### 3.18 $$\int \frac{(a m x^m+b n q \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))}{x} \, dx$$

Optimal. Leaf size=22 $\frac{1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2$

[Out]

(a*x^m + b*Log[c*x^n]^q)^2/2

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Rubi [A]  time = 0.109131, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.024, Rules used = {2544} $\frac{1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2$

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))/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^2/2

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

Mathematica [A]  time = 0.032603, size = 22, normalized size = 1. $\frac{1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2$

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))/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^2/2

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Maple [C]  time = 0.162, size = 135, normalized size = 6.1 \begin{align*}{\frac{{a}^{2} \left ({x}^{m} \right ) ^{2}}{2}}+{\frac{{b}^{2} \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}}{2}}+ab{x}^{m} \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)/x,x)

[Out]

1/2*a^2*(x^m)^2+1/2*b^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)^2+a*b*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)+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)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.95908, size = 115, normalized size = 5.23 \begin{align*}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} a b x^{m} + \frac{1}{2} \,{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{2 \, q} b^{2} + \frac{1}{2} \, a^{2} x^{2 \, 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)/x,x, algorithm="fricas")

[Out]

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