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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0552597, 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 = {} \[ \int \frac{(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.43132, size = 0, normalized size = 0. \[ \int \frac{(g+h x)^m}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.649, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( hx+g \right ) ^{m}}{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(f*x + e)*(h*x + g)^m/(b^2*f*p*q*log(((f*x + e)^p)^q) + a*b*f*p*q + (f*p*q*log(c) + f*p*q*log(d^q))*b^2) + in
tegrate((f*h*(m + 1)*x + e*h*m + f*g)*(h*x + g)^m/(a*b*f*g*p*q + (f*g*p*q*log(c) + f*g*p*q*log(d^q))*b^2 + (a*
b*f*h*p*q + (f*h*p*q*log(c) + f*h*p*q*log(d^q))*b^2)*x + (b^2*f*h*p*q*x + b^2*f*g*p*q)*log(((f*x + e)^p)^q)),
x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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