*d^2*Sqrt[-c + (Sqrt[a]* 
d)/Sqrt[b]])/(c + d*x) + (288*a*b*c^4*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] 
)/(c + d*x) - (240*a*b*B*c^3*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]])/(c + d...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2355

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

\(\Big \downarrow \) 638

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7296

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

\(\Big \downarrow \) 2011

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

\(\Big \downarrow \) 2091

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

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

Input:

Int[((a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^5*Sqrt[c + d*x]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 638

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Unintegrable[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x] /; FreeQ 
[{a, b, c, d, e, m, n, p}, x]
 

rule 2011

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x 
, a + b*x])
 

rule 2091

Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex 
pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && PolyQ[Px, x] && BinomialQ[z, x 
] && TrinomialQ[u, x] &&  !(BinomialMatchQ[z, x] && TrinomialMatchQ[u, x])
 

rule 2248

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) 
^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 
+ c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; 
 FreeQ[{a, b, c, d, e, f, m}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2] && In 
tegerQ[q]
 

rule 2355

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(e*x)^m*(c + d* 
x)^(n + 1)*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x]   I 
nt[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p} 
, x] && PolynomialQ[Px, x] && LtQ[n, 0]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7296

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst]]
 

Maple [A] (verified)

Time = 7.38 (sec) , antiderivative size = 1135, normalized size of antiderivative = 1.50

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1135\)
risch	\(\text {Expression too large to display}\)	\(1532\)
default	\(\text {Expression too large to display}\)	\(6267\)

Input:

int((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(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)*(-1/4*A*a/c/x^4* 
(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+1/24*a*(7*A*d-8*B*c)/c^2*(-b*d*x^3-b*c* 
x^2+a*d*x+a*c)^(1/2)/x^3-1/96/c^3*(35*A*a*d^2-60*A*b*c^2-40*B*a*c*d+48*C*a 
*c^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x^2+1/192*(105*A*a*d^3-188*A*b*c^ 
2*d-120*B*a*c*d^2+256*B*b*c^3+144*C*a*c^2*d)/c^4*(-b*d*x^3-b*c*x^2+a*d*x+a 
*c)^(1/2)/x+2*(B*b^2+1/192*b*d*(35*A*a*d^2-60*A*b*c^2-40*B*a*c*d+48*C*a*c^ 
2)/c^3)*(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*(C*b^2+1/384*b*d*(105*A*a*d^3-188*A*b*c^2*d-120*B*a*c*d^2+25 
6*B*b*c^3+144*C*a*c^2*d)/c^4)*(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)))-1/64*(35*A*a^2*d^4-72*A*a*b*c^2*d^2+48*A*b^2 
*c^4-40*B*a^2*c*d^3+96*B*a*b*c^3*d+48*C*a^2*c^2*d^2-192*C*a*b*c^4)/c^5*(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)^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x, algorithm="f 
ricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )}{x^{5} \sqrt {c + d x}}\, dx \] Input:

integrate((-b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**5/(d*x+c)**(1/2),x)
 

Output:

Integral((a - b*x**2)**(3/2)*(A + B*x + C*x**2)/(x**5*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} x^{5}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x, algorithm="m 
axima")
 

Output:

integrate((C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(sqrt(d*x + c)*x^5), x)
 

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} x^{5}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^5/(d*x+c)^(1/2),x, algorithm="g 
iac")
 

Output:

integrate((C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(sqrt(d*x + c)*x^5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/2}\,\left (C\,x^2+B\,x+A\right )}{x^5\,\sqrt {c+d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^5 \sqrt {c+d x}} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (C \,x^{2}+B x +A \right )}{x^{5} \sqrt {d x +c}}d x \] Input:

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

Output:

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