3.39 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^8} \, dx\)

Optimal. Leaf size=59 \[ -\frac{A (b+c x)^4}{4 b x^4}-\frac{b^3 B}{3 x^3}-\frac{3 b^2 B c}{2 x^2}-\frac{3 b B c^2}{x}+B c^3 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0671478, antiderivative size = 59, 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 \[ -\frac{A (b+c x)^4}{4 b x^4}-\frac{b^3 B}{3 x^3}-\frac{3 b^2 B c}{2 x^2}-\frac{3 b B c^2}{x}+B c^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7718, size = 56, normalized size = 0.95 \[ - \frac{A \left (b + c x\right )^{4}}{4 b x^{4}} - \frac{B b^{3}}{3 x^{3}} - \frac{3 B b^{2} c}{2 x^{2}} - \frac{3 B b c^{2}}{x} + B c^{3} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.0660893, size = 71, normalized size = 1.2 \[ B c^3 \log (x)-\frac{3 A \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )+2 b B x \left (2 b^2+9 b c x+18 c^2 x^2\right )}{12 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^8,x]

[Out]

-(2*b*B*x*(2*b^2 + 9*b*c*x + 18*c^2*x^2) + 3*A*(b^3 + 4*b^2*c*x + 6*b*c^2*x^2 +
4*c^3*x^3))/(12*x^4) + B*c^3*Log[x]

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Maple [A]  time = 0.009, size = 76, normalized size = 1.3 \[ B{c}^{3}\ln \left ( x \right ) -{\frac{A{b}^{3}}{4\,{x}^{4}}}-{\frac{A{b}^{2}c}{{x}^{3}}}-{\frac{B{b}^{3}}{3\,{x}^{3}}}-{\frac{3\,Ab{c}^{2}}{2\,{x}^{2}}}-{\frac{3\,B{b}^{2}c}{2\,{x}^{2}}}-{\frac{A{c}^{3}}{x}}-3\,{\frac{Bb{c}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^8,x)

[Out]

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

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Maxima [A]  time = 0.70031, size = 97, normalized size = 1.64 \[ B c^{3} \log \left (x\right ) - \frac{3 \, A b^{3} + 12 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^8,x, algorithm="maxima")

[Out]

B*c^3*log(x) - 1/12*(3*A*b^3 + 12*(3*B*b*c^2 + A*c^3)*x^3 + 18*(B*b^2*c + A*b*c^
2)*x^2 + 4*(B*b^3 + 3*A*b^2*c)*x)/x^4

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Fricas [A]  time = 0.265271, size = 101, normalized size = 1.71 \[ \frac{12 \, B c^{3} x^{4} \log \left (x\right ) - 3 \, A b^{3} - 12 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} - 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} - 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^8,x, algorithm="fricas")

[Out]

1/12*(12*B*c^3*x^4*log(x) - 3*A*b^3 - 12*(3*B*b*c^2 + A*c^3)*x^3 - 18*(B*b^2*c +
A*b*c^2)*x^2 - 4*(B*b^3 + 3*A*b^2*c)*x)/x^4

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Sympy [A]  time = 4.48803, size = 75, normalized size = 1.27 \[ B c^{3} \log{\left (x \right )} - \frac{3 A b^{3} + x^{3} \left (12 A c^{3} + 36 B b c^{2}\right ) + x^{2} \left (18 A b c^{2} + 18 B b^{2} c\right ) + x \left (12 A b^{2} c + 4 B b^{3}\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**8,x)

[Out]

B*c**3*log(x) - (3*A*b**3 + x**3*(12*A*c**3 + 36*B*b*c**2) + x**2*(18*A*b*c**2 +
18*B*b**2*c) + x*(12*A*b**2*c + 4*B*b**3))/(12*x**4)

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GIAC/XCAS [A]  time = 0.271532, size = 99, normalized size = 1.68 \[ B c^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, A b^{3} + 12 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^8,x, algorithm="giac")

[Out]

B*c^3*ln(abs(x)) - 1/12*(3*A*b^3 + 12*(3*B*b*c^2 + A*c^3)*x^3 + 18*(B*b^2*c + A*
b*c^2)*x^2 + 4*(B*b^3 + 3*A*b^2*c)*x)/x^4