Optimal. Leaf size=224 \[ \frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{2} \sqrt {\pi } a^2 b \sqrt {d} e^{-\frac {c}{d}} \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+3 a^2 b e^x \sqrt {c+d x}-\frac {3}{4} \sqrt {\frac {\pi }{2}} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}-\frac {1}{6} \sqrt {\frac {\pi }{3}} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x} \]
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Rubi [A] time = 0.26, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2183, 2176, 2180, 2204} \[ -\frac {3}{2} \sqrt {\pi } a^2 b \sqrt {d} e^{-\frac {c}{d}} \text {Erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+3 a^2 b e^x \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{4} \sqrt {\frac {\pi }{2}} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}-\frac {1}{6} \sqrt {\frac {\pi }{3}} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2183
Rule 2204
Rubi steps
\begin {align*} \int \left (a+b e^x\right )^3 \sqrt {c+d x} \, dx &=\int \left (a^3 \sqrt {c+d x}+3 a^2 b e^x \sqrt {c+d x}+3 a b^2 e^{2 x} \sqrt {c+d x}+b^3 e^{3 x} \sqrt {c+d x}\right ) \, dx\\ &=\frac {2 a^3 (c+d x)^{3/2}}{3 d}+\left (3 a^2 b\right ) \int e^x \sqrt {c+d x} \, dx+\left (3 a b^2\right ) \int e^{2 x} \sqrt {c+d x} \, dx+b^3 \int e^{3 x} \sqrt {c+d x} \, dx\\ &=3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {1}{2} \left (3 a^2 b d\right ) \int \frac {e^x}{\sqrt {c+d x}} \, dx-\frac {1}{4} \left (3 a b^2 d\right ) \int \frac {e^{2 x}}{\sqrt {c+d x}} \, dx-\frac {1}{6} \left (b^3 d\right ) \int \frac {e^{3 x}}{\sqrt {c+d x}} \, dx\\ &=3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\left (3 a^2 b\right ) \operatorname {Subst}\left (\int e^{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )-\frac {1}{2} \left (3 a b^2\right ) \operatorname {Subst}\left (\int e^{-\frac {2 c}{d}+\frac {2 x^2}{d}} \, dx,x,\sqrt {c+d x}\right )-\frac {1}{3} b^3 \operatorname {Subst}\left (\int e^{-\frac {3 c}{d}+\frac {3 x^2}{d}} \, dx,x,\sqrt {c+d x}\right )\\ &=3 a^2 b e^x \sqrt {c+d x}+\frac {3}{2} a b^2 e^{2 x} \sqrt {c+d x}+\frac {1}{3} b^3 e^{3 x} \sqrt {c+d x}+\frac {2 a^3 (c+d x)^{3/2}}{3 d}-\frac {3}{2} a^2 b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )-\frac {3}{4} a b^2 \sqrt {d} e^{-\frac {2 c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )-\frac {1}{6} b^3 \sqrt {d} e^{-\frac {3 c}{d}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )\\ \end {align*}
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Mathematica [A] time = 0.80, size = 196, normalized size = 0.88 \[ -\frac {108 \sqrt {\pi } a^2 b d^{3/2} e^{-\frac {c}{d}} \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )-12 \sqrt {c+d x} \left (4 a^3 (c+d x)+18 a^2 b d e^x+9 a b^2 d e^{2 x}+2 b^3 d e^{3 x}\right )+27 \sqrt {2 \pi } a b^2 d^{3/2} e^{-\frac {2 c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+4 \sqrt {3 \pi } b^3 d^{3/2} e^{-\frac {3 c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {c+d x}}{\sqrt {d}}\right )}{72 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 196, normalized size = 0.88 \[ \frac {27 \, \sqrt {2} \sqrt {\pi } a b^{2} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )} + 4 \, \sqrt {3} \sqrt {\pi } b^{3} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {3 \, c}{d}\right )} + 108 \, \sqrt {\pi } a^{2} b d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )} + 12 \, {\left (4 \, a^{3} d x + 2 \, b^{3} d e^{\left (3 \, x\right )} + 9 \, a b^{2} d e^{\left (2 \, x\right )} + 18 \, a^{2} b d e^{x} + 4 \, a^{3} c\right )} \sqrt {d x + c}}{72 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 368, normalized