3.3.14 \(\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx\) [214]

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x)

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Rubi [A]
time = 0.09, 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 (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)),x]

[Out]

Defer[Int][(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)), x]

Rubi steps

\begin {align*} \int \frac {(a+b \log (c (e+f x)))^p}{(h+214 x) (d e+d f x)} \, dx &=\int \frac {(a+b \log (c (e+f x)))^p}{(h+214 x) (d e+d f x)} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)),x]

[Out]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)), x]

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Maple [A]
time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{\left (d f x +e d \right ) \left (i x +h \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x)

[Out]

int((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),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*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x, algorithm="maxima")

[Out]

integrate((b*log((f*x + e)*c) + a)^p/((d*f*x + d*e)*(h + I*x)), 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*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x, algorithm="fricas")

[Out]

integral(-I*(b*log(c*f*x + c*e) + a)^p/(-I*d*f*h*x + d*f*x^2 + (-I*d*h + d*x)*e), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e h + e i x + f h x + f i x^{2}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(f*x+e)))**p/(d*f*x+d*e)/(i*x+h),x)

[Out]

Integral((a + b*log(c*e + c*f*x))**p/(e*h + e*i*x + f*h*x + f*i*x**2), x)/d

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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*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x, algorithm="giac")

[Out]

integrate((b*log((f*x + e)*c) + a)^p/((d*f*x + d*e)*(h + I*x)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{\left (h+i\,x\right )\,\left (d\,e+d\,f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(e + f*x)))^p/((h + i*x)*(d*e + d*f*x)),x)

[Out]

int((a + b*log(c*(e + f*x)))^p/((h + i*x)*(d*e + d*f*x)), x)

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