3.81 \(\int \frac{(a+b x)^3 (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=64 \[ -\frac{a^3 A}{3 x^3}-\frac{a^2 (a B+3 A b)}{2 x^2}+b^2 \log (x) (3 a B+A b)-\frac{3 a b (a B+A b)}{x}+b^3 B x \]

[Out]

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

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Rubi [A]  time = 0.101765, antiderivative size = 64, 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 \[ -\frac{a^3 A}{3 x^3}-\frac{a^2 (a B+3 A b)}{2 x^2}+b^2 \log (x) (3 a B+A b)-\frac{3 a b (a B+A b)}{x}+b^3 B x \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.060854, size = 67, normalized size = 1.05 \[ b^2 \log (x) (3 a B+A b)-\frac{a^3 (2 A+3 B x)+9 a^2 b x (A+2 B x)+18 a A b^2 x^2-6 b^3 B x^4}{6 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 72, normalized size = 1.1 \[{b}^{3}Bx+A\ln \left ( x \right ){b}^{3}+3\,B\ln \left ( x \right ) a{b}^{2}-{\frac{3\,{a}^{2}bA}{2\,{x}^{2}}}-{\frac{{a}^{3}B}{2\,{x}^{2}}}-3\,{\frac{a{b}^{2}A}{x}}-3\,{\frac{{a}^{2}bB}{x}}-{\frac{A{a}^{3}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.34797, size = 93, normalized size = 1.45 \[ B b^{3} x +{\left (3 \, B a b^{2} + A b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206592, size = 101, normalized size = 1.58 \[ \frac{6 \, B b^{3} x^{4} + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} \log \left (x\right ) - 2 \, A a^{3} - 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.39135, size = 70, normalized size = 1.09 \[ B b^{3} x + b^{2} \left (A b + 3 B a\right ) \log{\left (x \right )} - \frac{2 A a^{3} + x^{2} \left (18 A a b^{2} + 18 B a^{2} b\right ) + x \left (9 A a^{2} b + 3 B a^{3}\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**3*x + b**2*(A*b + 3*B*a)*log(x) - (2*A*a**3 + x**2*(18*A*a*b**2 + 18*B*a**2
*b) + x*(9*A*a**2*b + 3*B*a**3))/(6*x**3)

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GIAC/XCAS [A]  time = 0.293465, size = 95, normalized size = 1.48 \[ B b^{3} x +{\left (3 \, B a b^{2} + A b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A a^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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