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

Optimal. Leaf size=66 \[ b^2 x (3 A c+b B)+A b^3 \log (x)+\frac{1}{3} c^2 x^3 (A c+3 b B)+\frac{3}{2} b c x^2 (A c+b B)+\frac{1}{4} B c^3 x^4 \]

[Out]

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

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Rubi [A]  time = 0.0414899, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ b^2 x (3 A c+b B)+A b^3 \log (x)+\frac{1}{3} c^2 x^3 (A c+3 b B)+\frac{3}{2} b c x^2 (A c+b B)+\frac{1}{4} B c^3 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^4} \, dx &=\int \left (b^2 (b B+3 A c)+\frac{A b^3}{x}+3 b c (b B+A c) x+c^2 (3 b B+A c) x^2+B c^3 x^3\right ) \, dx\\ &=b^2 (b B+3 A c) x+\frac{3}{2} b c (b B+A c) x^2+\frac{1}{3} c^2 (3 b B+A c) x^3+\frac{1}{4} B c^3 x^4+A b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0245278, size = 63, normalized size = 0.95 \[ \frac{1}{12} x \left (18 b^2 c (2 A+B x)+6 b c^2 x (3 A+2 B x)+c^3 x^2 (4 A+3 B x)+12 b^3 B\right )+A b^3 \log (x) \]

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 = 0.001, size = 70, normalized size = 1.1 \begin{align*}{\frac{B{c}^{3}{x}^{4}}{4}}+{\frac{A{x}^{3}{c}^{3}}{3}}+B{x}^{3}b{c}^{2}+{\frac{3\,A{x}^{2}b{c}^{2}}{2}}+{\frac{3\,B{x}^{2}{b}^{2}c}{2}}+3\,A{b}^{2}cx+{b}^{3}Bx+A{b}^{3}\ln \left ( x \right ) \end{align*}

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

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Maxima [A]  time = 1.0865, size = 92, normalized size = 1.39 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/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 = 1.77382, size = 154, normalized size = 2.33 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/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 = 0.414911, size = 73, normalized size = 1.11 \begin{align*} 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 ) \end{align*}

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 [A]  time = 1.12582, size = 95, normalized size = 1.44 \begin{align*} \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} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/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*log(abs(x))