Integrand size = 33, antiderivative size = 115 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 e^{a+b x+c x^2}}{15 \sqrt {a+b x+c x^2}}+\frac {8}{15} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6839, 2208, 2211, 2235} \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {8}{15} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {8 e^{a+b x+c x^2}}{15 \sqrt {a+b x+c x^2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}} \]
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Rule 2208
Rule 2211
Rule 2235
Rule 6839
Rubi steps \begin{align*} \text {integral}& = \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}}{5 \left (a+b x+c x^2\right )^{5/2}}+\frac {2}{5} \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}}{5 \left (a+b x+c x^2\right )^{5/2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}+\frac {4}{15} \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}}{5 \left (a+b x+c x^2\right )^{5/2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 e^{a+b x+c x^2}}{15 \sqrt {a+b x+c x^2}}+\frac {8}{15} \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}}{5 \left (a+b x+c x^2\right )^{5/2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 e^{a+b x+c x^2}}{15 \sqrt {a+b x+c x^2}}+\frac {16}{15} \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}}{5 \left (a+b x+c x^2\right )^{5/2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 e^{a+b x+c x^2}}{15 \sqrt {a+b x+c x^2}}+\frac {8}{15} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \\ \end{align*}
Time = 4.79 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.79 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {-2 e^{a+x (b+c x)} \left (3+2 (a+x (b+c x))+4 (a+x (b+c x))^2\right )+8 (-a-x (b+c x))^{5/2} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )}{15 (a+x (b+c x))^{5/2}} \]
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Time = 0.49 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{5 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{15 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {8 \,\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{15}-\frac {8 \,{\mathrm e}^{c \,x^{2}+b x +a}}{15 \sqrt {c \,x^{2}+b x +a}}\) | \(95\) |
default | \(-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{5 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{15 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {8 \,\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{15}-\frac {8 \,{\mathrm e}^{c \,x^{2}+b x +a}}{15 \sqrt {c \,x^{2}+b x +a}}\) | \(95\) |
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\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{2}}} \,d x } \]
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Time = 1.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.12 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (6\,c\,x^2+6\,b\,x+6\,a\right )+4\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^2+8\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^3+8\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{7/2}}{15\,{\left (c\,x^2+b\,x+a\right )}^{7/2}} \]
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