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

Optimal. Leaf size=53 \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]

[Out]

a^2*B*x + a*A*c*x^2 + (2*a*B*c*x^3)/3 + (A*c^2*x^4)/4 + (B*c^2*x^5)/5 + a^2*A*Lo
g[x]

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Rubi [A]  time = 0.053328, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

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

[Out]

a^2*B*x + a*A*c*x^2 + (2*a*B*c*x^3)/3 + (A*c^2*x^4)/4 + (B*c^2*x^5)/5 + a^2*A*Lo
g[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a^{2} \log{\left (x \right )} + 2 A a c \int x\, dx + \frac{A c^{2} x^{4}}{4} + \frac{2 B a c x^{3}}{3} + \frac{B c^{2} x^{5}}{5} + a^{2} \int B\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*log(x) + 2*A*a*c*Integral(x, x) + A*c**2*x**4/4 + 2*B*a*c*x**3/3 + B*c**2
*x**5/5 + a**2*Integral(B, x)

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Mathematica [A]  time = 0.0107268, size = 53, normalized size = 1. \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

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

[Out]

a^2*B*x + a*A*c*x^2 + (2*a*B*c*x^3)/3 + (A*c^2*x^4)/4 + (B*c^2*x^5)/5 + a^2*A*Lo
g[x]

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Maple [A]  time = 0.003, size = 48, normalized size = 0.9 \[{a}^{2}Bx+aAc{x}^{2}+{\frac{2\,aBc{x}^{3}}{3}}+{\frac{A{c}^{2}{x}^{4}}{4}}+{\frac{B{c}^{2}{x}^{5}}{5}}+{a}^{2}A\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^2/x,x)

[Out]

a^2*B*x+a*A*c*x^2+2/3*a*B*c*x^3+1/4*A*c^2*x^4+1/5*B*c^2*x^5+a^2*A*ln(x)

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Maxima [A]  time = 0.682741, size = 63, normalized size = 1.19 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*c^2*x^5 + 1/4*A*c^2*x^4 + 2/3*B*a*c*x^3 + A*a*c*x^2 + B*a^2*x + A*a^2*log(
x)

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Fricas [A]  time = 0.287524, size = 63, normalized size = 1.19 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*c^2*x^5 + 1/4*A*c^2*x^4 + 2/3*B*a*c*x^3 + A*a*c*x^2 + B*a^2*x + A*a^2*log(
x)

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Sympy [A]  time = 1.16217, size = 54, normalized size = 1.02 \[ A a^{2} \log{\left (x \right )} + A a c x^{2} + \frac{A c^{2} x^{4}}{4} + B a^{2} x + \frac{2 B a c x^{3}}{3} + \frac{B c^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*log(x) + A*a*c*x**2 + A*c**2*x**4/4 + B*a**2*x + 2*B*a*c*x**3/3 + B*c**2*
x**5/5

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GIAC/XCAS [A]  time = 0.269609, size = 65, normalized size = 1.23 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*c^2*x^5 + 1/4*A*c^2*x^4 + 2/3*B*a*c*x^3 + A*a*c*x^2 + B*a^2*x + A*a^2*ln(a
bs(x))