Optimal. Leaf size=145 \[ \frac{16}{105} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )-\frac{16 e^{a+b x+c x^2}}{105 \sqrt{a+b x+c x^2}}-\frac{8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}-\frac{4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.379406, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {6707, 2177, 2180, 2204} \[ \frac{16}{105} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )-\frac{16 e^{a+b x+c x^2}}{105 \sqrt{a+b x+c x^2}}-\frac{8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}-\frac{4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6707
Rule 2177
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{9/2}} \, dx &=\operatorname{Subst}\left (\int \frac{e^x}{x^{9/2}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}+\frac{2}{7} \operatorname{Subst}\left (\int \frac{e^x}{x^{7/2}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}-\frac{4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}+\frac{4}{35} \operatorname{Subst}\left (\int \frac{e^x}{x^{5/2}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}-\frac{4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac{8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}+\frac{8}{105} \operatorname{Subst}\left (\int \frac{e^x}{x^{3/2}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}-\frac{4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac{8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 e^{a+b x+c x^2}}{105 \sqrt{a+b x+c x^2}}+\frac{16}{105} \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}-\frac{4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac{8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 e^{a+b x+c x^2}}{105 \sqrt{a+b x+c x^2}}+\frac{32}{105} \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )\\ &=-\frac{2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}-\frac{4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac{8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 e^{a+b x+c x^2}}{105 \sqrt{a+b x+c x^2}}+\frac{16}{105} \sqrt{\pi } \text{erfi}\left (\sqrt{a+b x+c x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.193919, size = 103, normalized size = 0.71 \[ -\frac{2 \left (8 (-a-x (b+c x))^{7/2} \text{Gamma}\left (\frac{1}{2},-a-x (b+c x)\right )+e^{a+x (b+c x)} \left (8 (a+x (b+c x))^3+4 (a+x (b+c x))^2+6 (a+x (b+c x))+15\right )\right )}{105 (a+x (b+c x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 120, normalized size = 0.8 \begin{align*} -{\frac{2\,{{\rm e}^{c{x}^{2}+bx+a}}}{7} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{2}}}}-{\frac{4\,{{\rm e}^{c{x}^{2}+bx+a}}}{35} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{5}{2}}}}-{\frac{8\,{{\rm e}^{c{x}^{2}+bx+a}}}{105} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,\sqrt{\pi }}{105}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{16\,{{\rm e}^{c{x}^{2}+bx+a}}}{105}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{c^{5} x^{10} + 5 \, b c^{4} x^{9} + 5 \,{\left (2 \, b^{2} c^{3} + a c^{4}\right )} x^{8} + 10 \,{\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} x^{7} + 5 \,{\left (b^{4} c + 6 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} x^{6} + 5 \, a^{4} b x +{\left (b^{5} + 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} x^{5} + a^{5} + 5 \,{\left (a b^{4} + 6 \, a^{2} b^{2} c + 2 \, a^{3} c^{2}\right )} x^{4} + 10 \,{\left (a^{2} b^{3} + 2 \, a^{3} b c\right )} x^{3} + 5 \,{\left (2 \, a^{3} b^{2} + a^{4} c\right )} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]