Optimal. Leaf size=289 \[ \frac{d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt{\pi } b^4}+\frac{d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^4}-\frac{(b c-a d)^4 \text{Erf}(a+b x)}{4 b^4 d}-\frac{3 d (b c-a d)^2 \text{Erf}(a+b x)}{4 b^4}+\frac{e^{-(a+b x)^2} (b c-a d)^3}{\sqrt{\pi } b^4}+\frac{3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt{\pi } b^4}-\frac{3 d^3 \text{Erf}(a+b x)}{16 b^4}+\frac{d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt{\pi } b^4}+\frac{3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt{\pi } b^4}+\frac{(c+d x)^4 \text{Erf}(a+b x)}{4 d} \]
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Rubi [A] time = 0.320618, antiderivative size = 289, 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 = {6361, 2226, 2205, 2209, 2212} \[ \frac{d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{\sqrt{\pi } b^4}+\frac{d^2 e^{-(a+b x)^2} (b c-a d)}{\sqrt{\pi } b^4}-\frac{(b c-a d)^4 \text{Erf}(a+b x)}{4 b^4 d}-\frac{3 d (b c-a d)^2 \text{Erf}(a+b x)}{4 b^4}+\frac{e^{-(a+b x)^2} (b c-a d)^3}{\sqrt{\pi } b^4}+\frac{3 d e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{2 \sqrt{\pi } b^4}-\frac{3 d^3 \text{Erf}(a+b x)}{16 b^4}+\frac{d^3 e^{-(a+b x)^2} (a+b x)^3}{4 \sqrt{\pi } b^4}+\frac{3 d^3 e^{-(a+b x)^2} (a+b x)}{8 \sqrt{\pi } b^4}+\frac{(c+d x)^4 \text{Erf}(a+b x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 6361
Rule 2226
Rule 2205
Rule 2209
Rule 2212
Rubi steps
\begin{align*} \int (c+d x)^3 \text{erf}(a+b x) \, dx &=\frac{(c+d x)^4 \text{erf}(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{erf}(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{erf}(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{erf}(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{erf}(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{erf}(a+b x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.304701, size = 248, normalized size = 0.86 \[ \frac{e^{-(a+b x)^2} \left (-\sqrt{\pi } e^{(a+b x)^2} \text{Erf}(a+b x) \left (12 a^2 \left (2 b^2 c^2 d+d^3\right )-16 a^3 b c d^2+4 a^4 d^3-8 a \left (2 b^3 c^3+3 b c d^2\right )-4 b^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )+12 b^2 c^2 d+3 d^3\right )+2 b d^2 \left (8 \left (a^2+1\right ) c+\left (2 a^2+3\right ) d x\right )-2 a \left (2 a^2+5\right ) d^3-4 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )+4 b^3 \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )}{16 \sqrt{\pi } b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 466, normalized size = 1.6 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx+a \right ) \left ( d \left ( bx+a \right ) -ad+bc \right ) ^{4}}{4\,d{b}^{3}}}-{\frac{1}{2\,\sqrt{\pi }{b}^{3}d} \left ({d}^{4} \left ( -{\frac{ \left ( bx+a \right ) ^{3}}{2\,{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-{\frac{3\,bx+3\,a}{4\,{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+{\frac{3\,\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{8}} \right ) +{\frac{{a}^{4}{d}^{4}\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{2}}+{\frac{{b}^{4}{c}^{4}\sqrt{\pi }{\it Erf} \left ( bx+a \right ) }{2}}+2\,{\frac{{a}^{3}{d}^{4}}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+6\,{a}^{2}{d}^{4} \left ( -1/2\,{\frac{bx+a}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+1/4\,\sqrt{\pi }{\it Erf} \left ( bx+a \right ) \right ) -4\,a{d}^{4} \left ( -1/2\,{\frac{ \left ( bx+a \right ) ^{2}}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-1/2\, \left ({{\rm e}^{ \left ( bx+a \right ) ^{2}}} \right ) ^{-1} \right ) -2\,{\frac{{b}^{3}{c}^{3}d}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+6\,{b}^{2}{c}^{2}{d}^{2} \left ( -1/2\,{\frac{bx+a}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+1/4\,\sqrt{\pi }{\it Erf} \left ( bx+a \right ) \right ) +4\,bc{d}^{3} \left ( -1/2\,{\frac{ \left ( bx+a \right ) ^{2}}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-1/2\, \left ({{\rm e}^{ \left ( bx+a \right ) ^{2}}} \right ) ^{-1} \right ) -2\,a{b}^{3}{c}^{3}d\sqrt{\pi }{\it Erf} \left ( bx+a \right ) +3\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}\sqrt{\pi }{\it Erf} \left ( bx+a \right ) -2\,{a}^{3}bc{d}^{3}\sqrt{\pi }{\it Erf} \left ( bx+a \right ) +6\,{\frac{a{b}^{2}{c}^{2}{d}^{2}}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-6\,{\frac{{a}^{2}bc{d}^{3}}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}-12\,abc{d}^{3} \left ( -1/2\,{\frac{bx+a}{{{\rm e}^{ \left ( bx+a \right ) ^{2}}}}}+1/4\,\sqrt{\pi }{\it Erf} \left ( bx+a \right ) \right ) \right ) } \right ) } \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*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58144, size = 579, normalized size = 2. \begin{align*} \frac{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 = 24.8021, size = 746, normalized size = 2.58 \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 = 1.31544, size = 540, normalized size = 1.87 \begin{align*} \frac{{\left (d x + c\right )}^{4} \operatorname{erf}\left (b x + a\right )}{4 \, d} + \frac{4 \, \pi c^{4} \operatorname{erf}\left (-b{\left (x + \frac{a}{b}\right )}\right ) - 16 \, \sqrt{\pi }{\left (\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}\right )} c^{3} d + \frac{12 \, \sqrt{\pi }{\left (\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}\right )} c^{2} d^{2}}{b} - \frac{8 \, \sqrt{\pi }{\left (\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}\right )} c d^{3}}{b^{2}} + \frac{\sqrt{\pi }{\left (\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}\right )} d^{4}}{b^{3}}}{16 \, \pi d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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