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

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

[Out]

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

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Rubi [A]  time = 0.152223, 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 = {} \[ \int \frac{1}{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right ) \left (d f+(e f+d g) x+e g x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))*(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]))*(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 ) \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 ) \left (d f+(e f+d g) x+e g x^2\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.378046, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right ) \left (d f+(e f+d g) x+e g x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

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Maple [A]  time = 0.076, 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 ) ^{-1}}\, 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)))/(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)))/(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*} \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 )}}\,{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)))/(d*f+(d*g+e*f)*x+e*g*x^2),x, algorithm="maxima")

[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)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a e g x^{2} + a d f +{\left (b e g x^{2} + b d f +{\left (b e f + b d g\right )} x\right )} F^{\frac{\sqrt{e x + d} c}{\sqrt{g x + f}}} +{\left (a e f + a d g\right )} x}, x\right ) \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)))/(d*f+(d*g+e*f)*x+e*g*x^2),x, algorithm="fricas")

[Out]

integral(1/(a*e*g*x^2 + a*d*f + (b*e*g*x^2 + b*d*f + (b*e*f + b*d*g)*x)*F^(sqrt(e*x + d)*c/sqrt(g*x + f)) + (a
*e*f + a*d*g)*x), 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(1/(a+b*F**(c*(e*x+d)**(1/2)/(g*x+f)**(1/2)))/(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 )}}\,{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)))/(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)), x)