Optimal. Leaf size=109 \[ \frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{4 d}+\frac{b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{16 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)}{32 d}-\frac{3 b e^3 \sin ^{-1}(c+d x)}{32 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0756311, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4805, 12, 4627, 321, 216} \[ \frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{4 d}+\frac{b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{16 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)}{32 d}-\frac{3 b e^3 \sin ^{-1}(c+d x)}{32 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4805
Rule 12
Rule 4627
Rule 321
Rule 216
Rubi steps
\begin{align*} \int (c e+d e x)^3 \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{4 d}-\frac{\left (b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{16 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{4 d}-\frac{\left (3 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{16 d}\\ &=\frac{3 b e^3 (c+d x) \sqrt{1-(c+d x)^2}}{32 d}+\frac{b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{16 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{4 d}-\frac{\left (3 b e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{32 d}\\ &=\frac{3 b e^3 (c+d x) \sqrt{1-(c+d x)^2}}{32 d}+\frac{b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{16 d}-\frac{3 b e^3 \sin ^{-1}(c+d x)}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.078866, size = 87, normalized size = 0.8 \[ \frac{e^3 \left (8 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )+2 b \sqrt{1-(c+d x)^2} (c+d x)^3+3 b \sqrt{1-(c+d x)^2} (c+d x)-3 b \sin ^{-1}(c+d x)\right )}{32 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 90, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{3} \left ( dx+c \right ) ^{4}a}{4}}+{e}^{3}b \left ({\frac{ \left ( dx+c \right ) ^{4}\arcsin \left ( dx+c \right ) }{4}}+{\frac{ \left ( dx+c \right ) ^{3}}{16}\sqrt{1- \left ( dx+c \right ) ^{2}}}+{\frac{3\,dx+3\,c}{32}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{3\,\arcsin \left ( dx+c \right ) }{32}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.39475, size = 452, normalized size = 4.15 \begin{align*} \frac{8 \, a d^{4} e^{3} x^{4} + 32 \, a c d^{3} e^{3} x^{3} + 48 \, a c^{2} d^{2} e^{3} x^{2} + 32 \, a c^{3} d e^{3} x +{\left (8 \, b d^{4} e^{3} x^{4} + 32 \, b c d^{3} e^{3} x^{3} + 48 \, b c^{2} d^{2} e^{3} x^{2} + 32 \, b c^{3} d e^{3} x +{\left (8 \, b c^{4} - 3 \, b\right )} e^{3}\right )} \arcsin \left (d x + c\right ) +{\left (2 \, b d^{3} e^{3} x^{3} + 6 \, b c d^{2} e^{3} x^{2} + 3 \,{\left (2 \, b c^{2} + b\right )} d e^{3} x +{\left (2 \, b c^{3} + 3 \, b c\right )} e^{3}\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.81556, size = 394, normalized size = 3.61 \begin{align*} \begin{cases} a c^{3} e^{3} x + \frac{3 a c^{2} d e^{3} x^{2}}{2} + a c d^{2} e^{3} x^{3} + \frac{a d^{3} e^{3} x^{4}}{4} + \frac{b c^{4} e^{3} \operatorname{asin}{\left (c + d x \right )}}{4 d} + b c^{3} e^{3} x \operatorname{asin}{\left (c + d x \right )} + \frac{b c^{3} e^{3} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{16 d} + \frac{3 b c^{2} d e^{3} x^{2} \operatorname{asin}{\left (c + d x \right )}}{2} + \frac{3 b c^{2} e^{3} x \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{16} + b c d^{2} e^{3} x^{3} \operatorname{asin}{\left (c + d x \right )} + \frac{3 b c d e^{3} x^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{16} + \frac{3 b c e^{3} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{32 d} + \frac{b d^{3} e^{3} x^{4} \operatorname{asin}{\left (c + d x \right )}}{4} + \frac{b d^{2} e^{3} x^{3} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{16} + \frac{3 b e^{3} x \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{32} - \frac{3 b e^{3} \operatorname{asin}{\left (c + d x \right )}}{32 d} & \text{for}\: d \neq 0 \\c^{3} e^{3} x \left (a + b \operatorname{asin}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.39028, size = 204, normalized size = 1.87 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b \arcsin \left (d x + c\right ) e^{3}}{4 \, d} - \frac{{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac{3}{2}}{\left (d x + c\right )} b e^{3}}{16 \, d} + \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a e^{3}}{4 \, d} + \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )} b \arcsin \left (d x + c\right ) e^{3}}{2 \, d} + \frac{5 \, \sqrt{-{\left (d x + c\right )}^{2} + 1}{\left (d x + c\right )} b e^{3}}{32 \, d} + \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )} a e^{3}}{2 \, d} + \frac{5 \, b \arcsin \left (d x + c\right ) e^{3}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]