3.1.69 \(\int \frac {F^{a+b (c+d x)} (e+f x)^2}{x} \, dx\) [69]

Optimal. Leaf size=96 \[ e^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {f^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x}{b d \log (F)} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 96, normalized size of antiderivative = 1.00, 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 = {2230, 2225, 2209, 2207} \begin {gather*} -\frac {f^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+e^2 F^{a+b c} \text {Ei}(b d x \log (F))+\frac {2 e f F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x F^{a+b c+b d x}}{b d \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2209

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

Rule 2225

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 2230

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] &&  !TrueQ[$UseGamma]

Rubi steps

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

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Mathematica [A]
time = 0.23, size = 54, normalized size = 0.56 \begin {gather*} F^{a+b c} \left (e^2 \text {Ei}(b d x \log (F))+\frac {f F^{b d x} (-f+b d (2 e+f x) \log (F))}{b^2 d^2 \log ^2(F)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.08, size = 118, normalized size = 1.23

method	result	size
meijerg	\(\frac {F^{c b +a} f^{2} \left (1-\frac {\left (-2 b d x \ln \left (F \right )+2\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{2}\right )}{b^{2} d^{2} \ln \left (F \right )^{2}}-\frac {2 F^{c b +a} f e \left (1-{\mathrm e}^{b d x \ln \left (F \right )}\right )}{b d \ln \left (F \right )}+F^{c b +a} e^{2} \left (-\ln \left (-b d x \ln \left (F \right )\right )-\expIntegral \left (1, -b d x \ln \left (F \right )\right )+\ln \left (x \right )+\ln \left (-b d \right )+\ln \left (\ln \left (F \right )\right )\right )\)	\(118\)
risch	\(-e^{2} F^{c b} F^{a} \expIntegral \left (1, c b \ln \left (F \right )+\ln \left (F \right ) a -b d x \ln \left (F \right )-\left (c b +a \right ) \ln \left (F \right )\right )+\frac {f^{2} F^{b d x} F^{c b +a} x}{d b \ln \left (F \right )}-\frac {f^{2} F^{b d x} F^{c b +a}}{d^{2} b^{2} \ln \left (F \right )^{2}}+\frac {2 e f \,F^{b d x} F^{c b +a}}{d b \ln \left (F \right )}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 87, normalized size = 0.91 \begin {gather*} F^{b c + a} {\rm Ei}\left (b d x \log \left (F\right )\right ) e^{2} + \frac {2 \, F^{b d x + b c + a} f e}{b d \log \left (F\right )} + \frac {{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} F^{b d x} f^{2}}{b^{2} d^{2} \log \left (F\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 75, normalized size = 0.78 \begin {gather*} \frac {F^{b c + a} b^{2} d^{2} {\rm Ei}\left (b d x \log \left (F\right )\right ) e^{2} \log \left (F\right )^{2} - {\left (f^{2} - {\left (b d f^{2} x + 2 \, b d f e\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{2} d^{2} \log \left (F\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} \left (e + f x\right )^{2}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 3.58, size = 80, normalized size = 0.83 \begin {gather*} \frac {F^{a+b\,c}\,\left (b^2\,d^2\,e^2\,\mathrm {ei}\left (b\,d\,x\,\ln \left (F\right )\right )\,{\ln \left (F\right )}^2-F^{b\,d\,x}\,f^2+F^{b\,d\,x}\,b\,d\,f^2\,x\,\ln \left (F\right )+2\,F^{b\,d\,x}\,b\,d\,e\,f\,\ln \left (F\right )\right )}{b^2\,d^2\,{\ln \left (F\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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