3.549 \(\int \frac{1}{(a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}})^2 (d f+(e f+d g) x+e g x^2)} \, dx\)

Optimal. Leaf size=52 \[ \text{Unintegrable}\left (\frac{1}{\left (x (d g+e f)+d f+e g x^2\right ) \left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.148908, 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 = {} \[ \int \frac{1}{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right )^2 \left (d f+(e f+d g) x+e g x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.2715, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right )^2 \left (d f+(e f+d g) x+e g x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{df+ \left ( dg+fe \right ) x+eg{x}^{2}} \left ( a+b{F}^{{c\sqrt{ex+d}{\frac{1}{\sqrt{gx+f}}}}} \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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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(1/(a+b*F**(c*(e*x+d)**(1/2)/(g*x+f)**(1/2)))**2/(d*f+(d*g+e*f)*x+e*g*x**2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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