3.103 \(\int \frac{1}{(f+g x) (a+b \log (c (d+e x)^n))^3} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3},x\right ) \]

[Out]

Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3), x]

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Rubi [A]  time = 0.03413, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Defer[Int][1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3), x]

Rubi steps

\begin{align*} \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=\int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx\\ \end{align*}

Mathematica [A]  time = 1.21911, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Integrate[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3), x]

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Maple [A]  time = 3.516, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^3,x)

[Out]

int(1/(g*x+f)/(a+b*ln(c*(e*x+d)^n))^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

-1/2*(b*e^2*g*n*x^2 + (d*e*f - d^2*g)*a + (d*e*f*n + (d*e*f - d^2*g)*log(c))*b + ((e^2*f - d*e*g)*a + (e^2*f*n
+ d*e*g*n + (e^2*f - d*e*g)*log(c))*b)*x + ((e^2*f - d*e*g)*b*x + (d*e*f - d^2*g)*b)*log((e*x + d)^n))/(b^4*e
^2*f^2*n^2*log(c)^2 + 2*a*b^3*e^2*f^2*n^2*log(c) + a^2*b^2*e^2*f^2*n^2 + (b^4*e^2*g^2*n^2*log(c)^2 + 2*a*b^3*e
^2*g^2*n^2*log(c) + a^2*b^2*e^2*g^2*n^2)*x^2 + (b^4*e^2*g^2*n^2*x^2 + 2*b^4*e^2*f*g*n^2*x + b^4*e^2*f^2*n^2)*l
og((e*x + d)^n)^2 + 2*(b^4*e^2*f*g*n^2*log(c)^2 + 2*a*b^3*e^2*f*g*n^2*log(c) + a^2*b^2*e^2*f*g*n^2)*x + 2*(b^4
*e^2*f^2*n^2*log(c) + a*b^3*e^2*f^2*n^2 + (b^4*e^2*g^2*n^2*log(c) + a*b^3*e^2*g^2*n^2)*x^2 + 2*(b^4*e^2*f*g*n^
2*log(c) + a*b^3*e^2*f*g*n^2)*x)*log((e*x + d)^n)) + integrate(1/2*(e^2*f^2 - 3*d*e*f*g + 2*d^2*g^2 - (e^2*f*g
- d*e*g^2)*x)/(b^3*e^2*f^3*n^2*log(c) + a*b^2*e^2*f^3*n^2 + (b^3*e^2*g^3*n^2*log(c) + a*b^2*e^2*g^3*n^2)*x^3
+ 3*(b^3*e^2*f*g^2*n^2*log(c) + a*b^2*e^2*f*g^2*n^2)*x^2 + 3*(b^3*e^2*f^2*g*n^2*log(c) + a*b^2*e^2*f^2*g*n^2)*
x + (b^3*e^2*g^3*n^2*x^3 + 3*b^3*e^2*f*g^2*n^2*x^2 + 3*b^3*e^2*f^2*g*n^2*x + b^3*e^2*f^3*n^2)*log((e*x + d)^n)
), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{3} g x + a^{3} f +{\left (b^{3} g x + b^{3} f\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \,{\left (a b^{2} g x + a b^{2} f\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \,{\left (a^{2} b g x + a^{2} b f\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

integral(1/(a^3*g*x + a^3*f + (b^3*g*x + b^3*f)*log((e*x + d)^n*c)^3 + 3*(a*b^2*g*x + a*b^2*f)*log((e*x + d)^n
*c)^2 + 3*(a^2*b*g*x + a^2*b*f)*log((e*x + d)^n*c)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (f + g x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Integral(1/((a + b*log(c*(d + e*x)**n))**3*(f + g*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

integrate(1/((g*x + f)*(b*log((e*x + d)^n*c) + a)^3), x)