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

Optimal. Leaf size=109 \[ -\frac{2 b d \log (F) (d e-c f) \text{Int}\left (\frac{F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2}-\frac{F^{a+b (c+d x)^2}}{f (e+f x)}+\frac{\sqrt{\pi } \sqrt{b} d F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{f^2} \]

[Out]

-(F^(a + b*(c + d*x)^2)/(f*(e + f*x))) + (Sqrt[b]*d*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c
+ d*x)*Sqrt[Log[F]]]*Sqrt[Log[F]])/f^2 - (2*b*d*(d*e - c*f)*Log[F]*Unintegrable
[F^(a + b*(c + d*x)^2)/(e + f*x), x])/f^2

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Rubi [A]  time = 0.249194, 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. \[ \text{Int}\left (\frac{F^{a+b (c+d x)^2}}{(e+f x)^2},x\right ) \]

Verification is Not applicable to the result.

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

[Out]

-(F^(a + b*(c + d*x)^2)/(f*(e + f*x))) + (Sqrt[b]*d*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c
+ d*x)*Sqrt[Log[F]]]*Sqrt[Log[F]])/f^2 - (2*b*d*(d*e - c*f)*Log[F]*Defer[Int][F
^(a + b*(c + d*x)^2)/(e + f*x), x])/f^2

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{\pi } F^{a} \sqrt{b} d \sqrt{\log{\left (F \right )}} \operatorname{erfi}{\left (\sqrt{b} \left (c + d x\right ) \sqrt{\log{\left (F \right )}} \right )}}{f^{2}} - \frac{F^{a + b \left (c + d x\right )^{2}}}{f \left (e + f x\right )} + \frac{2 b d \left (c f - d e\right ) \log{\left (F \right )} \int \frac{F^{a + b \left (c + d x\right )^{2}}}{e + f x}\, dx}{f^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)/(f*x+e)**2,x)

[Out]

sqrt(pi)*F**a*sqrt(b)*d*sqrt(log(F))*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/f**2 -
F**(a + b*(c + d*x)**2)/(f*(e + f*x)) + 2*b*d*(c*f - d*e)*log(F)*Integral(F**(a
+ b*(c + d*x)**2)/(e + f*x), x)/f**2

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Mathematica [A]  time = 0.939131, size = 0, normalized size = 0. \[ \int \frac{F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{2}}}{ \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2)/(f*x+e)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^2*b + a)/(f*x + e)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c)**2)/(f*x+e)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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