size = 1.64 \[ -\frac {\frac {108 \, \sqrt {2} \sqrt {\pi } a b^{2} c d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + \frac {24 \, \sqrt {3} \sqrt {\pi } b^{3} c d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-d}} + \frac {216 \, \sqrt {\pi } a^{2} b c d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} - 144 \, \sqrt {d x + c} a^{3} c - 48 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} - 108 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} + 2 \, \sqrt {d x + c} d e^{x}\right )} a^{2} b - 27 \, {\left (\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + 4 \, \sqrt {d x + c} d e^{\left (2 \, x\right )}\right )} a b^{2} - 4 \, {\left (\frac {\sqrt {3} \sqrt {\pi } {\left (6 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-d}} + 6 \, \sqrt {d x + c} d e^{\left (3 \, x\right )}\right )} b^{3}}{72 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 211, normalized size = 0.94 \[ \frac {6 \left (-\frac {\sqrt {\pi }\, d \erf \left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}+\frac {\sqrt {d x +c}\, d \,{\mathrm e}^{\frac {d x +c}{d}}}{2}\right ) a^{2} b \,{\mathrm e}^{-\frac {c}{d}}+6 \left (-\frac {\sqrt {\pi }\, d \erf \left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}+\frac {\sqrt {d x +c}\, d \,{\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}\right ) a \,b^{2} {\mathrm e}^{-\frac {2 c}{d}}+2 \left (-\frac {\sqrt {\pi }\, d \erf \left (\sqrt {-\frac {3}{d}}\, \sqrt {d x +c}\right )}{12 \sqrt {-\frac {3}{d}}}+\frac {\sqrt {d x +c}\, d \,{\mathrm e}^{\frac {3 d x +3 c}{d}}}{6}\right ) b^{3} {\mathrm e}^{-\frac {3 c}{d}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{3}}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 238, normalized size = 1.06 \[ \frac {48 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} - 108 \, {\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 2 \, \sqrt {d x + c} d e^{\left (\frac {d x + c}{d} - \frac {c}{d}\right )}\right )} a^{2} b - 27 \, {\left (\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 4 \, \sqrt {d x + c} d e^{\left (\frac {2 \, {\left (d x + c\right )}}{d} - \frac {2 \, c}{d}\right )}\right )} a b^{2} - 4 \, {\left (\frac {\sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {3} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {3 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 6 \, \sqrt {d x + c} d e^{\left (\frac {3 \, {\left (d x + c\right )}}{d} - \frac {3 \, c}{d}\right )}\right )} b^{3}}{72 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,{\mathrm {e}}^x\right )}^3\,\sqrt {c+d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.36, size = 291, normalized size = 1.30 \[ \frac {2 a^{3} \left (c + d x\right )^{\frac {3}{2}}}{3 d} - \frac {3 \sqrt {\pi } a^{2} b \sqrt {d} e^{- \frac {c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{2} + \frac {3 a^{2} b \sqrt {c + d x} e^{- \frac {c}{d}} e^{\frac {c}{d} + x}}{\sqrt {d} \sqrt {\frac {1}{d}}} - \frac {3 \sqrt {2} \sqrt {\pi } a b^{2} \sqrt {d} e^{- \frac {2 c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {2} \sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{8} + \frac {3 a b^{2} \sqrt {c + d x} e^{- \frac {2 c}{d}} e^{\frac {2 c}{d} + 2 x}}{2 \sqrt {d} \sqrt {\frac {1}{d}}} - \frac {\sqrt {3} \sqrt {\pi } b^{3} \sqrt {d} e^{- \frac {3 c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {3} \sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{18} + \frac {b^{3} \sqrt {c + d x} e^{- \frac {3 c}{d}} e^{\frac {3 c}{d} + 3 x}}{3 \sqrt {d} \sqrt {\frac {1}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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