3.216 \(\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx\)

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {(a+b \log (c (e+f x)))^p}{(h+i x)^3 (d e+d f x)},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^3,x)

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Rubi [A]  time = 0.13, 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 = {} \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx \]

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)^3),x]

[Out]

Defer[Int][(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^3), x]

Rubi steps

\begin {align*} \int \frac {(a+b \log (c (e+f x)))^p}{(h+216 x)^3 (d e+d f x)} \, dx &=\int \frac {(a+b \log (c (e+f x)))^p}{(h+216 x)^3 (d e+d f x)} \, dx\\ \end {align*}

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Mathematica [A]  time = 3.85, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx \]

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)^3),x]

[Out]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^3), x]

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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c f x + c e\right ) + a\right )}^{p}}{d f i^{3} x^{4} + d e h^{3} + {\left (3 \, d f h i^{2} + d e i^{3}\right )} x^{3} + 3 \, {\left (d f h^{2} i + d e h i^{2}\right )} x^{2} + {\left (d f h^{3} + 3 \, d e h^{2} i\right )} x}, x\right ) \]

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)^3,x, algorithm="fricas")

[Out]

integral((b*log(c*f*x + c*e) + a)^p/(d*f*i^3*x^4 + d*e*h^3 + (3*d*f*h*i^2 + d*e*i^3)*x^3 + 3*(d*f*h^2*i + d*e*
h*i^2)*x^2 + (d*f*h^3 + 3*d*e*h^2*i)*x), x)

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

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)^3,x, algorithm="giac")

[Out]

integrate((b*log((f*x + e)*c) + a)^p/((d*f*x + d*e)*(i*x + h)^3), x)

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maple [A]  time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\left (f x +e \right ) c \right )+a \right )^{p}}{\left (d f x +d e \right ) \left (i x +h \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln((f*x+e)*c)+a)^p/(d*f*x+d*e)/(i*x+h)^3,x)

[Out]

int((b*ln((f*x+e)*c)+a)^p/(d*f*x+d*e)/(i*x+h)^3,x)

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

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)^3,x, algorithm="maxima")

[Out]

integrate((b*log((f*x + e)*c) + a)^p/((d*f*x + d*e)*(i*x + h)^3), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{{\left (h+i\,x\right )}^3\,\left (d\,e+d\,f\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(e + f*x)))^p/((h + i*x)^3*(d*e + d*f*x)),x)

[Out]

int((a + b*log(c*(e + f*x)))^p/((h + i*x)^3*(d*e + d*f*x)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**3,x)

[Out]

Timed out

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