3.21 \(\int \frac{\left (a+b x^3\right )^2 \left (A+B x^3\right )}{x^8} \, dx\)

Optimal. Leaf size=53 \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{x}+\frac{1}{2} b^2 B x^2 \]

[Out]

-(a^2*A)/(7*x^7) - (a*(2*A*b + a*B))/(4*x^4) - (b*(A*b + 2*a*B))/x + (b^2*B*x^2)
/2

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Rubi [A]  time = 0.0948535, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{x}+\frac{1}{2} b^2 B x^2 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^2*(A + B*x^3))/x^8,x]

[Out]

-(a^2*A)/(7*x^7) - (a*(2*A*b + a*B))/(4*x^4) - (b*(A*b + 2*a*B))/x + (b^2*B*x^2)
/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{7 x^{7}} + B b^{2} \int x\, dx - \frac{a \left (2 A b + B a\right )}{4 x^{4}} - \frac{b \left (A b + 2 B a\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2*(B*x**3+A)/x**8,x)

[Out]

-A*a**2/(7*x**7) + B*b**2*Integral(x, x) - a*(2*A*b + B*a)/(4*x**4) - b*(A*b + 2
*B*a)/x

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Mathematica [A]  time = 0.0301731, size = 54, normalized size = 1.02 \[ -\frac{a^2 \left (4 A+7 B x^3\right )+14 a b x^3 \left (A+4 B x^3\right )-14 b^2 x^6 \left (B x^3-2 A\right )}{28 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^2*(A + B*x^3))/x^8,x]

[Out]

-(-14*b^2*x^6*(-2*A + B*x^3) + 14*a*b*x^3*(A + 4*B*x^3) + a^2*(4*A + 7*B*x^3))/(
28*x^7)

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Maple [A]  time = 0.008, size = 48, normalized size = 0.9 \[ -{\frac{A{a}^{2}}{7\,{x}^{7}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{4\,{x}^{4}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{x}}+{\frac{{b}^{2}B{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2*(B*x^3+A)/x^8,x)

[Out]

-1/7*a^2*A/x^7-1/4*a*(2*A*b+B*a)/x^4-b*(A*b+2*B*a)/x+1/2*b^2*B*x^2

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Maxima [A]  time = 1.41352, size = 73, normalized size = 1.38 \[ \frac{1}{2} \, B b^{2} x^{2} - \frac{28 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 7 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} + 4 \, A a^{2}}{28 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2/x^8,x, algorithm="maxima")

[Out]

1/2*B*b^2*x^2 - 1/28*(28*(2*B*a*b + A*b^2)*x^6 + 7*(B*a^2 + 2*A*a*b)*x^3 + 4*A*a
^2)/x^7

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Fricas [A]  time = 0.213405, size = 72, normalized size = 1.36 \[ \frac{14 \, B b^{2} x^{9} - 28 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} - 7 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} - 4 \, A a^{2}}{28 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2/x^8,x, algorithm="fricas")

[Out]

1/28*(14*B*b^2*x^9 - 28*(2*B*a*b + A*b^2)*x^6 - 7*(B*a^2 + 2*A*a*b)*x^3 - 4*A*a^
2)/x^7

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Sympy [A]  time = 4.04729, size = 54, normalized size = 1.02 \[ \frac{B b^{2} x^{2}}{2} - \frac{4 A a^{2} + x^{6} \left (28 A b^{2} + 56 B a b\right ) + x^{3} \left (14 A a b + 7 B a^{2}\right )}{28 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2*(B*x**3+A)/x**8,x)

[Out]

B*b**2*x**2/2 - (4*A*a**2 + x**6*(28*A*b**2 + 56*B*a*b) + x**3*(14*A*a*b + 7*B*a
**2))/(28*x**7)

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GIAC/XCAS [A]  time = 0.213396, size = 76, normalized size = 1.43 \[ \frac{1}{2} \, B b^{2} x^{2} - \frac{56 \, B a b x^{6} + 28 \, A b^{2} x^{6} + 7 \, B a^{2} x^{3} + 14 \, A a b x^{3} + 4 \, A a^{2}}{28 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^2/x^8,x, algorithm="giac")

[Out]

1/2*B*b^2*x^2 - 1/28*(56*B*a*b*x^6 + 28*A*b^2*x^6 + 7*B*a^2*x^3 + 14*A*a*b*x^3 +
4*A*a^2)/x^7