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

Optimal. Leaf size=119 \[ -\frac{(f x)^{-(q+1) r} \left (d+e x^r\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d f (q+1) r}-\frac{b n (f x)^{-(q+1) r} \left (d+e x^r\right )^q \left (\frac{e x^r}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac{e x^r}{d}\right )}{f (q+1)^2 r^2} \]

[Out]

-((b*n*(d + e*x^r)^q*Hypergeometric2F1[-1 - q, -1 - q, -q, -((e*x^r)/d)])/(f*(1 + q)^2*r^2*(f*x)^((1 + q)*r)*(
1 + (e*x^r)/d)^q)) - ((d + e*x^r)^(1 + q)*(a + b*Log[c*x^n]))/(d*f*(1 + q)*r*(f*x)^((1 + q)*r))

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Rubi [A]  time = 0.13262, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {2335, 365, 364} \[ -\frac{(f x)^{-(q+1) r} \left (d+e x^r\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d f (q+1) r}-\frac{b n (f x)^{-(q+1) r} \left (d+e x^r\right )^q \left (\frac{e x^r}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac{e x^r}{d}\right )}{f (q+1)^2 r^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*n*(d + e*x^r)^q*Hypergeometric2F1[-1 - q, -1 - q, -q, -((e*x^r)/d)])/(f*(1 + q)^2*r^2*(f*x)^((1 + q)*r)*(
1 + (e*x^r)/d)^q)) - ((d + e*x^r)^(1 + q)*(a + b*Log[c*x^n]))/(d*f*(1 + q)*r*(f*x)^((1 + q)*r))

Rule 2335

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

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

Rubi steps

\begin{align*} \int (f x)^{-1-(1+q) r} \left (d+e x^r\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{(f x)^{-(1+q) r} \left (d+e x^r\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d f (1+q) r}+\frac{(b n) \int (f x)^{-1-(1+q) r} \left (d+e x^r\right )^{1+q} \, dx}{d (1+q) r}\\ &=-\frac{(f x)^{-(1+q) r} \left (d+e x^r\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d f (1+q) r}+\frac{\left (b n \left (d+e x^r\right )^q \left (1+\frac{e x^r}{d}\right )^{-q}\right ) \int (f x)^{-1-(1+q) r} \left (1+\frac{e x^r}{d}\right )^{1+q} \, dx}{(1+q) r}\\ &=-\frac{b n (f x)^{-(1+q) r} \left (d+e x^r\right )^q \left (1+\frac{e x^r}{d}\right )^{-q} \, _2F_1\left (-1-q,-1-q;-q;-\frac{e x^r}{d}\right )}{f (1+q)^2 r^2}-\frac{(f x)^{-(1+q) r} \left (d+e x^r\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d f (1+q) r}\\ \end{align*}

Mathematica [A]  time = 0.340544, size = 98, normalized size = 0.82 \[ -\frac{(f x)^{-(q+1) r} \left (d+e x^r\right )^q \left (\frac{(q+1) r \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+b n \left (\frac{e x^r}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac{e x^r}{d}\right )\right )}{f (q+1)^2 r^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d + e*x^r)^q*((b*n*Hypergeometric2F1[-1 - q, -1 - q, -q, -((e*x^r)/d)])/(1 + (e*x^r)/d)^q + ((1 + q)*r*(d
+ e*x^r)*(a + b*Log[c*x^n]))/d))/(f*(1 + q)^2*r^2*(f*x)^((1 + q)*r)))

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Maple [F]  time = 0.779, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1- \left ( 1+q \right ) r} \left ( d+e{x}^{r} \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((f*x)^(-1-(1+q)*r)*(d+e*x^r)^q*(a+b*ln(c*x^n)),x)

[Out]

int((f*x)^(-1-(1+q)*r)*(d+e*x^r)^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 (e x^{r} + d\right )}^{q} \left (f x\right )^{-{\left (q + 1\right )} r - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((f*x)^(-(q + 1)*r - 1)*b*log(c*x^n) + (f*x)^(-(q + 1)*r - 1)*a)*(e*x^r + d)^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((f*x)**(-1-(1+q)*r)*(d+e*x**r)**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 (e x^{r} + d\right )}^{q} \left (f x\right )^{-{\left (q + 1\right )} r - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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