3.4.22 \(\int \frac {(f x)^m (a+b \log (c x^n))}{d+e x^2} \, dx\) [322]

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*ln(c*x^n))/(e*x^2+d),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 = {} \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. \(108\) vs. \(2(28)=56\).
time = 0.15, size = 108, normalized size = 3.86 \begin {gather*} \frac {x (f x)^m \left (-b n \, _3F_2\left (1,\frac {1}{2}+\frac {m}{2},\frac {1}{2}+\frac {m}{2};\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};-\frac {e x^2}{d}\right )+(1+m) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{d (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[{1, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 + m/2}, -((e*x^2)/d)]) + (1 + m
)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d*(1 + m)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

[Out]

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