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

Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.122677, 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 = {} \[ \int \frac{1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.454745, size = 0, normalized size = 0. \[ \int \frac{1}{(g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.676, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{hx+g} \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (h x + g\right )}{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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