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

Optimal. Leaf size=28 \[ a A \log (x)+a B x+\frac{1}{2} A c x^2+\frac{1}{3} B c x^3 \]

[Out]

a*B*x + (A*c*x^2)/2 + (B*c*x^3)/3 + a*A*Log[x]

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Rubi [A]  time = 0.0319317, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ a A \log (x)+a B x+\frac{1}{2} A c x^2+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

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

[Out]

a*B*x + (A*c*x^2)/2 + (B*c*x^3)/3 + a*A*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*log(x) + A*c*Integral(x, x) + B*c*x**3/3 + a*Integral(B, x)

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Mathematica [A]  time = 0.00583937, size = 28, normalized size = 1. \[ a A \log (x)+a B x+\frac{1}{2} A c x^2+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

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

[Out]

a*B*x + (A*c*x^2)/2 + (B*c*x^3)/3 + a*A*Log[x]

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Maple [A]  time = 0.003, size = 25, normalized size = 0.9 \[ aBx+{\frac{Ac{x}^{2}}{2}}+{\frac{Bc{x}^{3}}{3}}+aA\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)/x,x)

[Out]

a*B*x+1/2*A*c*x^2+1/3*B*c*x^3+a*A*ln(x)

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Maxima [A]  time = 0.684511, size = 32, normalized size = 1.14 \[ \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, A c x^{2} + B a x + A a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*c*x^3 + 1/2*A*c*x^2 + B*a*x + A*a*log(x)

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Fricas [A]  time = 0.29187, size = 32, normalized size = 1.14 \[ \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, A c x^{2} + B a x + A a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*c*x^3 + 1/2*A*c*x^2 + B*a*x + A*a*log(x)

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Sympy [A]  time = 1.05257, size = 27, normalized size = 0.96 \[ A a \log{\left (x \right )} + \frac{A c x^{2}}{2} + B a x + \frac{B c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*log(x) + A*c*x**2/2 + B*a*x + B*c*x**3/3

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GIAC/XCAS [A]  time = 0.266766, size = 34, normalized size = 1.21 \[ \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, A c x^{2} + B a x + A a{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*c*x^3 + 1/2*A*c*x^2 + B*a*x + A*a*ln(abs(x))