3.37 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{14}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{A b^3}{7 x^7}-\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x \]

[Out]

-(A*b^3)/(7*x^7) - (b^2*(b*B + 3*A*c))/(5*x^5) - (b*c*(b*B + A*c))/x^3 - (c^2*(3
*b*B + A*c))/x + B*c^3*x

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Rubi [A]  time = 0.127157, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{A b^3}{7 x^7}-\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^14,x]

[Out]

-(A*b^3)/(7*x^7) - (b^2*(b*B + 3*A*c))/(5*x^5) - (b*c*(b*B + A*c))/x^3 - (c^2*(3
*b*B + A*c))/x + B*c^3*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A b^{3}}{7 x^{7}} - \frac{b^{2} \left (3 A c + B b\right )}{5 x^{5}} - \frac{b c \left (A c + B b\right )}{x^{3}} + c^{3} \int B\, dx - \frac{c^{2} \left (A c + 3 B b\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**3/x**14,x)

[Out]

-A*b**3/(7*x**7) - b**2*(3*A*c + B*b)/(5*x**5) - b*c*(A*c + B*b)/x**3 + c**3*Int
egral(B, x) - c**2*(A*c + 3*B*b)/x

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Mathematica [A]  time = 0.0488057, size = 66, normalized size = 1. \[ -\frac{A b^3}{7 x^7}-\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{c^2 (A c+3 b B)}{x}-\frac{b c (A c+b B)}{x^3}+B c^3 x \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^14,x]

[Out]

-(A*b^3)/(7*x^7) - (b^2*(b*B + 3*A*c))/(5*x^5) - (b*c*(b*B + A*c))/x^3 - (c^2*(3
*b*B + A*c))/x + B*c^3*x

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Maple [A]  time = 0.009, size = 63, normalized size = 1. \[ -{\frac{A{b}^{3}}{7\,{x}^{7}}}-{\frac{{b}^{2} \left ( 3\,Ac+Bb \right ) }{5\,{x}^{5}}}-{\frac{bc \left ( Ac+Bb \right ) }{{x}^{3}}}-{\frac{{c}^{2} \left ( Ac+3\,Bb \right ) }{x}}+B{c}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^3/x^14,x)

[Out]

-1/7*A*b^3/x^7-1/5*b^2*(3*A*c+B*b)/x^5-b*c*(A*c+B*b)/x^3-c^2*(A*c+3*B*b)/x+B*c^3
*x

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Maxima [A]  time = 1.38035, size = 99, normalized size = 1.5 \[ B c^{3} x - \frac{35 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 35 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + 5 \, A b^{3} + 7 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^14,x, algorithm="maxima")

[Out]

B*c^3*x - 1/35*(35*(3*B*b*c^2 + A*c^3)*x^6 + 35*(B*b^2*c + A*b*c^2)*x^4 + 5*A*b^
3 + 7*(B*b^3 + 3*A*b^2*c)*x^2)/x^7

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Fricas [A]  time = 0.197807, size = 101, normalized size = 1.53 \[ \frac{35 \, B c^{3} x^{8} - 35 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} - 35 \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} - 5 \, A b^{3} - 7 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^14,x, algorithm="fricas")

[Out]

1/35*(35*B*c^3*x^8 - 35*(3*B*b*c^2 + A*c^3)*x^6 - 35*(B*b^2*c + A*b*c^2)*x^4 - 5
*A*b^3 - 7*(B*b^3 + 3*A*b^2*c)*x^2)/x^7

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Sympy [A]  time = 2.95958, size = 75, normalized size = 1.14 \[ B c^{3} x - \frac{5 A b^{3} + x^{6} \left (35 A c^{3} + 105 B b c^{2}\right ) + x^{4} \left (35 A b c^{2} + 35 B b^{2} c\right ) + x^{2} \left (21 A b^{2} c + 7 B b^{3}\right )}{35 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**3/x**14,x)

[Out]

B*c**3*x - (5*A*b**3 + x**6*(35*A*c**3 + 105*B*b*c**2) + x**4*(35*A*b*c**2 + 35*
B*b**2*c) + x**2*(21*A*b**2*c + 7*B*b**3))/(35*x**7)

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GIAC/XCAS [A]  time = 0.210697, size = 104, normalized size = 1.58 \[ B c^{3} x - \frac{105 \, B b c^{2} x^{6} + 35 \, A c^{3} x^{6} + 35 \, B b^{2} c x^{4} + 35 \, A b c^{2} x^{4} + 7 \, B b^{3} x^{2} + 21 \, A b^{2} c x^{2} + 5 \, A b^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3*(B*x^2 + A)/x^14,x, algorithm="giac")

[Out]

B*c^3*x - 1/35*(105*B*b*c^2*x^6 + 35*A*c^3*x^6 + 35*B*b^2*c*x^4 + 35*A*b*c^2*x^4
+ 7*B*b^3*x^2 + 21*A*b^2*c*x^2 + 5*A*b^3)/x^7