Optimal. Leaf size=145 \[ -\frac {2 e^{a+b x+c x^2}}{7 \left (a+b x+c x^2\right )^{7/2}}-\frac {4 e^{a+b x+c x^2}}{35 \left (a+b x+c x^2\right )^{5/2}}-\frac {8 e^{a+b x+c x^2}}{105 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 e^{a+b x+c x^2}}{105 \sqrt {a+b x+c x^2}}+\frac {16}{105} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
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Rubi [A]
time = 0.26, antiderivative size = 145, normalized size of antiderivative = 1.00, 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 = {6839, 2208,
2211, 2235} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2208
Rule 2211
Rule 2235
Rule 6839
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 &=\text {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} \text {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} \text {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} \text {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} \text {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} \text {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*}
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Mathematica [A]
time = 5.95, size = 103, normalized size = 0.71 \begin {gather*} -\frac {2 \left (e^{a+x (b+c x)} \left (15+6 (a+x (b+c x))+4 (a+x (b+c x))^2+8 (a+x (b+c x))^3\right )+8 (-a-x (b+c x))^{7/2} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )\right )}{105 (a+x (b+c x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 120, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{7 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{35 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {8 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 \erfi \left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{105}-\frac {16 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \sqrt {c \,x^{2}+b x +a}}\) | \(120\) |
default | \(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{7 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{35 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {8 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 \erfi \left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{105}-\frac {16 \,{\mathrm e}^{c \,x^{2}+b x +a}}{105 \sqrt {c \,x^{2}+b x +a}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.70, size = 154, normalized size = 1.06 \begin {gather*} -\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (30\,c\,x^2+30\,b\,x+30\,a\right )+12\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^2+8\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^3+16\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^4-16\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{9/2}}{105\,{\left (c\,x^2+b\,x+a\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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