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

Optimal. Leaf size=43 \[ a^2 A \log (x)+a A b x^2+\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} A b^2 x^4 \]

[Out]

a*A*b*x^2 + (A*b^2*x^4)/4 + (B*(a + b*x^2)^3)/(6*b) + a^2*A*Log[x]

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Rubi [A]  time = 0.0807077, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ a^2 A \log (x)+a A b x^2+\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} A b^2 x^4 \]

Antiderivative was successfully verified.

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

[Out]

a*A*b*x^2 + (A*b^2*x^4)/4 + (B*(a + b*x^2)^3)/(6*b) + a^2*A*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A a^{2} \log{\left (x^{2} \right )}}{2} + A a b x^{2} + \frac{A b^{2} \int ^{x^{2}} x\, dx}{2} + \frac{B \left (a + b x^{2}\right )^{3}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*log(x**2)/2 + A*a*b*x**2 + A*b**2*Integral(x, (x, x**2))/2 + B*(a + b*x**
2)**3/(6*b)

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Mathematica [A]  time = 0.0258223, size = 51, normalized size = 1.19 \[ a^2 A \log (x)+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 51, normalized size = 1.2 \[{\frac{B{b}^{2}{x}^{6}}{6}}+{\frac{A{b}^{2}{x}^{4}}{4}}+{\frac{B{x}^{4}ab}{2}}+aAb{x}^{2}+{\frac{B{x}^{2}{a}^{2}}{2}}+{a}^{2}A\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(B*x^2+A)/x,x)

[Out]

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

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Maxima [A]  time = 1.35194, size = 70, normalized size = 1.63 \[ \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{4} \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + \frac{1}{2} \, A a^{2} \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244783, size = 66, normalized size = 1.53 \[ \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{4} \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.18992, size = 49, normalized size = 1.14 \[ A a^{2} \log{\left (x \right )} + \frac{B b^{2} x^{6}}{6} + x^{4} \left (\frac{A b^{2}}{4} + \frac{B a b}{2}\right ) + x^{2} \left (A a b + \frac{B a^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227323, size = 72, normalized size = 1.67 \[ \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{2} \, B a b x^{4} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{2} \, B a^{2} x^{2} + A a b x^{2} + \frac{1}{2} \, A a^{2}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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