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

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {1}{(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {1}{(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {g x + f}}{a g^{2} x^{2} + 2 \, a f g x + a f^{2} + {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(g*x + f)/(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 + 2*b*f*g*x + 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 \[ \int \frac {1}{{\left (g x + f\right )}^{\frac {3}{2}} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (g x +f \right )^{\frac {3}{2}} \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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