3.5.16 \(\int \frac {a+b \log (c x^n)}{x^3 (d+e x^r)^2} \, dx\) [416]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*x^n))/x^3/(d+e*x^r)^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 {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2),x]

[Out]

Defer[Int][(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2), x]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx &=\int \frac {a+b \log \left (c x^n\right )}{x^3 \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. \(139\) vs. \(2(26)=52\).
time = 0.15, size = 139, normalized size = 5.35 \begin {gather*} -\frac {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 )}{4 d^2 r x^2 \left (d+e x^r\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2),x]

[Out]

-1/4*(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]))/(d^2*r*x^2*(d + e*x^r))

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{3} \left (d +e \,x^{r}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^3/(d+e*x^r)^2,x)

[Out]

int((a+b*ln(c*x^n))/x^3/(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((a+b*log(c*x^n))/x^3/(d+e*x^r)^2,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)^2*x^3), 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((a+b*log(c*x^n))/x^3/(d+e*x^r)^2,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(2*d*x^3*x^r*e + d^2*x^3 + x^3*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 {a + b \log {\left (c x^{n} \right )}}{x^{3} \left (d + e x^{r}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**3/(d+e*x**r)**2,x)

[Out]

Integral((a + b*log(c*x**n))/(x**3*(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((a+b*log(c*x^n))/x^3/(d+e*x^r)^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((x^r*e + d)^2*x^3), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (d+e\,x^r\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(x^3*(d + e*x^r)^2),x)

[Out]

int((a + b*log(c*x^n))/(x^3*(d + e*x^r)^2), x)

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