3.2.4 \(\int \frac {1}{(f+g x)^2 (a+b \log (c (d+e x)^n))^3} \, dx\) [104]

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 = 0.02, 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 {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \end {gather*}

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 = 3.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{3}}\, dx\]

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 = 0.00, 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(1/(g*x+f)^2/(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

1/2*(2*b*d^2*g*log(c) + 2*a*d^2*g - ((g*n - g*log(c))*b - a*g)*x^2*e^2 - (((f*n + f*log(c))*b + a*f)*e^2 - (3*
a*d*g - (d*g*n - 3*d*g*log(c))*b)*e)*x - (a*d*f + (d*f*n + d*f*log(c))*b)*e + (b*g*x^2*e^2 + 2*b*d^2*g - b*d*f
*e + (3*b*d*g*e - b*f*e^2)*x)*log((x*e + d)^n))/((b^4*g^3*n^2*log(c)^2 + 2*a*b^3*g^3*n^2*log(c) + a^2*b^2*g^3*
n^2)*x^3*e^2 + 3*(b^4*f*g^2*n^2*log(c)^2 + 2*a*b^3*f*g^2*n^2*log(c) + a^2*b^2*f*g^2*n^2)*x^2*e^2 + 3*(b^4*f^2*
g*n^2*log(c)^2 + 2*a*b^3*f^2*g*n^2*log(c) + a^2*b^2*f^2*g*n^2)*x*e^2 + (b^4*g^3*n^2*x^3*e^2 + 3*b^4*f*g^2*n^2*
x^2*e^2 + 3*b^4*f^2*g*n^2*x*e^2 + b^4*f^3*n^2*e^2)*log((x*e + d)^n)^2 + (b^4*f^3*n^2*log(c)^2 + 2*a*b^3*f^3*n^
2*log(c) + a^2*b^2*f^3*n^2)*e^2 + 2*((b^4*g^3*n^2*log(c) + a*b^3*g^3*n^2)*x^3*e^2 + 3*(b^4*f*g^2*n^2*log(c) +
a*b^3*f*g^2*n^2)*x^2*e^2 + 3*(b^4*f^2*g*n^2*log(c) + a*b^3*f^2*g*n^2)*x*e^2 + (b^4*f^3*n^2*log(c) + a*b^3*f^3*
n^2)*e^2)*log((x*e + d)^n)) + integrate(1/2*(g^2*x^2*e^2 + 6*d^2*g^2 - 6*d*f*g*e + f^2*e^2 + 2*(3*d*g^2*e - 2*
f*g*e^2)*x)/((b^3*g^4*n^2*log(c) + a*b^2*g^4*n^2)*x^4*e^2 + 4*(b^3*f*g^3*n^2*log(c) + a*b^2*f*g^3*n^2)*x^3*e^2
+ 6*(b^3*f^2*g^2*n^2*log(c) + a*b^2*f^2*g^2*n^2)*x^2*e^2 + 4*(b^3*f^3*g*n^2*log(c) + a*b^2*f^3*g*n^2)*x*e^2 +
(b^3*f^4*n^2*log(c) + a*b^2*f^4*n^2)*e^2 + (b^3*g^4*n^2*x^4*e^2 + 4*b^3*f*g^3*n^2*x^3*e^2 + 6*b^3*f^2*g^2*n^2
*x^2*e^2 + 4*b^3*f^3*g*n^2*x*e^2 + b^3*f^4*n^2*e^2)*log((x*e + d)^n)), x)

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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(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((x*e + 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((x*e + 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((x*e + d)^n*c)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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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Giac [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(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((x*e + d)^n*c) + a)^3), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \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 \end {gather*}

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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