Optimal. Leaf size=322 \[ \frac{2 \sqrt{\pi } b^2 e^{3/2} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{2 b^2 e (c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}+\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^3 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b e (b c-a d)^2 \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{b^3 e^2 \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}+\frac{b^3 (c+d x)^4 e^{\frac{e}{(c+d x)^2}}}{4 d^4}+\frac{b^3 e (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{4 d^4} \]
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Rubi [A] time = 0.338777, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2226, 2206, 2211, 2204, 2214, 2210} \[ \frac{2 \sqrt{\pi } b^2 e^{3/2} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{2 b^2 e (c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}+\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^3 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b e (b c-a d)^2 \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{b^3 e^2 \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}+\frac{b^3 (c+d x)^4 e^{\frac{e}{(c+d x)^2}}}{4 d^4}+\frac{b^3 e (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{4 d^4} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2206
Rule 2211
Rule 2204
Rule 2214
Rule 2210
Rubi steps
\begin{align*} \int e^{\frac{e}{(c+d x)^2}} (a+b x)^3 \, dx &=\int \left (\frac{(-b c+a d)^3 e^{\frac{e}{(c+d x)^2}}}{d^3}+\frac{3 b (b c-a d)^2 e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^3}-\frac{3 b^2 (b c-a d) e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac{b^3 e^{\frac{e}{(c+d x)^2}} (c+d x)^3}{d^3}\right ) \, dx\\ &=\frac{b^3 \int e^{\frac{e}{(c+d x)^2}} (c+d x)^3 \, dx}{d^3}-\frac{\left (3 b^2 (b c-a d)\right ) \int e^{\frac{e}{(c+d x)^2}} (c+d x)^2 \, dx}{d^3}+\frac{\left (3 b (b c-a d)^2\right ) \int e^{\frac{e}{(c+d x)^2}} (c+d x) \, dx}{d^3}-\frac{(b c-a d)^3 \int e^{\frac{e}{(c+d x)^2}} \, dx}{d^3}\\ &=-\frac{(b c-a d)^3 e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^4}+\frac{3 b (b c-a d)^2 e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{2 d^4}-\frac{b^2 (b c-a d) e^{\frac{e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac{b^3 e^{\frac{e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac{\left (b^3 e\right ) \int e^{\frac{e}{(c+d x)^2}} (c+d x) \, dx}{2 d^3}-\frac{\left (2 b^2 (b c-a d) e\right ) \int e^{\frac{e}{(c+d x)^2}} \, dx}{d^3}+\frac{\left (3 b (b c-a d)^2 e\right ) \int \frac{e^{\frac{e}{(c+d x)^2}}}{c+d x} \, dx}{d^3}-\frac{\left (2 (b c-a d)^3 e\right ) \int \frac{e^{\frac{e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d^3}\\ &=-\frac{(b c-a d)^3 e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^4}-\frac{2 b^2 (b c-a d) e e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^4}+\frac{3 b (b c-a d)^2 e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac{b^3 e e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac{b^2 (b c-a d) e^{\frac{e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac{b^3 e^{\frac{e}{(c+d x)^2}} (c+d x)^4}{4 d^4}-\frac{3 b (b c-a d)^2 e \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}+\frac{\left (2 (b c-a d)^3 e\right ) \operatorname{Subst}\left (\int e^{e x^2} \, dx,x,\frac{1}{c+d x}\right )}{d^4}+\frac{\left (b^3 e^2\right ) \int \frac{e^{\frac{e}{(c+d x)^2}}}{c+d x} \, dx}{2 d^3}-\frac{\left (4 b^2 (b c-a d) e^2\right ) \int \frac{e^{\frac{e}{(c+d x)^2}}}{(c+d x)^2} \, dx}{d^3}\\ &=-\frac{(b c-a d)^3 e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^4}-\frac{2 b^2 (b c-a d) e e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^4}+\frac{3 b (b c-a d)^2 