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

Optimal. Leaf size=29 \[ \frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4 \]

[Out]

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

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Rubi [A]  time = 0.0699793, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0145397, size = 29, normalized size = 1. \[ \frac{1}{2} x^2 (A c+b B)+A b \log (x)+\frac{1}{4} B c x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 1. \[{\frac{Bc{x}^{4}}{4}}+{\frac{A{x}^{2}c}{2}}+{\frac{Bb{x}^{2}}{2}}+Ab\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37615, size = 38, normalized size = 1.31 \[ \frac{1}{4} \, B c x^{4} + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} + \frac{1}{2} \, A b \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224134, size = 34, normalized size = 1.17 \[ \frac{1}{4} \, B c x^{4} + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} + A b \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.527099, size = 27, normalized size = 0.93 \[ A b \log{\left (x \right )} + \frac{B c x^{4}}{4} + x^{2} \left (\frac{A c}{2} + \frac{B b}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208314, size = 41, normalized size = 1.41 \[ \frac{1}{4} \, B c x^{4} + \frac{1}{2} \, B b x^{2} + \frac{1}{2} \, A c x^{2} + \frac{1}{2} \, A b{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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