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

Optimal. Leaf size=53 \[ \frac {(c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{2 d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2214, 2210} \[ \frac {(c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{2 d}-\frac {b F^a \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2210

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]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 47, normalized size = 0.89 \[ \frac {F^a \left ((c+d x)^2 F^{\frac {b}{(c+d x)^2}}-b \log (F) \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 96, normalized size = 1.81 \[ -\frac {F^{a} b {\rm Ei}\left (\frac {b \log \relax (F)}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \relax (F) - {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, d} \]

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

________________________________________________________________________________________

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

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((d*x + c)*F^(a + b/(d*x + c)^2), x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 86, normalized size = 1.62 \[ \frac {d \,x^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2}+c x \,F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}+\frac {b \,F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{\left (d x +c \right )^{2}}\right ) \ln \relax (F )}{2 d}+\frac {c^{2} F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^2*b)*(d*x+c),x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (F^{a} d x^{2} + 2 \, F^{a} c x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac {{\left (F^{a} b d^{2} x^{2} \log \relax (F) + 2 \, F^{a} b c d x \log \relax (F)\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} \]

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]

1/2*(F^a*d*x^2 + 2*F^a*c*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2)) + integrate((F^a*b*d^2*x^2*log(F) + 2*F^a*b*c*d*x*
log(F))*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

________________________________________________________________________________________

mupad [B]  time = 5.45, size = 51, normalized size = 0.96 \[ \frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^2}{2\,d}+\frac {F^a\,b\,\ln \relax (F)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^2}\right )}{2\,d} \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________