3.3.15 \(\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx\) [215]

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^2,x)

________________________________________________________________________________________

Rubi [A]
time = 0.08, 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)^2} \, 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)^2),x]

[Out]

Defer[Int][(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^2), x]

Rubi steps

\begin {align*} \int \frac {(a+b \log (c (e+f x)))^p}{(h+215 x)^2 (d e+d f x)} \, dx &=\int \frac {(a+b \log (c (e+f x)))^p}{(h+215 x)^2 (d e+d f x)} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.04, 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)^2} \, 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)^2),x]

[Out]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^2), x]

________________________________________________________________________________________

Maple [A]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{\left (d f x +e d \right ) \left (i x +h \right )^{2}}\, dx\]

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)^2,x)

[Out]

int((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^2,x)

________________________________________________________________________________________

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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[0,1,0,0,0,0]%%%} / %%%{1,[0,0,1,1,0,0]%%%}+%%%{i,[0,0
,1,0,1,1]%%

________________________________________________________________________________________

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 )}^2\,\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)^2*(d*e + d*f*x)),x)

[Out]

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

________________________________________________________________________________________