3.446 \(\int (d+e x^{-\frac{1}{1+q}})^q (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=102 \[ \frac{x \left (d+e x^{-\frac{1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}-b n x \left (d+e x^{-\frac{1}{q+1}}\right )^q \left (\frac{e x^{-\frac{1}{q+1}}}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac{e x^{-\frac{1}{q+1}}}{d}\right ) \]

[Out]

-((b*n*x*(d + e/x^(1 + q)^(-1))^q*Hypergeometric2F1[-1 - q, -1 - q, -q, -(e/(d*x^(1 + q)^(-1)))])/(1 + e/(d*x^
(1 + q)^(-1)))^q) + (x*(d + e/x^(1 + q)^(-1))^(1 + q)*(a + b*Log[c*x^n]))/d

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Rubi [A]  time = 0.0440655, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2314, 246, 245} \[ \frac{x \left (d+e x^{-\frac{1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}-b n x \left (d+e x^{-\frac{1}{q+1}}\right )^q \left (\frac{e x^{-\frac{1}{q+1}}}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac{e x^{-\frac{1}{q+1}}}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*n*x*(d + e/x^(1 + q)^(-1))^q*Hypergeometric2F1[-1 - q, -1 - q, -q, -(e/(d*x^(1 + q)^(-1)))])/(1 + e/(d*x^
(1 + q)^(-1)))^q) + (x*(d + e/x^(1 + q)^(-1))^(1 + q)*(a + b*Log[c*x^n]))/d

Rule 2314

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (d+e x^{-\frac{1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{x \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{(b n) \int \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \, dx}{d}\\ &=\frac{x \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}-\left (b n \left (d+e x^{-\frac{1}{1+q}}\right )^q \left (1+\frac{e x^{-\frac{1}{1+q}}}{d}\right )^{-q}\right ) \int \left (1+\frac{e x^{-\frac{1}{1+q}}}{d}\right )^{1+q} \, dx\\ &=-b n x \left (d+e x^{-\frac{1}{1+q}}\right )^q \left (1+\frac{e x^{-\frac{1}{1+q}}}{d}\right )^{-q} \, _2F_1\left (-1-q,-1-q;-q;-\frac{e x^{-\frac{1}{1+q}}}{d}\right )+\frac{x \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.571386, size = 143, normalized size = 1.4 \[ \frac{x^{-\frac{1}{q+1}} \left (d+e x^{-\frac{1}{q+1}}\right )^q \left (\frac{d x^{\frac{1}{q+1}}}{e}+1\right )^{-q} \left (-b d n (q+1)^2 x^{\frac{q+2}{q+1}} \, _3F_2\left (1,1,-q;2,2;-\frac{d x^{\frac{1}{q+1}}}{e}\right )+\left (d x^{\frac{q+2}{q+1}}+e x\right ) \left (\frac{d x^{\frac{1}{q+1}}}{e}+1\right )^q \left (a+b \log \left (c x^n\right )\right )-b e n x \log (x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e/x^(1 + q)^(-1))^q*(-(b*d*n*(1 + q)^2*x^((2 + q)/(1 + q))*HypergeometricPFQ[{1, 1, -q}, {2, 2}, -((d*x^
(1 + q)^(-1))/e)]) - b*e*n*x*Log[x] + (1 + (d*x^(1 + q)^(-1))/e)^q*(e*x + d*x^((2 + q)/(1 + q)))*(a + b*Log[c*
x^n])))/(d*x^(1 + q)^(-1)*(1 + (d*x^(1 + q)^(-1))/e)^q)

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Maple [F]  time = 0.829, size = 0, normalized size = 0. \begin{align*} \int \left ( d+{\frac{e}{{x}^{ \left ( 1+q \right ) ^{-1}}}} \right ) ^{q} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e/(x^(1/(1+q))))^q*(a+b*ln(c*x^n)),x)

[Out]

int((d+e/(x^(1/(1+q))))^q*(a+b*ln(c*x^n)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*(d + e/x^(1/(q + 1)))^q, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} \left (\frac{d x^{\left (\frac{1}{q + 1}\right )} + e}{x^{\left (\frac{1}{q + 1}\right )}}\right )^{q}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)*((d*x^(1/(q + 1)) + e)/x^(1/(q + 1)))^q, x)

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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((d+e/(x**(1/(1+q))))**q*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*(d + e/x^(1/(q + 1)))^q, x)