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

Optimal. Leaf size=33 \[ \text {Int}\left (\frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.96, size = 0, normalized size = 0.00 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {h x + g}}{a h^{2} x^{2} + 2 \, a g h x + a g^{2} + {\left (b h^{2} x^{2} + 2 \, b g h x + b g^{2}\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (h x +g \right )^{\frac {3}{2}} \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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