### 3.16 $$\int \frac{(a m x^m+b n q \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))^p}{x} \, dx$$

Optimal. Leaf size=26 $\frac{\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1}$

[Out]

(a*x^m + b*Log[c*x^n]^q)^(1 + p)/(1 + p)

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Rubi [A]  time = 0.171346, antiderivative size = 26, 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} $\frac{\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1}$

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)^p)/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^(1 + p)/(1 + p)

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

Mathematica [A]  time = 0.0893974, size = 26, normalized size = 1. $\frac{\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1}$

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)^p)/x,x]

[Out]

(a*x^m + b*Log[c*x^n]^q)^(1 + p)/(1 + p)

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Maple [F]  time = 0.5, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( am{x}^{m}+bnq \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q} \right ) \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) ^{p}}{x}}\, dx \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)^p/x,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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)^p/x,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.99164, size = 107, normalized size = 4.12 \begin{align*} \frac{{\left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right )}{\left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right )}^{p}}{p + 1} \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)^p/x,x, algorithm="fricas")

[Out]

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

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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)**p/x,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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)^p/x,x, algorithm="giac")

[Out]

Exception raised: RuntimeError