Optimal. Leaf size=112 \[ -\frac {15}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac {5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac {15}{4} e^{a+b x+c x^2} \sqrt {a+b x+c x^2} \]
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Rubi [A] time = 0.46, antiderivative size = 112, 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, 2176, 2180, 2204} \[ -\frac {15}{8} \sqrt {\pi } \text {Erfi}\left (\sqrt {a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac {5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac {15}{4} e^{a+b x+c x^2} \sqrt {a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2204
Rule 6707
Rubi steps
\begin {align*} \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\operatorname {Subst}\left (\int e^x x^{5/2} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac {5}{2} \operatorname {Subst}\left (\int e^x x^{3/2} \, dx,x,a+b x+c x^2\right )\\ &=-\frac {5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+\frac {15}{4} \operatorname {Subst}\left (\int e^x \sqrt {x} \, dx,x,a+b x+c x^2\right )\\ &=\frac {15}{4} e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac {15}{8} \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,a+b x+c x^2\right )\\ &=\frac {15}{4} e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac {15}{4} \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )\\ &=\frac {15}{4} e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {5}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}-\frac {15}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 46, normalized size = 0.41 \[ \frac {\sqrt {a+x (b+c x)} \Gamma \left (\frac {7}{2},-a-x (b+c x)\right )}{\sqrt {-a-x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, c^{3} x^{5} + 5 \, b c^{2} x^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} x^{3} + a^{2} b + {\left (b^{3} + 6 \, a b c\right )} x^{2} + 2 \, {\left (a b^{2} + a^{2} c\right )} x\right )} \sqrt {c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 79, normalized size = 0.71 \[ -\frac {15}{8} \, \sqrt {\pi } i \operatorname {erf}\left (-\sqrt {c x^{2} + b x + a} i\right ) + \frac {1}{4} \, {\left (4 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} + 15 \, \sqrt {c x^{2} + b x + a}\right )} e^{\left (c x^{2} + b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 94, normalized size = 0.84 \[ -\frac {15 \sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right )}{8}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} {\mathrm e}^{c \,x^{2}+b x +a}}{2}+\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} {\mathrm e}^{c \,x^{2}+b x +a}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, {\mathrm e}^{c \,x^{2}+b x +a}}{4} \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.91, size = 117, normalized size = 1.04 \[ \frac {\left ({\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (\frac {15\,\sqrt {-c\,x^2-b\,x-a}}{4}+\frac {5\,{\left (-c\,x^2-b\,x-a\right )}^{3/2}}{2}+{\left (-c\,x^2-b\,x-a\right )}^{5/2}\right )+\frac {15\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )}{8}\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (-c\,x^2-b\,x-a\right )}^{5/2}} \]
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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