3.36 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.11984, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{A b^3}{x}+b^2 \log (x) (3 A c+b B)+\frac{1}{2} c^2 x^2 (A c+3 b B)+3 b c x (A c+b B)+\frac{1}{3} B c^3 x^3 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0647985, size = 67, normalized size = 1.03 \[ -\frac{A b^3}{x}+\log (x) \left (3 A b^2 c+b^3 B\right )+\frac{1}{2} c^2 x^2 (A c+3 b B)+3 b c x (A c+b B)+\frac{1}{3} B c^3 x^3 \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^5,x]

[Out]

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

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Maple [A]  time = 0.009, size = 71, normalized size = 1.1 \[{\frac{B{c}^{3}{x}^{3}}{3}}+{\frac{A{x}^{2}{c}^{3}}{2}}+{\frac{3\,B{x}^{2}b{c}^{2}}{2}}+3\,Axb{c}^{2}+3\,Bx{b}^{2}c+3\,A\ln \left ( x \right ){b}^{2}c+B\ln \left ( x \right ){b}^{3}-{\frac{A{b}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^5,x)

[Out]

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

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Maxima [A]  time = 0.700581, size = 93, normalized size = 1.43 \[ \frac{1}{3} \, B c^{3} x^{3} - \frac{A b^{3}}{x} + \frac{1}{2} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{2} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} x +{\left (B b^{3} + 3 \, A b^{2} c\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.284685, size = 101, normalized size = 1.55 \[ \frac{2 \, B c^{3} x^{4} - 6 \, A b^{3} + 3 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 6 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x \log \left (x\right )}{6 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.51782, size = 70, normalized size = 1.08 \[ - \frac{A b^{3}}{x} + \frac{B c^{3} x^{3}}{3} + b^{2} \left (3 A c + B b\right ) \log{\left (x \right )} + x^{2} \left (\frac{A c^{3}}{2} + \frac{3 B b c^{2}}{2}\right ) + x \left (3 A b c^{2} + 3 B b^{2} c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.270629, size = 96, normalized size = 1.48 \[ \frac{1}{3} \, B c^{3} x^{3} + \frac{3}{2} \, B b c^{2} x^{2} + \frac{1}{2} \, A c^{3} x^{2} + 3 \, B b^{2} c x + 3 \, A b c^{2} x - \frac{A b^{3}}{x} +{\left (B b^{3} + 3 \, A b^{2} c\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^5,x, algorithm="giac")

[Out]

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