3.71 \(\int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx\)

Optimal. Leaf size=136 \[ \frac {1}{2} b^2 d^2 e^2 \log ^2(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {b d e^2 \log (F) F^{a+b c+b d x}}{2 x}+2 b d e f \log (F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F)) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.36, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2199, 2177, 2178} \[ \frac {1}{2} b^2 d^2 e^2 \log ^2(F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {b d e^2 \log (F) F^{a+b c+b d x}}{2 x}+2 b d e f \log (F) F^{a+b c} \text {Ei}(b d x \log (F))-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F)) \]

Antiderivative was successfully verified.

[In]

Int[(F^(a + b*(c + d*x))*(e + f*x)^2)/x^3,x]

[Out]

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

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

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

Rubi steps

\begin {align*} \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx &=\int \left (\frac {e^2 F^{a+b c+b d x}}{x^3}+\frac {2 e f F^{a+b c+b d x}}{x^2}+\frac {f^2 F^{a+b c+b d x}}{x}\right ) \, dx\\ &=e^2 \int \frac {F^{a+b c+b d x}}{x^3} \, dx+(2 e f) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+f^2 \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))+\frac {1}{2} \left (b d e^2 \log (F)\right ) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+(2 b d e f \log (F)) \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{2 x}+2 b d e f F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)+\frac {1}{2} \left (b^2 d^2 e^2 \log ^2(F)\right ) \int \frac {F^{a+b c+b d x}}{x} \, dx\\ &=-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{2 x}+2 b d e f F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)+\frac {1}{2} b^2 d^2 e^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log ^2(F)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 76, normalized size = 0.56 \[ \frac {F^{a+b c} \left (x^2 \left (b^2 d^2 e^2 \log ^2(F)+4 b d e f \log (F)+2 f^2\right ) \text {Ei}(b d x \log (F))-e F^{b d x} (b d e x \log (F)+e+4 f x)\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(F^(a + b*(c + d*x))*(e + f*x)^2)/x^3,x]

[Out]

(F^(a + b*c)*(-(e*F^(b*d*x)*(e + 4*f*x + b*d*e*x*Log[F])) + x^2*ExpIntegralEi[b*d*x*Log[F]]*(2*f^2 + 4*b*d*e*f
*Log[F] + b^2*d^2*e^2*Log[F]^2)))/(2*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 89, normalized size = 0.65 \[ \frac {{\left (b^{2} d^{2} e^{2} x^{2} \log \relax (F)^{2} + 4 \, b d e f x^{2} \log \relax (F) + 2 \, f^{2} x^{2}\right )} F^{b c + a} {\rm Ei}\left (b d x \log \relax (F)\right ) - {\left (b d e^{2} x \log \relax (F) + 4 \, e f x + e^{2}\right )} F^{b d x + b c + a}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*(f*x+e)^2/x^3,x, algorithm="fricas")

[Out]

1/2*((b^2*d^2*e^2*x^2*log(F)^2 + 4*b*d*e*f*x^2*log(F) + 2*f^2*x^2)*F^(b*c + a)*Ei(b*d*x*log(F)) - (b*d*e^2*x*l
og(F) + 4*e*f*x + e^2)*F^(b*d*x + b*c + a))/x^2

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*(f*x+e)^2/x^3,x, algorithm="giac")

[Out]

integrate((f*x + e)^2*F^((d*x + c)*b + a)/x^3, x)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 204, normalized size = 1.50 \[ -\frac {b^{2} d^{2} e^{2} F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right ) \ln \relax (F )^{2}}{2}-2 b d e f \,F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right ) \ln \relax (F )-\frac {b d \,e^{2} F^{b d x} F^{b c +a} \ln \relax (F )}{2 x}-f^{2} F^{a} F^{b c} \Ei \left (1, -b d x \ln \relax (F )+b c \ln \relax (F )+a \ln \relax (F )-\left (b c +a \right ) \ln \relax (F )\right )-\frac {2 e f \,F^{b d x} F^{b c +a}}{x}-\frac {e^{2} F^{b d x} F^{b c +a}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c))*(f*x+e)^2/x^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.23, size = 74, normalized size = 0.54 \[ -F^{b c + a} b^{2} d^{2} e^{2} \Gamma \left (-2, -b d x \log \relax (F)\right ) \log \relax (F)^{2} + 2 \, F^{b c + a} b d e f \Gamma \left (-1, -b d x \log \relax (F)\right ) \log \relax (F) + F^{b c + a} f^{2} {\rm Ei}\left (b d x \log \relax (F)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c))*(f*x+e)^2/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.60, size = 133, normalized size = 0.98 \[ F^{a+b\,c}\,f^2\,\mathrm {ei}\left (b\,d\,x\,\ln \relax (F)\right )-\frac {2\,F^{b\,d\,x}\,F^{a+b\,c}\,e\,f}{x}-F^{a+b\,c}\,b^2\,d^2\,e^2\,{\ln \relax (F)}^2\,\left (\frac {\mathrm {expint}\left (-b\,d\,x\,\ln \relax (F)\right )}{2}+F^{b\,d\,x}\,\left (\frac {1}{2\,b\,d\,x\,\ln \relax (F)}+\frac {1}{2\,b^2\,d^2\,x^2\,{\ln \relax (F)}^2}\right )\right )-2\,F^{a+b\,c}\,b\,d\,e\,f\,\ln \relax (F)\,\mathrm {expint}\left (-b\,d\,x\,\ln \relax (F)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((F^(a + b*(c + d*x))*(e + f*x)^2)/x^3,x)

[Out]

F^(a + b*c)*f^2*ei(b*d*x*log(F)) - (2*F^(b*d*x)*F^(a + b*c)*e*f)/x - F^(a + b*c)*b^2*d^2*e^2*log(F)^2*(expint(
-b*d*x*log(F))/2 + F^(b*d*x)*(1/(2*b*d*x*log(F)) + 1/(2*b^2*d^2*x^2*log(F)^2))) - 2*F^(a + b*c)*b*d*e*f*log(F)
*expint(-b*d*x*log(F))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c))*(f*x+e)**2/x**3,x)

[Out]

Integral(F**(a + b*(c + d*x))*(e + f*x)**2/x**3, x)

________________________________________________________________________________________