Optimal. Leaf size=71 \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0881561, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2183, 2176, 2180, 2204} \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2183
Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \left (a+b e^x\right ) \sqrt{c+d x} \, dx &=\int \left (a \sqrt{c+d x}+b e^x \sqrt{c+d x}\right ) \, dx\\ &=\frac{2 a (c+d x)^{3/2}}{3 d}+b \int e^x \sqrt{c+d x} \, dx\\ &=b e^x \sqrt{c+d x}+\frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} (b d) \int \frac{e^x}{\sqrt{c+d x}} \, dx\\ &=b e^x \sqrt{c+d x}+\frac{2 a (c+d x)^{3/2}}{3 d}-b \operatorname{Subst}\left (\int e^{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )\\ &=b e^x \sqrt{c+d x}+\frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} b \sqrt{d} e^{-\frac{c}{d}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )\\ \end{align*}
Mathematica [A] time = 0.0940573, size = 71, normalized size = 1. \[ \frac{2 a (c+d x)^{3/2}}{3 d}-\frac{1}{2} \sqrt{\pi } b \sqrt{d} e^{-\frac{c}{d}} \text{Erfi}\left (\frac{\sqrt{c+d x}}{\sqrt{d}}\right )+b e^x \sqrt{c+d x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 77, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( 1/3\,a \left ( dx+c \right ) ^{3/2}+{b \left ( 1/2\,\sqrt{dx+c}{{\rm e}^{{\frac{dx+c}{d}}}}d-1/4\,{d\sqrt{\pi }{\it Erf} \left ( \sqrt{-{d}^{-1}}\sqrt{dx+c} \right ){\frac{1}{\sqrt{-{d}^{-1}}}}} \right ) \left ({{\rm e}^{{\frac{c}{d}}}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.19239, size = 111, normalized size = 1.56 \begin{align*} \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a - 3 \,{\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 )} b}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.636, size = 167, normalized size = 2.35 \begin{align*} \frac{3 \, \sqrt{\pi } 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 )} + 2 \,{\left (2 \, a d x + 3 \, b d e^{x} + 2 \, a c\right )} \sqrt{d x + c}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.89973, size = 85, normalized size = 1.2 \begin{align*} \frac{2 a \left (c + d x\right )^{\frac{3}{2}}}{3 d} + b \sqrt{d} \sqrt{c + d x} \sqrt{\frac{1}{d}} e^{- \frac{c}{d}} e^{\frac{c}{d} + x} - \frac{\sqrt{\pi } b \sqrt{d} e^{- \frac{c}{d}} \operatorname{erfi}{\left (\frac{\sqrt{c + d x}}{d \sqrt{\frac{1}{d}}} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.3116, size = 93, normalized size = 1.31 \begin{align*} \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a + 3 \,{\left (\frac{\sqrt{\pi } d^{2} \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 )} b}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]