Integrand size = 33, antiderivative size = 85 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e^{a+b x+c x^2}}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
-2/3*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(3/2)+4/3*erfi((c*x^2+b*x+a)^(1/2))*Pi ^(1/2)-4/3*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(1/2)
Time = 3.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (e^{a+x (b+c x)} (1+2 (a+x (b+c x)))+2 (-a-x (b+c x))^{3/2} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )\right )}{3 (a+x (b+c x))^{3/2}} \]
(-2*(E^(a + x*(b + c*x))*(1 + 2*(a + x*(b + c*x))) + 2*(-a - x*(b + c*x))^ (3/2)*Gamma[1/2, -a - x*(b + c*x)]))/(3*(a + x*(b + c*x))^(3/2))
Time = 0.54 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {7258, 2608, 2608, 2611, 2633}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(b+2 c x) e^{a+b x+c x^2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 7258 |
\(\displaystyle \int \frac {e^{a+b x+c x^2}}{\left (a+b x+c x^2\right )^{5/2}}d\left (a+b x+c x^2\right )\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {2}{3} \int \frac {e^{c x^2+b x+a}}{\left (c x^2+b x+a\right )^{3/2}}d\left (c x^2+b x+a\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {2}{3} \left (2 \int \frac {e^{c x^2+b x+a}}{\sqrt {c x^2+b x+a}}d\left (c x^2+b x+a\right )-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {2}{3} \left (4 \int e^{c x^2+b x+a}d\sqrt {c x^2+b x+a}-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2}{3} \left (2 \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}\right )-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
(-2*E^(a + b*x + c*x^2))/(3*(a + b*x + c*x^2)^(3/2)) + (2*((-2*E^(a + b*x + c*x^2))/Sqrt[a + b*x + c*x^2] + 2*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]])) /3
3.7.33.3.1 Defintions of rubi rules used
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] - Simp[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] && In tegerQ[2*m] && !TrueQ[$UseGamma]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^(v_)*(u_)*(w_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q Subst[Int[x^m*F^x, x], x, v], x] /; !FalseQ[q]] /; FreeQ[ {F, m}, x] && EqQ[w, v]
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {4 \,\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{3}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{3 \sqrt {c \,x^{2}+b x +a}}\) | \(70\) |
default | \(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {4 \,\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{3}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{3 \sqrt {c \,x^{2}+b x +a}}\) | \(70\) |
-2/3*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(3/2)+4/3*erfi((c*x^2+b*x+a)^(1/2))*Pi ^(1/2)-4/3*exp(c*x^2+b*x+a)/(c*x^2+b*x+a)^(1/2)
\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(c*x^2 + b*x + a)*(2*c*x + b)*e^(c*x^2 + b*x + a)/(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)*x^3 + a^ 3 + 3*(a*b^2 + a^2*c)*x^2), x)
Time = 17.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.24 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {\left (\frac {4 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )}}{3} - \frac {\left (- \frac {4 a}{3} - \frac {4 b x}{3} - \frac {4 c x^{2}}{3} - \frac {2}{3}\right ) e^{a + b x + c x^{2}}}{\left (- a - b x - c x^{2}\right )^{\frac {3}{2}}}\right ) \left (- a - b x - c x^{2}\right )^{\frac {5}{2}}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}} \]
(4*sqrt(pi)*erfc(sqrt(-a - b*x - c*x**2))/3 - (-4*a/3 - 4*b*x/3 - 4*c*x**2 /3 - 2/3)*exp(a + b*x + c*x**2)/(-a - b*x - c*x**2)**(3/2))*(-a - b*x - c* x**2)**(5/2)/(a + b*x + c*x**2)**(5/2)
\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
Time = 1.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.22 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (2\,c\,x^2+2\,b\,x+2\,a\right )+4\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^2-4\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{5/2}}{3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \]