3.5.14 \(\int \frac {x (a+b \log (c x^n))}{(d+e x^r)^2} \, dx\) [414]

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

Unintegrable(x*(a+b*ln(c*x^n))/(d+e*x^r)^2,x)

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Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

Defer[Int][(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x]

Rubi steps

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

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(24)=48\).
time = 0.15, size = 140, normalized size = 5.83 \begin {gather*} \frac {x^2 \left (-b n (-2+r) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {2}{r},\frac {2}{r};1+\frac {2}{r},1+\frac {2}{r};-\frac {e x^r}{d}\right )+4 d \left (a+b \log \left (c x^n\right )\right )+2 \left (d+e x^r\right ) \, _2F_1\left (1,\frac {2}{r};\frac {2+r}{r};-\frac {e x^r}{d}\right ) \left (-b n+a (-2+r)+b (-2+r) \log \left (c x^n\right )\right )\right )}{4 d^2 r \left (d+e x^r\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

(x^2*(-(b*n*(-2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 + 2/r}, -((e*x^r)/d)]) + 4*d*(a
+ b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-2 + r) + b*(-
2 + r)*Log[c*x^n])))/(4*d^2*r*(d + e*x^r))

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (d +e \,x^{r}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*ln(c*x^n))/(d+e*x^r)^2,x)

[Out]

int(x*(a+b*ln(c*x^n))/(d+e*x^r)^2,x)

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Maxima [A]
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(x*(a+b*log(c*x^n))/(d+e*x^r)^2,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)*x/(e*x^r + d)^2, x)

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Fricas [A]
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(x*(a+b*log(c*x^n))/(d+e*x^r)^2,x, algorithm="fricas")

[Out]

integral((b*x*log(c*x^n) + a*x)/(2*d*x^r*e + d^2 + x^(2*r)*e^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{r}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*ln(c*x**n))/(d+e*x**r)**2,x)

[Out]

Integral(x*(a + b*log(c*x**n))/(d + e*x**r)**2, x)

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Giac [A]
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(x*(a+b*log(c*x^n))/(d+e*x^r)^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x/(x^r*e + d)^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^r\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*log(c*x^n)))/(d + e*x^r)^2,x)

[Out]

int((x*(a + b*log(c*x^n)))/(d + e*x^r)^2, x)

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