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

Optimal. Leaf size=108 \[ -\frac{2 b d \log (F) (d e-c f) \text{Unintegrable}\left (\frac{F^{a+b (c+d x)^2}}{e+f x},x\right )}{f^2}-\frac{F^{a+b (c+d x)^2}}{f (e+f x)}+\frac{\sqrt{\pi } \sqrt{b} d F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{f^2} \]

[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 = 0.156402, 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 = {} \[ \int \frac{F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]

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

Rubi steps

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

Mathematica [A]  time = 0.747803, 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 = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{2}}}{ \left ( fx+e \right ) ^{2}}}\, dx \end{align*}

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. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)/(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. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)/(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. \begin{align*} \int \frac{F^{a + b \left (c + d x\right )^{2}}}{\left (e + f x\right )^{2}}\, dx \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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