3.409 \(\int \frac {a+b \log (c x^n)}{x^3 (d+e x^r)} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*x^n))/x^3/(d+e*x^r),x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)),x]

[Out]

Defer[Int][(a + b*Log[c*x^n])/(x^3*(d + e*x^r)), x]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx &=\int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 86, normalized size = 3.31 \[ -\frac {b n \, _3F_2\left (1,-\frac {2}{r},-\frac {2}{r};1-\frac {2}{r},1-\frac {2}{r};-\frac {e x^r}{d}\right )+2 \, _2F_1\left (1,-\frac {2}{r};\frac {r-2}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)),x]

[Out]

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

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{3} x^{r} + d x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(d+e*x^r),x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e*x^3*x^r + d*x^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(d+e*x^r),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)*x^3), x)

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maple [A]  time = 0.76, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \,x^{r}+d \right ) x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)/x^3/(e*x^r+d),x)

[Out]

int((b*ln(c*x^n)+a)/x^3/(e*x^r+d),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(d+e*x^r),x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)*x^3), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\left (d+e\,x^r\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(x^3*(d + e*x^r)),x)

[Out]

int((a + b*log(c*x^n))/(x^3*(d + e*x^r)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \log {\left (c x^{n} \right )}}{x^{3} \left (d + e x^{r}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**3/(d+e*x**r),x)

[Out]

Integral((a + b*log(c*x**n))/(x**3*(d + e*x**r)), x)

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