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

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

[Out]

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

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Rubi [A]  time = 0.149229, 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 = {} \[ \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.663491, size = 0, normalized size = 0. \[ \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{f+g x}}}\right )^n}{d f+(e f+d g) x+e g x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.092, 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 ) ^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((F^(sqrt(e*x + d)*c/sqrt(g*x + f))*b + a)^n/(e*g*x^2 + d*f + (e*f + d*g)*x), 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((a+b*F^(c*(e*x+d)^(1/2)/(g*x+f)^(1/2)))^n/(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((a+b*F**(c*(e*x+d)**(1/2)/(g*x+f)**(1/2)))**n/(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{{\left (F^{\frac{\sqrt{e x + d} c}{\sqrt{g x + f}}} b + a\right )}^{n}}{e g x^{2} + d f +{\left (e f + d g\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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