3.18 \(\int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^5} \, dx\)

Optimal. Leaf size=51 \[ -\frac{a^2 A}{4 x^4}-\frac{a (a B+2 A b)}{2 x^2}+b \log (x) (2 a B+A b)+\frac{1}{2} b^2 B x^2 \]

[Out]

-(a^2*A)/(4*x^4) - (a*(2*A*b + a*B))/(2*x^2) + (b^2*B*x^2)/2 + b*(A*b + 2*a*B)*L
og[x]

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Rubi [A]  time = 0.122924, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2 A}{4 x^4}-\frac{a (a B+2 A b)}{2 x^2}+b \log (x) (2 a B+A b)+\frac{1}{2} b^2 B x^2 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(A + B*x^2))/x^5,x]

[Out]

-(a^2*A)/(4*x^4) - (a*(2*A*b + a*B))/(2*x^2) + (b^2*B*x^2)/2 + b*(A*b + 2*a*B)*L
og[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(B*x**2+A)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0411089, size = 50, normalized size = 0.98 \[ b \log (x) (2 a B+A b)-\frac{a^2 \left (A+2 B x^2\right )+4 a A b x^2-2 b^2 B x^6}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(A + B*x^2))/x^5,x]

[Out]

-(4*a*A*b*x^2 - 2*b^2*B*x^6 + a^2*(A + 2*B*x^2))/(4*x^4) + b*(A*b + 2*a*B)*Log[x
]

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Maple [A]  time = 0.008, size = 51, normalized size = 1. \[{\frac{{b}^{2}B{x}^{2}}{2}}+A\ln \left ( x \right ){b}^{2}+2\,B\ln \left ( x \right ) ab-{\frac{A{a}^{2}}{4\,{x}^{4}}}-{\frac{abA}{{x}^{2}}}-{\frac{{a}^{2}B}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(B*x^2+A)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.35162, size = 73, normalized size = 1.43 \[ \frac{1}{2} \, B b^{2} x^{2} + \frac{1}{2} \,{\left (2 \, B a b + A b^{2}\right )} \log \left (x^{2}\right ) - \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235866, size = 74, normalized size = 1.45 \[ \frac{2 \, B b^{2} x^{6} + 4 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.77345, size = 49, normalized size = 0.96 \[ \frac{B b^{2} x^{2}}{2} + b \left (A b + 2 B a\right ) \log{\left (x \right )} - \frac{A a^{2} + x^{2} \left (4 A a b + 2 B a^{2}\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(B*x**2+A)/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.237539, size = 97, normalized size = 1.9 \[ \frac{1}{2} \, B b^{2} x^{2} + \frac{1}{2} \,{\left (2 \, B a b + A b^{2}\right )}{\rm ln}\left (x^{2}\right ) - \frac{6 \, B a b x^{4} + 3 \, A b^{2} x^{4} + 2 \, B a^{2} x^{2} + 4 \, A a b x^{2} + A a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^5,x, algorithm="giac")

[Out]

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