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

Optimal. Leaf size=22 \[ \log (x) (a B+A b)-\frac{a A}{x}+b B x \]

[Out]

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

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Rubi [A]  time = 0.0393829, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \log (x) (a B+A b)-\frac{a A}{x}+b B x \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0140815, size = 22, normalized size = 1. \[ \log (x) (a B+A b)-\frac{a A}{x}+b B x \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 23, normalized size = 1.1 \[ bBx+A\ln \left ( x \right ) b+B\ln \left ( x \right ) a-{\frac{Aa}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.34666, size = 30, normalized size = 1.36 \[ B b x +{\left (B a + A b\right )} \log \left (x\right ) - \frac{A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b*x + (B*a + A*b)*log(x) - A*a/x

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Fricas [A]  time = 0.201067, size = 35, normalized size = 1.59 \[ \frac{B b x^{2} +{\left (B a + A b\right )} x \log \left (x\right ) - A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.32623, size = 19, normalized size = 0.86 \[ - \frac{A a}{x} + B b x + \left (A b + B a\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/x + B*b*x + (A*b + B*a)*log(x)

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GIAC/XCAS [A]  time = 0.285383, size = 31, normalized size = 1.41 \[ B b x +{\left (B a + A b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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