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*(d*x+c)^2)/f/(f*x+e)+d*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*b^(1/2)*Pi^(1/2)*ln(F)^(1/2)/f^2-2*b*d*(-
c*f+d*e)*ln(F)*Unintegrable(F^(a+b*(d*x+c)^2)/(f*x+e),x)/f^2

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Rubi [A]  time = 0.16, 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 = {} \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]

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*}

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Mathematica [A]  time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {F^{a+b (c+d x)^2}}{(e+f x)^2} \, dx \]

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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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(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)

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{a +\left (d x +c \right )^{2} b}}{\left (f x +e \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{a+b\,{\left (c+d\,x\right )}^2}}{{\left (e+f\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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