Optimal. Leaf size=292 \[ -\frac{d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt{\pi } b^4}-\frac{d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^4}+\frac{(b c-a d)^4 \text{Erf}(a+b x)}{4 b^4 d}+\frac{3 d (b c-a d)^2 \text{Erf}(a+b x)}{4 b^4}-\frac{e^{-(a+b x)^2} (b c-a d)^3}{\sqrt{\pi } b^4}-\frac{3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt{\pi } b^4}+\frac{3 d^3 \text{Erf}(a+b x)}{16 b^4}-\frac{d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt{\pi } b^4}-\frac{3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt{\pi } b^4}+\frac{(c+d x)^4 \text{Erfc}(a+b x)}{4 d} \]
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Rubi [A] time = 0.272314, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6362, 2226, 2205, 2209, 2212} \[ -\frac{d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt{\pi } b^4}-\frac{d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^4}+\frac{(b c-a d)^4 \text{Erf}(a+b x)}{4 b^4 d}+\frac{3 d (b c-a d)^2 \text{Erf}(a+b x)}{4 b^4}-\frac{e^{-(a+b x)^2} (b c-a d)^3}{\sqrt{\pi } b^4}-\frac{3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt{\pi } b^4}+\frac{3 d^3 \text{Erf}(a+b x)}{16 b^4}-\frac{d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt{\pi } b^4}-\frac{3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt{\pi } b^4}+\frac{(c+d x)^4 \text{Erfc}(a+b x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 6362
Rule 2226
Rule 2205
Rule 2209
Rule 2212
Rubi steps
\begin{align*} \int (c+d x)^3 \text{erfc}(a+b x) \, dx &=\frac{(c+d x)^4 \text{erfc}(a+b x)}{4 d}+\frac{b \int e^{-(a+b x)^2} (c+d x)^4 \, dx}{2 d \sqrt{\pi }}\\ &=\frac{(c+d x)^4 \text{erfc}(a+b x)}{4 d}+\frac{b \int \left (\frac{(b c-a d)^4 e^{-(a+b x)^2}}{b^4}+\frac{4 d (b c-a d)^3 e^{-(a+b x)^2} (a+b x)}{b^4}+\frac{6 d^2 (b c-a d)^2 e^{-(a+b x)^2} (a+b x)^2}{b^4}+\frac{4 d^3 (b c-a d) e^{-(a+b x)^2} (a+b x)^3}{b^4}+\frac{d^4 e^{-(a+b x)^2} (a+b x)^4}{b^4}\right ) \, dx}{2 d \sqrt{\pi }}\\ &=\frac{(c+d x)^4 \text{erfc}(a+b x)}{4 d}+\frac{d^3 \int e^{-(a+b x)^2} (a+b x)^4 \, dx}{2 b^3 \sqrt{\pi }}+\frac{\left (2 d^2 (b c-a d)\right ) \int e^{-(a+b x)^2} (a+b x)^3 \, dx}{b^3 \sqrt{\pi }}+\frac{\left (3 d (b c-a d)^2\right ) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{b^3 \sqrt{\pi }}+\frac{\left (2 (b c-a d)^3\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt{\pi }}+\frac{(b c-a d)^4 \int e^{-(a+b x)^2} \, dx}{2 b^3 d \sqrt{\pi }}\\ &=-\frac{(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt{\pi }}-\frac{3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt{\pi }}-\frac{d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt{\pi }}-\frac{d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt{\pi }}+\frac{(b c-a d)^4 \text{erf}(a+b x)}{4 b^4 d}+\frac{(c+d x)^4 \text{erfc}(a+b x)}{4 d}+\frac{\left (3 d^3\right ) \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{4 b^3 \sqrt{\pi }}+\frac{\left (2 d^2 (b c-a d)\right ) \int e^{-(a+b x)^2} (a+b x) \, dx}{b^3 \sqrt{\pi }}+\frac{\left (3 d (b c-a d)^2\right ) \int e^{-(a+b x)^2} \, dx}{2 b^3 \sqrt{\pi }}\\ &=-\frac{d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt{\pi }}-\frac{(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt{\pi }}-\frac{3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt{\pi }}-\frac{3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt{\pi }}-\frac{d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt{\pi }}-\frac{d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt{\pi }}+\frac{3 d (b c-a d)^2 \text{erf}(a+b x)}{4 b^4}+\frac{(b c-a d)^4 \text{erf}(a+b x)}{4 b^4 d}+\frac{(c+d x)^4 \text{erfc}(a+b x)}{4 d}+\frac{\left (3 d^3\right ) \int e^{-(a+b x)^2} \, dx}{8 b^3 \sqrt{\pi }}\\ &=-\frac{d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt{\pi }}-\frac{(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt{\pi }}-\frac{3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt{\pi }}-\frac{3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt{\pi }}-\frac{d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt{\pi }}-\frac{d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt{\pi }}+\frac{3 d^3 \text{erf}(a+b x)}{16 b^4}+\frac{3 d (b c-a d)^2 \text{erf}(a+b x)}{4 b^4}+\frac{(b c-a d)^4 \text{erf}(a+b x)}{4 b^4 d}+\frac{(c+d x)^4 \text{erfc}(a+b x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.359714, size = 268, normalized size = 0.92 \[ \frac{e^{-(a+b x)^2} \left (\sqrt{\pi } e^{(a+b x)^2} \left (12 a^2 \left (2 b^2 c^2 d+d^3\right )-16 a^3 b c d^2+4 a^4 d^3-8 a \left (2 b^3 c^3+3 b c d^2\right )+3 \left (4 b^2 c^2 d+d^3\right )\right ) \text{Erf}(a+b x)-2 b d^2 \left (8 \left (a^2+1\right ) c+\left (2 a^2+3\right ) d x\right )+2 a \left (2 a^2+5\right ) d^3+4 \sqrt{\pi } b^4 x e^{(a+b x)^2} \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \text{Erfc}(a+b x)+4 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )-4 b^3 \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )}{16 \sqrt{\pi } b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 729, normalized size = 2.5 \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*} \int{\left (d x + c\right )}^{3} \operatorname{erfc}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09864, size = 689, normalized size = 2.36 \begin{align*} \frac{4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x - 2 \, \sqrt{\pi }{\left (2 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 8 \,{\left (a^{2} + 1\right )} b c d^{2} -{\left (2 \, a^{3} + 5 \, a\right )} d^{3} + 2 \,{\left (4 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} +{\left (12 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} +{\left (2 \, a^{2} + 3\right )} b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} -{\left (4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x + \pi{\left (16 \, a b^{3} c^{3} - 12 \,{\left (2 \, a^{2} + 1\right )} b^{2} c^{2} d + 8 \,{\left (2 \, a^{3} + 3 \, a\right )} b c d^{2} -{\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} d^{3}\right )}\right )} \operatorname{erf}\left (b x + a\right )}{16 \, \pi b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.4494, size = 746, normalized size = 2.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50791, size = 587, normalized size = 2.01 \begin{align*} \frac{1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac{3}{2} \, c^{2} d x^{2} -{\left (x \operatorname{erf}\left (b x + a\right ) - \frac{\frac{\sqrt{\pi } a \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right )}{b} - \frac{e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt{\pi }}\right )} c^{3} - \frac{3}{4} \,{\left (2 \, x^{2} \operatorname{erf}\left (b x + a\right ) + \frac{\frac{\sqrt{\pi }{\left (2 \, a^{2} + 1\right )} \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right )}{b} + \frac{2 \,{\left (b{\left (x + \frac{a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt{\pi } b}\right )} c^{2} d - \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{erf}\left (b x + a\right ) - \frac{\frac{\sqrt{\pi }{\left (2 \, a^{3} + 3 \, a\right )} \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right )}{b} - \frac{2 \,{\left (b^{2}{\left (x + \frac{a}{b}\right )}^{2} - 3 \, a b{\left (x + \frac{a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt{\pi } b^{2}}\right )} c d^{2} - \frac{1}{16} \,{\left (4 \, x^{4} \operatorname{erf}\left (b x + a\right ) + \frac{\frac{\sqrt{\pi }{\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right )}{b} + \frac{2 \,{\left (2 \, b^{3}{\left (x + \frac{a}{b}\right )}^{3} - 8 \, a b^{2}{\left (x + \frac{a}{b}\right )}^{2} + 12 \, a^{2} b{\left (x + \frac{a}{b}\right )} - 8 \, a^{3} + 3 \, b{\left (x + \frac{a}{b}\right )} - 8 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt{\pi } b^{3}}\right )} d^{3} + c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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