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

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{1}{(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0424962, 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 = {} \[ \int \frac{1}{(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

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*}

Mathematica [A]  time = 1.3744, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

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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Maple [A]  time = 0.668, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) } \left ( gx+f \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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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 ) \left (f + g x\right )^{\frac{3}{2}}}\, dx \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)