3.35 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^4} \, dx\)

Optimal. Leaf size=54 \[ A b^3 \log (x)+3 A b^2 c x+\frac{3}{2} A b c^2 x^2+\frac{1}{3} A c^3 x^3+\frac{B (b+c x)^4}{4 c} \]

[Out]

3*A*b^2*c*x + (3*A*b*c^2*x^2)/2 + (A*c^3*x^3)/3 + (B*(b + c*x)^4)/(4*c) + A*b^3*
Log[x]

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Rubi [A]  time = 0.0668134, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ A b^3 \log (x)+3 A b^2 c x+\frac{3}{2} A b c^2 x^2+\frac{1}{3} A c^3 x^3+\frac{B (b+c x)^4}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^4,x]

[Out]

3*A*b^2*c*x + (3*A*b*c^2*x^2)/2 + (A*c^3*x^3)/3 + (B*(b + c*x)^4)/(4*c) + A*b^3*
Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A b^{3} \log{\left (x \right )} + 3 A b^{2} c x + 3 A b c^{2} \int x\, dx + \frac{A c^{3} x^{3}}{3} + \frac{B \left (b + c x\right )^{4}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**4,x)

[Out]

A*b**3*log(x) + 3*A*b**2*c*x + 3*A*b*c**2*Integral(x, x) + A*c**3*x**3/3 + B*(b
+ c*x)**4/(4*c)

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Mathematica [A]  time = 0.0519115, size = 63, normalized size = 1.17 \[ A b^3 \log (x)+\frac{1}{12} x \left (18 b^2 c (2 A+B x)+6 b c^2 x (3 A+2 B x)+c^3 x^2 (4 A+3 B x)+12 b^3 B\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^4,x]

[Out]

(x*(12*b^3*B + 18*b^2*c*(2*A + B*x) + 6*b*c^2*x*(3*A + 2*B*x) + c^3*x^2*(4*A + 3
*B*x)))/12 + A*b^3*Log[x]

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Maple [A]  time = 0.004, size = 70, normalized size = 1.3 \[{\frac{B{c}^{3}{x}^{4}}{4}}+{\frac{A{c}^{3}{x}^{3}}{3}}+B{x}^{3}b{c}^{2}+{\frac{3\,Ab{c}^{2}{x}^{2}}{2}}+{\frac{3\,B{x}^{2}{b}^{2}c}{2}}+3\,A{b}^{2}cx+Bx{b}^{3}+A{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^4,x)

[Out]

1/4*B*c^3*x^4+1/3*A*c^3*x^3+B*x^3*b*c^2+3/2*A*b*c^2*x^2+3/2*B*x^2*b^2*c+3*A*b^2*
c*x+B*x*b^3+A*b^3*ln(x)

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Maxima [A]  time = 0.698485, size = 92, normalized size = 1.7 \[ \frac{1}{4} \, B c^{3} x^{4} + A b^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} +{\left (B b^{3} + 3 \, A b^{2} c\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.267989, size = 92, normalized size = 1.7 \[ \frac{1}{4} \, B c^{3} x^{4} + A b^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} +{\left (B b^{3} + 3 \, A b^{2} c\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.32454, size = 73, normalized size = 1.35 \[ A b^{3} \log{\left (x \right )} + \frac{B c^{3} x^{4}}{4} + x^{3} \left (\frac{A c^{3}}{3} + B b c^{2}\right ) + x^{2} \left (\frac{3 A b c^{2}}{2} + \frac{3 B b^{2} c}{2}\right ) + x \left (3 A b^{2} c + B b^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**4,x)

[Out]

A*b**3*log(x) + B*c**3*x**4/4 + x**3*(A*c**3/3 + B*b*c**2) + x**2*(3*A*b*c**2/2
+ 3*B*b**2*c/2) + x*(3*A*b**2*c + B*b**3)

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GIAC/XCAS [A]  time = 0.268423, size = 95, normalized size = 1.76 \[ \frac{1}{4} \, B c^{3} x^{4} + B b c^{2} x^{3} + \frac{1}{3} \, A c^{3} x^{3} + \frac{3}{2} \, B b^{2} c x^{2} + \frac{3}{2} \, A b c^{2} x^{2} + B b^{3} x + 3 \, A b^{2} c x + A b^{3}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^4,x, algorithm="giac")

[Out]

1/4*B*c^3*x^4 + B*b*c^2*x^3 + 1/3*A*c^3*x^3 + 3/2*B*b^2*c*x^2 + 3/2*A*b*c^2*x^2
+ B*b^3*x + 3*A*b^2*c*x + A*b^3*ln(abs(x))