3.3.61 \(\int \frac {F^{a+b (c+d x)^2}}{c+d x} \, dx\) [261]

Optimal. Leaf size=22 \[ \frac {F^a \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{2 d} \]

[Out]

1/2*F^a*Ei(b*(d*x+c)^2*ln(F))/d

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Rubi [A]
time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2241} \begin {gather*} \frac {F^a \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^2*Log[F]])/(2*d)

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

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Mathematica [A]
time = 0.17, size = 22, normalized size = 1.00 \begin {gather*} \frac {F^a \text {Ei}\left (b (c+d x)^2 \log (F)\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^2*Log[F]])/(2*d)

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Maple [A]
time = 0.07, size = 23, normalized size = 1.05

method	result	size
risch	\(-\frac {F^{a} \expIntegral \left (1, -b \left (d x +c \right )^{2} \ln \left (F \right )\right )}{2 d}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^2)/(d*x+c),x,method=_RETURNVERBOSE)

[Out]

-1/2/d*F^a*Ei(1,-b*(d*x+c)^2*ln(F))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 32, normalized size = 1.45 \begin {gather*} \frac {F^{a} {\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*F^a*Ei((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a + b \left (c + d x\right )^{2}}}{c + d x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.68, size = 20, normalized size = 0.91 \begin {gather*} \frac {F^a\,\mathrm {ei}\left (b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2\right )}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^a*ei(b*log(F)*(c + d*x)^2))/(2*d)

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