Optimal. Leaf size=52 \[ e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6707, 2176, 2180, 2204} \[ e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \sqrt {\pi } \text {Erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2176
Rule 2180
Rule 2204
Rule 6707
Rubi steps
\begin {align*} \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx &=\operatorname {Subst}\left (\int e^x \sqrt {x} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )\\ &=e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 46, normalized size = 0.88 \[ \frac {\sqrt {a+x (b+c x)} \Gamma \left (\frac {3}{2},-a-x (b+c x)\right )}{\sqrt {-a-x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 47, normalized size = 0.90 \[ -\frac {1}{2} \, \sqrt {\pi } i \operatorname {erf}\left (-\sqrt {c x^{2} + b x + a} i\right ) + \sqrt {c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 44, normalized size = 0.85 \[ -\frac {\sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right )}{2}+\sqrt {c \,x^{2}+b x +a}\, {\mathrm e}^{c \,x^{2}+b x +a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.51, size = 76, normalized size = 1.46 \[ \frac {\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,\sqrt {c\,x^2+b\,x+a}}{2\,\sqrt {-c\,x^2-b\,x-a}}+{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,\sqrt {c\,x^2+b\,x+a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.80, size = 78, normalized size = 1.50 \[ \frac {\left (\sqrt {- a - b x - c x^{2}} e^{a + b x + c x^{2}} + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )}}{2}\right ) \sqrt {a + b x + c x^{2}}}{\sqrt {- a - b x - c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________