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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

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*}

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Mathematica [A]  time = 5.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \]

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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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{3} g^{2} x^{2} + 2 \, a^{3} f g x + a^{3} f^{2} + {\left (b^{3} g^{2} x^{2} + 2 \, b^{3} f g x + b^{3} f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{3} + 3 \, {\left (a b^{2} g^{2} x^{2} + 2 \, a b^{2} f g x + a b^{2} f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, {\left (a^{2} b g^{2} x^{2} + 2 \, a^{2} b f g x + a^{2} b f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}\,{d x} \]

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)

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maple [A]  time = 9.46, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (g x +f \right )^{2} \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a e^{2} g - {\left (e^{2} g n - e^{2} g \log \relax (c)\right )} b\right )} x^{2} - {\left (d e f - 2 \, d^{2} g\right )} a - {\left (d e f n + {\left (d e f - 2 \, d^{2} g\right )} \log \relax (c)\right )} b - {\left ({\left (e^{2} f - 3 \, d e g\right )} a + {\left (e^{2} f n + d e g n + {\left (e^{2} f - 3 \, d e g\right )} \log \relax (c)\right )} b\right )} x + {\left (b e^{2} g x^{2} - {\left (e^{2} f - 3 \, d e g\right )} b x - {\left (d e f - 2 \, d^{2} g\right )} b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{2 \, {\left (b^{4} e^{2} f^{3} n^{2} \log \relax (c)^{2} + 2 \, a b^{3} e^{2} f^{3} n^{2} \log \relax (c) + a^{2} b^{2} e^{2} f^{3} n^{2} + {\left (b^{4} e^{2} g^{3} n^{2} \log \relax (c)^{2} + 2 \, a b^{3} e^{2} g^{3} n^{2} \log \relax (c) + a^{2} b^{2} e^{2} g^{3} n^{2}\right )} x^{3} + 3 \, {\left (b^{4} e^{2} f g^{2} n^{2} \log \relax (c)^{2} + 2 \, a b^{3} e^{2} f g^{2} n^{2} \log \relax (c) + a^{2} b^{2} e^{2} f g^{2} n^{2}\right )} x^{2} + {\left (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}\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 3 \, {\left (b^{4} e^{2} f^{2} g n^{2} \log \relax (c)^{2} + 2 \, a b^{3} e^{2} f^{2} g n^{2} \log \relax (c) + a^{2} b^{2} e^{2} f^{2} g n^{2}\right )} x + 2 \, {\left (b^{4} e^{2} f^{3} n^{2} \log \relax (c) + a b^{3} e^{2} f^{3} n^{2} + {\left (b^{4} e^{2} g^{3} n^{2} \log \relax (c) + a b^{3} e^{2} g^{3} n^{2}\right )} x^{3} + 3 \, {\left (b^{4} e^{2} f g^{2} n^{2} \log \relax (c) + a b^{3} e^{2} f g^{2} n^{2}\right )} x^{2} + 3 \, {\left (b^{4} e^{2} f^{2} g n^{2} \log \relax (c) + a b^{3} e^{2} f^{2} g n^{2}\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}} + \int \frac {e^{2} g^{2} x^{2} + e^{2} f^{2} - 6 \, d e f g + 6 \, d^{2} g^{2} - 2 \, {\left (2 \, e^{2} f g - 3 \, d e g^{2}\right )} x}{2 \, {\left (b^{3} e^{2} f^{4} n^{2} \log \relax (c) + a b^{2} e^{2} f^{4} n^{2} + {\left (b^{3} e^{2} g^{4} n^{2} \log \relax (c) + a b^{2} e^{2} g^{4} n^{2}\right )} x^{4} + 4 \, {\left (b^{3} e^{2} f g^{3} n^{2} \log \relax (c) + a b^{2} e^{2} f g^{3} n^{2}\right )} x^{3} + 6 \, {\left (b^{3} e^{2} f^{2} g^{2} n^{2} \log \relax (c) + a b^{2} e^{2} f^{2} g^{2} n^{2}\right )} x^{2} + 4 \, {\left (b^{3} e^{2} f^{3} g n^{2} \log \relax (c) + a b^{2} e^{2} f^{3} g n^{2}\right )} x + {\left (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}\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (f+g\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (f + g x\right )^{2}}\, dx \]

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]

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

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