3.517 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0704439, 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 \[ -\frac{a^2 A}{2 x^2}-\frac{a (a B+2 A b)}{x}+b \log (x) (2 a B+A b)+b^2 B x \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.6977, size = 41, normalized size = 0.93 \[ - \frac{A a^{2}}{2 x^{2}} + B b^{2} x - \frac{a \left (2 A b + B a\right )}{x} + b \left (A b + 2 B a\right ) \log{\left (x \right )} \]

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**3,x)

[Out]

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

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Mathematica [A]  time = 0.0441653, size = 43, normalized size = 0.98 \[ -\frac{a^2 (A+2 B x)}{2 x^2}+b \log (x) (2 a B+A b)-\frac{2 a A b}{x}+b^2 B x \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 48, normalized size = 1.1 \[{b}^{2}Bx+A{b}^{2}\ln \left ( x \right ) +2\,B\ln \left ( x \right ) ab-{\frac{A{a}^{2}}{2\,{x}^{2}}}-2\,{\frac{abA}{x}}-{\frac{{a}^{2}B}{x}} \]

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^3,x)

[Out]

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

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Maxima [A]  time = 0.675507, size = 62, normalized size = 1.41 \[ B b^{2} x +{\left (2 \, B a b + A b^{2}\right )} \log \left (x\right ) - \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \]

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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.277055, size = 72, normalized size = 1.64 \[ \frac{2 \, B b^{2} x^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \]

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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.94092, size = 44, normalized size = 1. \[ B b^{2} x + b \left (A b + 2 B a\right ) \log{\left (x \right )} - \frac{A a^{2} + x \left (4 A a b + 2 B a^{2}\right )}{2 x^{2}} \]

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**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.267242, size = 63, normalized size = 1.43 \[ B b^{2} x +{\left (2 \, B a b + A b^{2}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \]

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^3,x, algorithm="giac")

[Out]

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