e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac{b^3 e e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac{b^2 (b c-a d) e^{\frac{e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac{b^3 e^{\frac{e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac{(b c-a d)^3 \sqrt{e} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b (b c-a d)^2 e \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}-\frac{b^3 e^2 \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}+\frac{\left (4 b^2 (b c-a d) e^2\right ) \operatorname{Subst}\left (\int e^{e x^2} \, dx,x,\frac{1}{c+d x}\right )}{d^4}\\ &=-\frac{(b c-a d)^3 e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^4}-\frac{2 b^2 (b c-a d) e e^{\frac{e}{(c+d x)^2}} (c+d x)}{d^4}+\frac{3 b (b c-a d)^2 e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{2 d^4}+\frac{b^3 e e^{\frac{e}{(c+d x)^2}} (c+d x)^2}{4 d^4}-\frac{b^2 (b c-a d) e^{\frac{e}{(c+d x)^2}} (c+d x)^3}{d^4}+\frac{b^3 e^{\frac{e}{(c+d x)^2}} (c+d x)^4}{4 d^4}+\frac{(b c-a d)^3 \sqrt{e} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}+\frac{2 b^2 (b c-a d) e^{3/2} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b (b c-a d)^2 e \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}-\frac{b^3 e^2 \text{Ei}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}\\ \end{align*}
Mathematica [A] time = 0.307828, size = 243, normalized size = 0.75 \[ \frac{4 \sqrt{\pi } \sqrt{e} (b c-a d) \left (a^2 d^2-2 a b c d+b^2 \left (c^2+2 e\right )\right ) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )-b e \left (6 a^2 d^2-12 a b c d+b^2 \left (6 c^2+e\right )\right ) \text{Ei}\left (\frac{e}{(c+d x)^2}\right )+d x e^{\frac{e}{(c+d x)^2}} \left (6 a^2 b d^3 x+4 a^3 d^3+4 a b^2 d \left (d^2 x^2+2 e\right )+b^3 \left (-6 c e+d^3 x^3+d e x\right )\right )}{4 d^4}-\frac{c e^{\frac{e}{(c+d x)^2}} \left (6 a^2 b c d^2-4 a^3 d^3-4 a b^2 d \left (c^2+2 e\right )+b^3 \left (c^3+7 c e\right )\right )}{4 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 560, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b^{3} d^{3} x^{4} + 4 \, a b^{2} d^{3} x^{3} +{\left (6 \, a^{2} b d^{3} + b^{3} d e\right )} x^{2} + 2 \,{\left (2 \, a^{3} d^{3} - 3 \, b^{3} c e + 4 \, a b^{2} d e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{3}} + \int \frac{{\left (3 \, b^{3} c^{4} e - 4 \, a b^{2} c^{3} d e -{\left (12 \, a b^{2} c d^{3} e - 6 \, a^{2} b d^{4} e -{\left (6 \, c^{2} d^{2} e + d^{2} e^{2}\right )} b^{3}\right )} x^{2} + 2 \,{\left (2 \, a^{3} d^{4} e - 2 \,{\left (3 \, c^{2} d^{2} e - 2 \, d^{2} e^{2}\right )} a b^{2} +{\left (4 \, c^{3} d e - 3 \, c d e^{2}\right )} b^{3}\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{2 \,{\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64628, size = 635, normalized size = 1.97 \begin{align*} -\frac{4 \, \sqrt{\pi }{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4} + 2 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} e\right )} \sqrt{-\frac{e}{d^{2}}} \operatorname{erf}\left (\frac{d \sqrt{-\frac{e}{d^{2}}}}{d x + c}\right ) +{\left (b^{3} e^{2} + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} e\right )}{\rm Ei}\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (b^{3} d^{4} x^{4} + 4 \, a b^{2} d^{4} x^{3} - b^{3} c^{4} + 4 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} +{\left (6 \, a^{2} b d^{4} + b^{3} d^{2} e\right )} x^{2} -{\left (7 \, b^{3} c^{2} - 8 \, a b^{2} c d\right )} e + 2 \,{\left (2 \, a^{3} d^{4} -{\left (3 \, b^{3} c d - 4 \, a b^{2} d^{2}\right )} e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{3} e^{\left (\frac{e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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