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

Optimal. Leaf size=85 \[ b d e^2 \log (F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \text{Ei}(b d x \log (F))+\frac{f^2 F^{a+b c+b d x}}{b d \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.276619, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2199, 2194, 2177, 2178} \[ b d e^2 \log (F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \text{Ei}(b d x \log (F))+\frac{f^2 F^{a+b c+b d x}}{b d \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.149333, size = 58, normalized size = 0.68 \[ F^{a+b c} \left (F^{b d x} \left (\frac{f^2}{b d \log (F)}-\frac{e^2}{x}\right )+e (b d e \log (F)+2 f) \text{Ei}(b d x \log (F))\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.053, size = 135, normalized size = 1.6 \begin{align*} -{\frac{{e}^{2}{F}^{bdx}{F}^{bc+a}}{x}}+{\frac{{f}^{2}{F}^{bdx}{F}^{bc+a}}{bd\ln \left ( F \right ) }}-\ln \left ( F \right ) bd{e}^{2}{F}^{bc}{F}^{a}{\it Ei} \left ( 1,bc\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) - \left ( bc+a \right ) \ln \left ( F \right ) \right ) -2\,fe{F}^{bc}{F}^{a}{\it Ei} \left ( 1,bc\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) - \left ( bc+a \right ) \ln \left ( F \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.19312, size = 92, normalized size = 1.08 \begin{align*} F^{b c + a} b d e^{2} \Gamma \left (-1, -b d x \log \left (F\right )\right ) \log \left (F\right ) + 2 \, F^{b c + a} e f{\rm Ei}\left (b d x \log \left (F\right )\right ) + \frac{F^{b d x + b c + a} f^{2}}{b d \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(b*c + a)*b*d*e^2*gamma(-1, -b*d*x*log(F))*log(F) + 2*F^(b*c + a)*e*f*Ei(b*d*x*log(F)) + F^(b*d*x + b*c + a)
*f^2/(b*d*log(F))

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Fricas [A]  time = 1.49197, size = 189, normalized size = 2.22 \begin{align*} \frac{{\left (b^{2} d^{2} e^{2} x \log \left (F\right )^{2} + 2 \, b d e f x \log \left (F\right )\right )} F^{b c + a}{\rm Ei}\left (b d x \log \left (F\right )\right ) -{\left (b d e^{2} \log \left (F\right ) - f^{2} x\right )} F^{b d x + b c + a}}{b d x \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*d^2*e^2*x*log(F)^2 + 2*b*d*e*f*x*log(F))*F^(b*c + a)*Ei(b*d*x*log(F)) - (b*d*e^2*log(F) - f^2*x)*F^(b*d*
x + b*c + a))/(b*d*x*log(F))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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