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

Optimal. Leaf size=44 \[ -\frac{a^2 A}{x}+b x (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{2} b^2 B x^2 \]

[Out]

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

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Rubi [A]  time = 0.0773728, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{a^2 A}{x}+b x (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{2} b^2 B x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0431667, size = 43, normalized size = 0.98 \[ -\frac{a^2 A}{x}+a \log (x) (a B+2 A b)+2 a b B x+\frac{1}{2} b^2 x (2 A+B x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 46, normalized size = 1.1 \[{\frac{{b}^{2}B{x}^{2}}{2}}+Ax{b}^{2}+2\,Bxab+2\,A\ln \left ( x \right ) ab+{a}^{2}B\ln \left ( x \right ) -{\frac{A{a}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.684156, size = 62, normalized size = 1.41 \[ \frac{1}{2} \, B b^{2} x^{2} - \frac{A a^{2}}{x} +{\left (2 \, B a b + A b^{2}\right )} x +{\left (B a^{2} + 2 \, A a b\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.284031, size = 70, normalized size = 1.59 \[ \frac{B b^{2} x^{3} - 2 \, A a^{2} + 2 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x \log \left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.33059, size = 42, normalized size = 0.95 \[ - \frac{A a^{2}}{x} + \frac{B b^{2} x^{2}}{2} + a \left (2 A b + B a\right ) \log{\left (x \right )} + x \left (A b^{2} + 2 B a b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269805, size = 62, normalized size = 1.41 \[ \frac{1}{2} \, B b^{2} x^{2} + 2 \, B a b x + A b^{2} x - \frac{A a^{2}}{x} +{\left (B a^{2} + 2 \, A a b\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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