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

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 = 0.05, 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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx \]

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]  time = 0.11, size = 72, normalized size = 2.77 \[ \frac {x (f x)^m \left ((m+1) \, _2F_1\left (2,m+1;m+2;-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (2,m+1,m+1;m+2,m+2;-\frac {e x}{d}\right )\right )}{d^2 (m+1)^2} \]

Warning: Unable to verify antiderivative.

[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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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (f x\right )^{m} b \log \left (c x^{n}\right ) + \left (f x\right )^{m} a}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]

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)/(e^2*x^2 + 2*d*e*x + d^2), x)

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

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/(e*x + d)^2, x)

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maple [A]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \left (f x \right )^{m}}{\left (e x +d \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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/(e*x + d)^2, x)

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

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

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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