3.115 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 c \left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{315 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{63 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \]

[Out]

-(A*(b*x^2 + c*x^4)^(5/2))/(9*b*x^14) - ((9*b*B - 4*A*c)*(b*x^2 + c*x^4)^(5/2))/
(63*b^2*x^12) + (2*c*(9*b*B - 4*A*c)*(b*x^2 + c*x^4)^(5/2))/(315*b^3*x^10)

_______________________________________________________________________________________

Rubi [A]  time = 0.424995, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 c \left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{315 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{63 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^13,x]

[Out]

-(A*(b*x^2 + c*x^4)^(5/2))/(9*b*x^14) - ((9*b*B - 4*A*c)*(b*x^2 + c*x^4)^(5/2))/
(63*b^2*x^12) + (2*c*(9*b*B - 4*A*c)*(b*x^2 + c*x^4)^(5/2))/(315*b^3*x^10)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.0346, size = 88, normalized size = 0.92 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{9 b x^{14}} + \frac{\left (4 A c - 9 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{63 b^{2} x^{12}} - \frac{2 c \left (4 A c - 9 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{315 b^{3} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**13,x)

[Out]

-A*(b*x**2 + c*x**4)**(5/2)/(9*b*x**14) + (4*A*c - 9*B*b)*(b*x**2 + c*x**4)**(5/
2)/(63*b**2*x**12) - 2*c*(4*A*c - 9*B*b)*(b*x**2 + c*x**4)**(5/2)/(315*b**3*x**1
0)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0890842, size = 66, normalized size = 0.69 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (A \left (-35 b^2+20 b c x^2-8 c^2 x^4\right )+9 b B x^2 \left (2 c x^2-5 b\right )\right )}{315 b^3 x^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^13,x]

[Out]

((x^2*(b + c*x^2))^(5/2)*(9*b*B*x^2*(-5*b + 2*c*x^2) + A*(-35*b^2 + 20*b*c*x^2 -
8*c^2*x^4)))/(315*b^3*x^14)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 70, normalized size = 0.7 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 8\,A{c}^{2}{x}^{4}-18\,B{x}^{4}bc-20\,Abc{x}^{2}+45\,B{b}^{2}{x}^{2}+35\,{b}^{2}A \right ) }{315\,{x}^{12}{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^13,x)

[Out]

-1/315*(c*x^2+b)*(8*A*c^2*x^4-18*B*b*c*x^4-20*A*b*c*x^2+45*B*b^2*x^2+35*A*b^2)*(
c*x^4+b*x^2)^(3/2)/x^12/b^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.330922, size = 147, normalized size = 1.53 \[ \frac{{\left (2 \,{\left (9 \, B b c^{3} - 4 \, A c^{4}\right )} x^{8} -{\left (9 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} x^{6} - 35 \, A b^{4} - 3 \,{\left (24 \, B b^{3} c + A b^{2} c^{2}\right )} x^{4} - 5 \,{\left (9 \, B b^{4} + 10 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, b^{3} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^13,x, algorithm="fricas")

[Out]

1/315*(2*(9*B*b*c^3 - 4*A*c^4)*x^8 - (9*B*b^2*c^2 - 4*A*b*c^3)*x^6 - 35*A*b^4 -
3*(24*B*b^3*c + A*b^2*c^2)*x^4 - 5*(9*B*b^4 + 10*A*b^3*c)*x^2)*sqrt(c*x^4 + b*x^
2)/(b^3*x^10)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**13,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**13, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.981075, size = 581, normalized size = 6.05 \[ \frac{4 \,{\left (315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{14} B c^{\frac{7}{2}}{\rm sign}\left (x\right ) - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} B b c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 840 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} A c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} B b^{2} c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 1260 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} A b c^{\frac{9}{2}}{\rm sign}\left (x\right ) - 819 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} B b^{3} c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} A b^{2} c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 441 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} B b^{4} c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 504 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} A b^{3} c^{\frac{9}{2}}{\rm sign}\left (x\right ) - 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b^{5} c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 144 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A b^{4} c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 81 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{6} c^{\frac{7}{2}}{\rm sign}\left (x\right ) - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} A b^{5} c^{\frac{9}{2}}{\rm sign}\left (x\right ) - 9 \, B b^{7} c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 4 \, A b^{6} c^{\frac{9}{2}}{\rm sign}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^13,x, algorithm="giac")

[Out]

4/315*(315*(sqrt(c)*x - sqrt(c*x^2 + b))^14*B*c^(7/2)*sign(x) - 315*(sqrt(c)*x -
sqrt(c*x^2 + b))^12*B*b*c^(7/2)*sign(x) + 840*(sqrt(c)*x - sqrt(c*x^2 + b))^12*
A*c^(9/2)*sign(x) + 315*(sqrt(c)*x - sqrt(c*x^2 + b))^10*B*b^2*c^(7/2)*sign(x) +
1260*(sqrt(c)*x - sqrt(c*x^2 + b))^10*A*b*c^(9/2)*sign(x) - 819*(sqrt(c)*x - sq
rt(c*x^2 + b))^8*B*b^3*c^(7/2)*sign(x) + 1764*(sqrt(c)*x - sqrt(c*x^2 + b))^8*A*
b^2*c^(9/2)*sign(x) + 441*(sqrt(c)*x - sqrt(c*x^2 + b))^6*B*b^4*c^(7/2)*sign(x)
+ 504*(sqrt(c)*x - sqrt(c*x^2 + b))^6*A*b^3*c^(9/2)*sign(x) - 9*(sqrt(c)*x - sqr
t(c*x^2 + b))^4*B*b^5*c^(7/2)*sign(x) + 144*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*b^
4*c^(9/2)*sign(x) + 81*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^6*c^(7/2)*sign(x) - 3
6*(sqrt(c)*x - sqrt(c*x^2 + b))^2*A*b^5*c^(9/2)*sign(x) - 9*B*b^7*c^(7/2)*sign(x
) + 4*A*b^6*c^(9/2)*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^9