3.17 \(\int \frac{(a m x^m+b n q \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))^2}{x} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{3} \left (a x^m+b \log ^q\left (c x^n\right )\right )^3 \]

[Out]

(a*x^m + b*Log[c*x^n]^q)^3/3

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Rubi [A]  time = 0.172719, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {2544} \[ \frac{1}{3} \left (a x^m+b \log ^q\left (c x^n\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[((a*m*x^m + b*n*q*Log[c*x^n]^(-1 + q))*(a*x^m + b*Log[c*x^n]^q)^2)/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^3/3

Rule 2544

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] /; FreeQ[{a, b, c,
d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]

Rubi steps

\begin{align*} \int \frac{\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^2}{x} \, dx &=\frac{1}{3} \left (a x^m+b \log ^q\left (c x^n\right )\right )^3\\ \end{align*}

Mathematica [A]  time = 0.0381848, size = 22, normalized size = 1. \[ \frac{1}{3} \left (a x^m+b \log ^q\left (c x^n\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

Integrate[((a*m*x^m + b*n*q*Log[c*x^n]^(-1 + q))*(a*x^m + b*Log[c*x^n]^q)^2)/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^3/3

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Maple [C]  time = 0.197, size = 204, normalized size = 9.3 \begin{align*}{\frac{{a}^{3} \left ({x}^{m} \right ) ^{3}}{3}}+{\frac{{b}^{3} \left ( \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q} \right ) ^{3}}{3}}+a{b}^{2}{x}^{m} \left ( \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q} \right ) ^{2}+{a}^{2}b \left ({x}^{m} \right ) ^{2} \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*m*x^m+b*n*q*ln(c*x^n)^(-1+q))*(a*x^m+b*ln(c*x^n)^q)^2/x,x)

[Out]

1/3*a^3*(x^m)^3+1/3*b^3*((ln(c)+ln(x^n)-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*c*x^n)+csgn(I*c))*(-csgn(I*c*x^n)+csgn
(I*x^n)))^q)^3+a*b^2*x^m*((ln(c)+ln(x^n)-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*c*x^n)+csgn(I*c))*(-csgn(I*c*x^n)+csg
n(I*x^n)))^q)^2+a^2*b*(x^m)^2*(ln(c)+ln(x^n)-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*c*x^n)+csgn(I*c))*(-csgn(I*c*x^n)
+csgn(I*x^n)))^q

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^2/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.05207, size = 174, normalized size = 7.91 \begin{align*}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} a^{2} b x^{2 \, m} +{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{2 \, q} a b^{2} x^{m} + \frac{1}{3} \,{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{3 \, q} b^{3} + \frac{1}{3} \, a^{3} x^{3 \, m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^2/x,x, algorithm="fricas")

[Out]

(n*log(x) + log(c))^q*a^2*b*x^(2*m) + (n*log(x) + log(c))^(2*q)*a*b^2*x^m + 1/3*(n*log(x) + log(c))^(3*q)*b^3
+ 1/3*a^3*x^(3*m)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x**m+b*n*q*ln(c*x**n)**(-1+q))*(a*x**m+b*ln(c*x**n)**q)**2/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}\right )}{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^2/x,x, algorithm="giac")

[Out]

integrate((b*n*q*log(c*x^n)^(q - 1) + a*m*x^m)*(a*x^m + b*log(c*x^n)^q)^2/x, x)