\(\int \frac {(a-b x^2)^{3/2} (A+B x+C x^2)}{x^6 \sqrt {c+d x}} \, dx\) [184]

Optimal result

Integrand size = 35, antiderivative size = 893 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=-\frac {a A \sqrt {c+d x} \sqrt {a-b x^2}}{5 c x^5}-\frac {a (10 B c-9 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{40 c^2 x^4}-\frac {\left (10 a c (8 c C-7 B d)-A \left (96 b c^2-63 a d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{240 c^3 x^3}+\frac {\left (12 b c^2 (50 B c-41 A d)+5 a d \left (80 c^2 C-70 B c d+63 A d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{960 c^4 x^2}-\frac {\left (3 A \left (128 b^2 c^4-516 a b c^2 d^2+315 a^2 d^4\right )-10 a c \left (4 b c^2 (64 c C-47 B d)-15 a d^2 (8 c C-7 B d)\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{1920 a c^5 x}+\frac {\sqrt {b} \left (3 A \left (128 b^2 c^4-516 a b c^2 d^2+315 a^2 d^4\right )-10 a c \left (4 b c^2 (64 c C-47 B d)-15 a d^2 (8 c C-7 B d)\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 )}{1920 \sqrt {a} c^5 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \left (A \left (384 b^2 c^4-564 a b c^2 d^2+315 a^2 d^4\right )+10 a c \left (5 a d^2 (8 c C-7 B d)+4 b c^2 (32 c C+17 B d)\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 )}{1920 \sqrt {a} c^4 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (48 b^2 c^4 (2 B c-A d)+24 a b c^2 d \left (8 c^2 C-6 B c d+5 A d^2\right )-a^2 d^3 \left (80 c^2 C-70 B c d+63 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 {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 )}{128 c^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-1/5*a*A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c/x^5-1/40*a*(-9*A*d+10*B*c)*(d*x+ 
c)^(1/2)*(-b*x^2+a)^(1/2)/c^2/x^4-1/240*(10*a*c*(-7*B*d+8*C*c)-A*(-63*a*d^ 
2+96*b*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c^3/x^3+1/960*(12*b*c^2*(-41*A 
*d+50*B*c)+5*a*d*(63*A*d^2-70*B*c*d+80*C*c^2))*(d*x+c)^(1/2)*(-b*x^2+a)^(1 
/2)/c^4/x^2-1/1920*(3*A*(315*a^2*d^4-516*a*b*c^2*d^2+128*b^2*c^4)-10*a*c*( 
4*b*c^2*(-47*B*d+64*C*c)-15*a*d^2*(-7*B*d+8*C*c)))*(d*x+c)^(1/2)*(-b*x^2+a 
)^(1/2)/a/c^5/x+1/1920*b^(1/2)*(3*A*(315*a^2*d^4-516*a*b*c^2*d^2+128*b^2*c 
^4)-10*a*c*(4*b*c^2*(-47*B*d+64*C*c)-15*a*d^2*(-7*B*d+8*C*c)))*(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))/a^(1/2)/c^5/((d*x+c)/(c+a 
^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/1920*b^(1/2)*(A*(315*a^2*d^4-5 
64*a*b*c^2*d^2+384*b^2*c^4)+10*a*c*(5*a*d^2*(-7*B*d+8*C*c)+4*b*c^2*(17*B*d 
+32*C*c)))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*Elli 
pticF(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))/a^(1/2)/c^4/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-1/128*(48* 
b^2*c^4*(-A*d+2*B*c)+24*a*b*c^2*d*(5*A*d^2-6*B*c*d+8*C*c^2)-a^2*d^3*(63*A* 
d^2-70*B*c*d+80*C*c^2))*((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^5/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.78 (sec) , antiderivative size = 3014, normalized size of antiderivative = 3.38 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=\text {Result too large to show} \] Input:

Integrate[((a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*Sqrt[c + d*x]),x]
 

Output:

