3.104 \(\int \frac{1}{(f+g x)^2 (a+b \log (c (d+e x)^n))^3} \, dx\)

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3},x\right ) \]

[Out]

Unintegrable[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3), x]

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Rubi [A]  time = 0.0336433, 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 = {} \[ \int \frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Defer[Int][1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3), x]

Rubi steps

\begin{align*} \int \frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=\int \frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx\\ \end{align*}

Mathematica [A]  time = 4.84387, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3),x]

[Out]

Integrate[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3), x]

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Maple [A]  time = 8.084, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{2} \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)^2/(a+b*ln(c*(e*x+d)^n))^3,x)

[Out]

int(1/(g*x+f)^2/(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)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

1/2*((a*e^2*g - (e^2*g*n - e^2*g*log(c))*b)*x^2 - (d*e*f - 2*d^2*g)*a - (d*e*f*n + (d*e*f - 2*d^2*g)*log(c))*b
- ((e^2*f - 3*d*e*g)*a + (e^2*f*n + d*e*g*n + (e^2*f - 3*d*e*g)*log(c))*b)*x + (b*e^2*g*x^2 - (e^2*f - 3*d*e*
g)*b*x - (d*e*f - 2*d^2*g)*b)*log((e*x + d)^n))/(b^4*e^2*f^3*n^2*log(c)^2 + 2*a*b^3*e^2*f^3*n^2*log(c) + a^2*b
^2*e^2*f^3*n^2 + (b^4*e^2*g^3*n^2*log(c)^2 + 2*a*b^3*e^2*g^3*n^2*log(c) + a^2*b^2*e^2*g^3*n^2)*x^3 + 3*(b^4*e^
2*f*g^2*n^2*log(c)^2 + 2*a*b^3*e^2*f*g^2*n^2*log(c) + a^2*b^2*e^2*f*g^2*n^2)*x^2 + (b^4*e^2*g^3*n^2*x^3 + 3*b^
4*e^2*f*g^2*n^2*x^2 + 3*b^4*e^2*f^2*g*n^2*x + b^4*e^2*f^3*n^2)*log((e*x + d)^n)^2 + 3*(b^4*e^2*f^2*g*n^2*log(c
)^2 + 2*a*b^3*e^2*f^2*g*n^2*log(c) + a^2*b^2*e^2*f^2*g*n^2)*x + 2*(b^4*e^2*f^3*n^2*log(c) + a*b^3*e^2*f^3*n^2
+ (b^4*e^2*g^3*n^2*log(c) + a*b^3*e^2*g^3*n^2)*x^3 + 3*(b^4*e^2*f*g^2*n^2*log(c) + a*b^3*e^2*f*g^2*n^2)*x^2 +
3*(b^4*e^2*f^2*g*n^2*log(c) + a*b^3*e^2*f^2*g*n^2)*x)*log((e*x + d)^n)) + integrate(1/2*(e^2*g^2*x^2 + e^2*f^2
- 6*d*e*f*g + 6*d^2*g^2 - 2*(2*e^2*f*g - 3*d*e*g^2)*x)/(b^3*e^2*f^4*n^2*log(c) + a*b^2*e^2*f^4*n^2 + (b^3*e^2
*g^4*n^2*log(c) + a*b^2*e^2*g^4*n^2)*x^4 + 4*(b^3*e^2*f*g^3*n^2*log(c) + a*b^2*e^2*f*g^3*n^2)*x^3 + 6*(b^3*e^2
*f^2*g^2*n^2*log(c) + a*b^2*e^2*f^2*g^2*n^2)*x^2 + 4*(b^3*e^2*f^3*g*n^2*log(c) + a*b^2*e^2*f^3*g*n^2)*x + (b^3
*e^2*g^4*n^2*x^4 + 4*b^3*e^2*f*g^3*n^2*x^3 + 6*b^3*e^2*f^2*g^2*n^2*x^2 + 4*b^3*e^2*f^3*g*n^2*x + b^3*e^2*f^4*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^{2} x^{2} + 2 \, a^{3} f g x + a^{3} f^{2} +{\left (b^{3} g^{2} x^{2} + 2 \, b^{3} f g x + b^{3} f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \,{\left (a b^{2} g^{2} x^{2} + 2 \, a b^{2} f g x + a b^{2} f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \,{\left (a^{2} b g^{2} x^{2} + 2 \, a^{2} b f g x + a^{2} b f^{2}\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)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)**2/(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}^{2}{\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)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

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