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

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

[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 = 0.0360875, 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, Rules used = {76} \[ -\frac{a^2 (a B+3 A b)}{2 x^2}-\frac{a^3 A}{3 x^3}+b^2 \log (x) (3 a B+A b)-\frac{3 a b (a B+A b)}{x}+b^3 B x \]

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]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(a+b x)^3 (A+B x)}{x^4} \, dx &=\int \left (b^3 B+\frac{a^3 A}{x^4}+\frac{a^2 (3 A b+a B)}{x^3}+\frac{3 a b (A b+a B)}{x^2}+\frac{b^2 (A b+3 a B)}{x}\right ) \, dx\\ &=-\frac{a^3 A}{3 x^3}-\frac{a^2 (3 A b+a B)}{2 x^2}-\frac{3 a b (A b+a B)}{x}+b^3 B x+b^2 (A b+3 a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0316233, size = 67, normalized size = 1.05 \[ b^2 \log (x) (3 a B+A b)-\frac{9 a^2 b x (A+2 B x)+a^3 (2 A+3 B x)+18 a A b^2 x^2-6 b^3 B x^4}{6 x^3} \]

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 = 0.005, size = 72, normalized size = 1.1 \begin{align*}{b}^{3}Bx+A\ln \left ( x \right ){b}^{3}+3\,B\ln \left ( x \right ) a{b}^{2}-{\frac{A{a}^{3}}{3\,{x}^{3}}}-{\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}} \end{align*}

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^2-1/3*a^3*A/x^3-3/2*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

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Maxima [A]  time = 1.00529, size = 93, normalized size = 1.45 \begin{align*} 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/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 = 1.70937, size = 166, normalized size = 2.59 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/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 = 0.907381, size = 70, normalized size = 1.09 \begin{align*} 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}} \end{align*}

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 [A]  time = 1.18388, size = 95, normalized size = 1.48 \begin{align*} B b^{3} x +{\left (3 \, B a b^{2} + A b^{3}\right )} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b^3*x + (3*B*a*b^2 + A*b^3)*log(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