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

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 111, normalized size = 3.96 \[ \frac {x (f x)^m \left ((m+1) \left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac {m+1}{r};\frac {m+r+1}{r};-\frac {e x^r}{d}\right )-b n \, _3F_2\left (1,\frac {m}{r}+\frac {1}{r},\frac {m}{r}+\frac {1}{r};\frac {m}{r}+\frac {1}{r}+1,\frac {m}{r}+\frac {1}{r}+1;-\frac {e x^r}{d}\right )\right )}{d (m+1)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (f x\right )^{m} b \log \left (c x^{n}\right ) + \left (f x\right )^{m} a}{e x^{r} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \left (f x \right )^{m}}{e \,x^{r}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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