3.5.46 \(\int (d+e x^{-\frac {1}{1+q}})^q (a+b \log (c x^n)) \, dx\) [446]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 102, normalized size of antiderivative = 1.00, 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 = {2351, 252, 251} \begin {gather*} \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 ) \end {gather*}

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 251

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])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[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 2351

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]

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*}

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Mathematica [A]
time = 0.43, size = 143, normalized size = 1.40 \begin {gather*} \frac {x^{-\frac {1}{1+q}} \left (d+e x^{-\frac {1}{1+q}}\right )^q \left (1+\frac {d x^{\frac {1}{1+q}}}{e}\right )^{-q} \left (-b d n (1+q)^2 x^{\frac {2+q}{1+q}} \, _3F_2\left (1,1,-q;2,2;-\frac {d x^{\frac {1}{1+q}}}{e}\right )-b e n x \log (x)+\left (1+\frac {d x^{\frac {1}{1+q}}}{e}\right )^q \left (e x+d x^{\frac {2+q}{1+q}}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d} \end {gather*}

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 = 0.09, size = 0, normalized size = 0.00 \[\int \left (d +e \,x^{-\frac {1}{1+q}}\right )^{q} \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d+\frac {e}{x^{\frac {1}{q+1}}}\right )}^q\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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