Optimal. Leaf size=115 \[ \frac {8}{15} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {8 e^{a+b x+c x^2}}{15 \sqrt {a+b x+c x^2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6707, 2177, 2180, 2204} \[ \frac {8}{15} \sqrt {\pi } \text {Erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {8 e^{a+b x+c x^2}}{15 \sqrt {a+b x+c x^2}}-\frac {4 e^{a+b x+c x^2}}{15 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e^{a+b x+c x^2}}{5 \left (a+b x+c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2180
Rule 2204
Rule 6707
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 )^{7/2}} \, dx &=\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}}{5 \left (a+b x+c x^2\right )^{5/2}}+\frac {2}{5} \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}}{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} \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}}{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} \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}}{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} \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}}{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*}
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Mathematica [A] time = 0.18, size = 91, normalized size = 0.79 \[ \frac {8 (-a-x (b+c x))^{5/2} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )-2 e^{a+x (b+c x)} \left (4 (a+x (b+c x))^2+2 (a+x (b+c x))+3\right )}{15 (a+x (b+c x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{c^{4} x^{8} + 4 \, b c^{3} x^{7} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{6} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{5} + 4 \, a^{3} b x + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{4} + a^{4} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 95, normalized size = 0.83 \[ \frac {8 \sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right )}{15}-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{5 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{15 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {8 \,{\mathrm e}^{c \,x^{2}+b x +a}}{15 \sqrt {c \,x^{2}+b x +a}} \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 129, normalized size = 1.12 \[ -\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (6\,c\,x^2+6\,b\,x+6\,a\right )+4\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^2+8\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^3+8\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{7/2}}{15\,{\left (c\,x^2+b\,x+a\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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