3.38 \(\int \frac{(A+B x) (b x+c x^2)^3}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.039532, antiderivative size = 64, 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} \[ -\frac{b^2 (3 A c+b B)}{2 x^2}-\frac{A b^3}{3 x^3}+c^2 \log (x) (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+B c^3 x \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x^7,x]

[Out]

-(A*b^3)/(3*x^3) - (b^2*(b*B + 3*A*c))/(2*x^2) - (3*b*c*(b*B + A*c))/x + B*c^3*x + c^2*(3*b*B + A*c)*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^7} \, dx &=\int \left (B c^3+\frac{A b^3}{x^4}+\frac{b^2 (b B+3 A c)}{x^3}+\frac{3 b c (b B+A c)}{x^2}+\frac{c^2 (3 b B+A c)}{x}\right ) \, dx\\ &=-\frac{A b^3}{3 x^3}-\frac{b^2 (b B+3 A c)}{2 x^2}-\frac{3 b c (b B+A c)}{x}+B c^3 x+c^2 (3 b B+A c) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0335994, size = 73, normalized size = 1.14 \[ -\frac{3 \left (A b c^2+b^2 B c\right )}{x}+\frac{b^3 (-B)-3 A b^2 c}{2 x^2}-\frac{A b^3}{3 x^3}+\log (x) \left (A c^3+3 b B c^2\right )+B c^3 x \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^7,x]

[Out]

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

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Maple [A]  time = 0.007, size = 72, normalized size = 1.1 \begin{align*} B{c}^{3}x+A\ln \left ( x \right ){c}^{3}+3\,B\ln \left ( x \right ) b{c}^{2}-{\frac{A{b}^{3}}{3\,{x}^{3}}}-{\frac{3\,A{b}^{2}c}{2\,{x}^{2}}}-{\frac{{b}^{3}B}{2\,{x}^{2}}}-3\,{\frac{Ab{c}^{2}}{x}}-3\,{\frac{B{b}^{2}c}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/x^7,x)

[Out]

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

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Maxima [A]  time = 1.14988, size = 93, normalized size = 1.45 \begin{align*} B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x\right ) - \frac{2 \, A b^{3} + 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.78562, size = 166, normalized size = 2.59 \begin{align*} \frac{6 \, B c^{3} x^{4} + 6 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} \log \left (x\right ) - 2 \, A b^{3} - 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} - 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.07059, size = 70, normalized size = 1.09 \begin{align*} B c^{3} x + c^{2} \left (A c + 3 B b\right ) \log{\left (x \right )} - \frac{2 A b^{3} + x^{2} \left (18 A b c^{2} + 18 B b^{2} c\right ) + x \left (9 A b^{2} c + 3 B b^{3}\right )}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**7,x)

[Out]

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

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Giac [A]  time = 1.15526, size = 95, normalized size = 1.48 \begin{align*} B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, A b^{3} + 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^7,x, algorithm="giac")

[Out]

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