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=53 \[ \text {Int}\left (\frac {1}{\left (x (d g+e f)+d f+e g x^2\right ) \left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*F^(c*(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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Rubi [A]  time = 0.15, 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 = {} \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^2 \left (d f+(e f+d g) x+e g x^2\right )} \, dx \]

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*}

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Mathematica [A]  time = 1.27, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}\right )^2 \left (d f+(e f+d g) x+e g x^2\right )} \, dx \]

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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fricas [A]  time = 13.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} e g x^{2} + a^{2} d f + {\left (b^{2} e g x^{2} + b^{2} d f + {\left (b^{2} e f + b^{2} d g\right )} x\right )} F^{\frac {2 \, \sqrt {e x + d} c}{\sqrt {g x + f}}} + 2 \, {\left (a b e g x^{2} + a b d f + {\left (a b e f + a b d g\right )} x\right )} F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} + {\left (a^{2} e f + a^{2} d g\right )} x}, x\right ) \]

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]

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

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

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)

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maple [A]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,F^{\frac {\sqrt {e x +d}\, c}{\sqrt {g x +f}}}+a \right )^{2} \left (e g \,x^{2}+d f +\left (d g +e f \right ) x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, \sqrt {g x + f}}{{\left (e f - d g\right )} \sqrt {e x + d} F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} a b c \log \relax (F) + {\left (e f - d g\right )} \sqrt {e x + d} a^{2} c \log \relax (F)} + \int \frac {\sqrt {e x + d} c \log \relax (F) + \sqrt {g x + f}}{{\left (a b c e g x^{2} \log \relax (F) + a b c d f \log \relax (F) + {\left (e f + d g\right )} a b c x \log \relax (F)\right )} \sqrt {e x + d} F^{\frac {\sqrt {e x + d} c}{\sqrt {g x + f}}} + {\left (a^{2} c e g x^{2} \log \relax (F) + a^{2} c d f \log \relax (F) + {\left (e f + d g\right )} a^{2} c x \log \relax (F)\right )} \sqrt {e x + d}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a+F^{\frac {c\,\sqrt {d+e\,x}}{\sqrt {f+g\,x}}}\,b\right )}^2\,\left (e\,g\,x^2+\left (d\,g+e\,f\right )\,x+d\,f\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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]

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

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