Optimal. Leaf size=145 \[ \frac {2 a^2 (c+d x)^{3/2}}{3 d}-\sqrt {\pi } a b \sqrt {d} e^{-\frac {c}{d}} \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+2 a b e^x \sqrt {c+d x}-\frac {1}{4} \sqrt {\frac {\pi }{2}} b^2 \sqrt {d} e^{-\frac {2 c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {1}{2} b^2 e^{2 x} \sqrt {c+d x} \]
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Rubi [A] time = 0.18, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2183, 2176, 2180, 2204} \[ \frac {2 a^2 (c+d x)^{3/2}}{3 d}-\sqrt {\pi } a b \sqrt {d} e^{-\frac {c}{d}} \text {Erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )+2 a b e^x \sqrt {c+d x}-\frac {1}{4} \sqrt {\frac {\pi }{2}} b^2 \sqrt {d} e^{-\frac {2 c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )+\frac {1}{2} b^2 e^{2 x} \sqrt {c+d x} \]
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 )^2 \sqrt {c+d x} \, dx &=\int \left (a^2 \sqrt {c+d x}+2 a b e^x \sqrt {c+d x}+b^2 e^{2 x} \sqrt {c+d x}\right ) \, dx\\ &=\frac {2 a^2 (c+d x)^{3/2}}{3 d}+(2 a b) \int e^x \sqrt {c+d x} \, dx+b^2 \int e^{2 x} \sqrt {c+d x} \, dx\\ &=2 a b e^x \sqrt {c+d x}+\frac {1}{2} b^2 e^{2 x} \sqrt {c+d x}+\frac {2 a^2 (c+d x)^{3/2}}{3 d}-(a b d) \int \frac {e^x}{\sqrt {c+d x}} \, dx-\frac {1}{4} \left (b^2 d\right ) \int \frac {e^{2 x}}{\sqrt {c+d x}} \, dx\\ &=2 a b e^x \sqrt {c+d x}+\frac {1}{2} b^2 e^{2 x} \sqrt {c+d x}+\frac {2 a^2 (c+d x)^{3/2}}{3 d}-(2 a b) \operatorname {Subst}\left (\int e^{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )-\frac {1}{2} b^2 \operatorname {Subst}\left (\int e^{-\frac {2 c}{d}+\frac {2 x^2}{d}} \, dx,x,\sqrt {c+d x}\right )\\ &=2 a b e^x \sqrt {c+d x}+\frac {1}{2} b^2 e^{2 x} \sqrt {c+d x}+\frac {2 a^2 (c+d x)^{3/2}}{3 d}-a b \sqrt {d} e^{-\frac {c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right )-\frac {1}{4} 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 )\\ \end {align*}
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Mathematica [A] time = 0.32, size = 134, normalized size = 0.92 \[ \frac {4 \sqrt {c+d x} \left (4 a^2 (c+d x)+12 a b d e^x+3 b^2 d e^{2 x}\right )-3 \sqrt {2 \pi } b^2 d^{3/2} e^{-\frac {2 c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 d}-\sqrt {\pi } a b \sqrt {d} e^{-\frac {c}{d}} \text {erfi}\left (\frac {\sqrt {c+d x}}{\sqrt {d}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 133, normalized size = 0.92 \[ \frac {3 \, \sqrt {2} \sqrt {\pi } b^{2} d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )} + 24 \, \sqrt {\pi } a b d^{2} \sqrt {-\frac {1}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )} + 4 \, {\left (4 \, a^{2} d x + 3 \, b^{2} d e^{\left (2 \, x\right )} + 12 \, a b d e^{x} + 4 \, a^{2} c\right )} \sqrt {d x + c}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 250, normalized size = 1.72 \[ -\frac {\frac {12 \, \sqrt {2} \sqrt {\pi } b^{2} c d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + \frac {48 \, \sqrt {\pi } a b c d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} - 48 \, \sqrt {d x + c} a^{2} c - 16 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} - 24 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-d}} + 2 \, \sqrt {d x + c} d e^{x}\right )} a b - 3 \, {\left (\frac {\sqrt {2} \sqrt {\pi } {\left (4 \, c + d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {d x + c} \sqrt {-d}}{d}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-d}} + 4 \, \sqrt {d x + c} d e^{\left (2 \, x\right )}\right )} b^{2}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 144, normalized size = 0.99 \[ \frac {4 \left (-\frac {\sqrt {\pi }\, d \erf \left (\sqrt {-\frac {1}{d}}\, \sqrt {d x +c}\right )}{4 \sqrt {-\frac {1}{d}}}+\frac {\sqrt {d x +c}\, d \,{\mathrm e}^{\frac {d x +c}{d}}}{2}\right ) a b \,{\mathrm e}^{-\frac {c}{d}}+2 \left (-\frac {\sqrt {\pi }\, d \erf \left (\sqrt {-\frac {2}{d}}\, \sqrt {d x +c}\right )}{8 \sqrt {-\frac {2}{d}}}+\frac {\sqrt {d x +c}\, d \,{\mathrm e}^{\frac {2 d x +2 c}{d}}}{4}\right ) b^{2} {\mathrm e}^{-\frac {2 c}{d}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2}}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.49, size = 160, normalized size = 1.10 \[ \frac {16 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} - 24 \, {\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 2 \, \sqrt {d x + c} d e^{\left (\frac {d x + c}{d} - \frac {c}{d}\right )}\right )} a b - 3 \, {\left (\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {1}{d}}\right ) e^{\left (-\frac {2 \, c}{d}\right )}}{\sqrt {-\frac {1}{d}}} - 4 \, \sqrt {d x + c} d e^{\left (\frac {2 \, {\left (d x + c\right )}}{d} - \frac {2 \, c}{d}\right )}\right )} b^{2}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,{\mathrm {e}}^x\right )}^2\,\sqrt {c+d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.21, size = 184, normalized size = 1.27 \[ \frac {2 a^{2} \left (c + d x\right )^{\frac {3}{2}}}{3 d} - \sqrt {\pi } a b \sqrt {d} e^{- \frac {c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )} + \frac {2 a b \sqrt {c + d x} e^{- \frac {c}{d}} e^{\frac {c}{d} + x}}{\sqrt {d} \sqrt {\frac {1}{d}}} - \frac {\sqrt {2} \sqrt {\pi } b^{2} \sqrt {d} e^{- \frac {2 c}{d}} \operatorname {erfi}{\left (\frac {\sqrt {2} \sqrt {c + d x}}{d \sqrt {\frac {1}{d}}} \right )}}{8} + \frac {b^{2} \sqrt {c + d x} e^{- \frac {2 c}{d}} e^{\frac {2 c}{d} + 2 x}}{2 \sqrt {d} \sqrt {\frac {1}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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