(Sqrt[c + d*x]*Sqrt[a - b*x^2]*(-384*A*b^2*c^4*x^4 + 4*a*b*c^2*x^2*(10*c*x 
*(30*B*c + 64*c*C*x - 47*B*d*x) + 3*A*(64*c^2 - 82*c*d*x + 129*d^2*x^2)) - 
 a^2*(A*(384*c^4 - 432*c^3*d*x + 504*c^2*d^2*x^2 - 630*c*d^3*x^3 + 945*d^4 
*x^4) + 10*c*x*(8*c*C*x*(8*c^2 - 10*c*d*x + 15*d^2*x^2) + B*(48*c^3 - 56*c 
^2*d*x + 70*c*d^2*x^2 - 105*d^3*x^3)))))/(1920*a*c^5*x^5) - (384*A*b^3*c^7 
*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 2560*a*b^2*c^7*C*Sqrt[-c + (Sqrt[a]*d)/S 
qrt[b]] + 1880*a*b^2*B*c^6*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 1932*a*A*b^2 
*c^5*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 3760*a^2*b*c^5*C*d^2*Sqrt[-c + ( 
Sqrt[a]*d)/Sqrt[b]] - 2930*a^2*b*B*c^4*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] 
+ 2493*a^2*A*b*c^3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 1200*a^3*c^3*C*d^4 
*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 1050*a^3*B*c^2*d^5*Sqrt[-c + (Sqrt[a]*d) 
/Sqrt[b]] - 945*a^3*A*c*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 768*A*b^3*c^6 
*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 5120*a*b^2*c^6*C*Sqrt[-c + (Sq 
rt[a]*d)/Sqrt[b]]*(c + d*x) - 3760*a*b^2*B*c^5*d*Sqrt[-c + (Sqrt[a]*d)/Sqr 
t[b]]*(c + d*x) + 3096*a*A*b^2*c^4*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + 
 d*x) - 2400*a^2*b*c^4*C*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 21 
00*a^2*b*B*c^3*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 1890*a^2*A*b 
*c^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 384*A*b^3*c^5*Sqrt[-c 
+ (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 2560*a*b^2*c^5*C*Sqrt[-c + (Sqrt[a]*d 
)/Sqrt[b]]*(c + d*x)^2 + 1880*a*b^2*B*c^4*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[...
 

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.

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

\(\Big \downarrow \) 2355

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

\(\Big \downarrow \) 638

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

\(\Big \downarrow \) 7293

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx+\int \left (\frac {C \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{d x^5}+\frac {(B d-c C) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{d^2 x^6}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx+\int \left (\frac {C \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{d x^5}-\frac {(c C-B d) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{d^2 x^6}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7293

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-\frac {\int \left (\frac {(c C-B d) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{x^6}-\frac {C d \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{x^5}\right )dx}{d^2}\)

\(\Big \downarrow \) 7296

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2011

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

\(\Big \downarrow \) 2091

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2248

\(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\left (a-b x^2\right )^{3/2}}{x^6 \sqrt {c+d x}}dx-2 d^3 \int \left (-\frac {B a^2}{d^4 x^5 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {c (c C-B d) a^2}{d^6 x^6 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {2 b B a}{d^4 x^3 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {\left (-a C d^2-2 b c (c C-B d)\right ) a}{d^6 x^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 B}{d^4 x \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}+\frac {b \left (2 a C d^2+b c (c C-B d)\right )}{d^6 x^2 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}-\frac {b^2 C}{d^4 \sqrt {-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 7239

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

Input:

Int[((a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*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 = 10.72 (sec) , antiderivative size = 1288, normalized size of antiderivative = 1.44

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1288\)
risch	\(\text {Expression too large to display}\)	\(1865\)
default	\(\text {Expression too large to display}\)	\(7291\)

Input:

int((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(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)*(-1/5*A*a/c/x^5* 
(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+1/40*a*(9*A*d-10*B*c)/c^2*(-b*d*x^3-b*c 
*x^2+a*d*x+a*c)^(1/2)/x^4-1/240/c^3*(63*A*a*d^2-96*A*b*c^2-70*B*a*c*d+80*C 
*a*c^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x^3+1/960*(315*A*a*d^3-492*A*b* 
c^2*d-350*B*a*c*d^2+600*B*b*c^3+400*C*a*c^2*d)/c^4*(-b*d*x^3-b*c*x^2+a*d*x 
+a*c)^(1/2)/x^2-1/1920/c^5/a*(945*A*a^2*d^4-1548*A*a*b*c^2*d^2+384*A*b^2*c 
^4-1050*B*a^2*c*d^3+1880*B*a*b*c^3*d+1200*C*a^2*c^2*d^2-2560*C*a*b*c^4)*(- 
b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x+2*(C*b^2-1/1920*b*d*(315*A*a*d^3-492*A* 
b*c^2*d-350*B*a*c*d^2+600*B*b*c^3+400*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)*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))-1/1920*(945*A*a^ 
2*d^4-1548*A*a*b*c^2*d^2+384*A*b^2*c^4-1050*B*a^2*c*d^3+1880*B*a*b*c^3*d+1 
200*C*a^2*c^2*d^2-2560*C*a*b*c^4)*b*d/a/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)^(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/...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c)^(1/2),x, algorithm="f 
ricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )}{x^{6} \sqrt {c + d x}}\, dx \] Input:

integrate((-b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**6/(d*x+c)**(1/2),x)
 

Output:

Integral((a - b*x**2)**(3/2)*(A + B*x + C*x**2)/(x**6*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} x^{6}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(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^6), x)
 

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x + c} x^{6}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(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^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/2}\,\left (C\,x^2+B\,x+A\right )}{x^6\,\sqrt {c+d\,x}} \,d x \] Input:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*(c + d*x)^(1/2)),x)
 

Output:

int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 \sqrt {c+d x}} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (C \,x^{2}+B x +A \right )}{x^{6} \sqrt {d x +c}}d x \] Input:

int((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c)^(1/2),x)
 

Output:

int((-b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c)^(1/2),x)