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

Optimal. Leaf size=65 \[ -\frac{a^3 A}{x}+a^2 \log (x) (a B+3 A b)+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]

[Out]

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

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Rubi [A]  time = 0.104044, antiderivative size = 65, 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 \[ -\frac{a^3 A}{x}+a^2 \log (x) (a B+3 A b)+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**3/x + B*b**3*x**3/3 + a**2*(3*A*b + B*a)*log(x) + 3*a*b*x*(A*b + B*a) + b*
*2*(A*b + 3*B*a)*Integral(x, x)

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Mathematica [A]  time = 0.042144, size = 67, normalized size = 1.03 \[ -\frac{a^3 A}{x}+\log (x) \left (a^3 B+3 a^2 A b\right )+\frac{1}{2} b^2 x^2 (3 a B+A b)+3 a b x (a B+A b)+\frac{1}{3} b^3 B x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 71, normalized size = 1.1 \[{\frac{{b}^{3}B{x}^{3}}{3}}+{\frac{A{x}^{2}{b}^{3}}{2}}+{\frac{3\,B{x}^{2}a{b}^{2}}{2}}+3\,Axa{b}^{2}+3\,Bx{a}^{2}b+3\,A\ln \left ( x \right ){a}^{2}b+B\ln \left ( x \right ){a}^{3}-{\frac{A{a}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.34158, size = 93, normalized size = 1.43 \[ \frac{1}{3} \, B b^{3} x^{3} - \frac{A a^{3}}{x} + \frac{1}{2} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} x +{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201134, size = 101, normalized size = 1.55 \[ \frac{2 \, B b^{3} x^{4} - 6 \, A a^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \left (x\right )}{6 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^2,x, algorithm="fricas")

[Out]

1/6*(2*B*b^3*x^4 - 6*A*a^3 + 3*(3*B*a*b^2 + A*b^3)*x^3 + 18*(B*a^2*b + A*a*b^2)*
x^2 + 6*(B*a^3 + 3*A*a^2*b)*x*log(x))/x

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Sympy [A]  time = 1.58298, size = 70, normalized size = 1.08 \[ - \frac{A a^{3}}{x} + \frac{B b^{3} x^{3}}{3} + a^{2} \left (3 A b + B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{3}}{2} + \frac{3 B a b^{2}}{2}\right ) + x \left (3 A a b^{2} + 3 B a^{2} b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28016, size = 96, normalized size = 1.48 \[ \frac{1}{3} \, B b^{3} x^{3} + \frac{3}{2} \, B a b^{2} x^{2} + \frac{1}{2} \, A b^{3} x^{2} + 3 \, B a^{2} b x + 3 \, A a b^{2} x - \frac{A a^{3}}{x} +{\left (B a^{3} + 3 \, A a^{2} b\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^2,x, algorithm="giac")

[Out]

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