3.2.67 \(\int \frac {(f x)^m (a+b \log (c x^n))}{(d+e x)^2} \, dx\) [167]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, 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 {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).
time = 0.06, size = 72, normalized size = 2.77 \begin {gather*} \frac {x (f x)^m \left (-b n \, _3F_2\left (2,1+m,1+m;2+m,2+m;-\frac {e x}{d}\right )+(1+m) \, _2F_1\left (2,1+m;2+m;-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d^2 (1+m)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1 + m, 1 + m}, {2 + m, 2 + m}, -((e*x)/d)]) + (1 + m)*Hypergeometric2F
1[2, 1 + m, 2 + m, -((e*x)/d)]*(a + b*Log[c*x^n])))/(d^2*(1 + m)^2)

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Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e x +d \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x)^m*(a+b*ln(c*x^n))/(e*x+d)^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((f*x)^m*(a+b*log(c*x^n))/(e*x+d)^2,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)*(f*x)^m/(x*e + 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((f*x)^m*(a+b*log(c*x^n))/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((f*x)**m*(a + b*log(c*x**n))/(d + e*x)**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((f*x)^m*(a+b*log(c*x^n))/(e*x+d)^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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