Optimal. Leaf size=179 \[ \frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{4 e}+\frac{b \sqrt{1-c^2 x^2} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{96 c^3}-\frac{b \left (24 c^2 d^2 e^2+8 c^4 d^4+3 e^4\right ) \sin ^{-1}(c x)}{32 c^4 e}+\frac{b \sqrt{1-c^2 x^2} (d+e x)^3}{16 c}+\frac{7 b d \sqrt{1-c^2 x^2} (d+e x)^2}{48 c} \]
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Rubi [A] time = 0.182325, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {4743, 743, 833, 780, 216} \[ \frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{4 e}+\frac{b \sqrt{1-c^2 x^2} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{96 c^3}-\frac{b \left (24 c^2 d^2 e^2+8 c^4 d^4+3 e^4\right ) \sin ^{-1}(c x)}{32 c^4 e}+\frac{b \sqrt{1-c^2 x^2} (d+e x)^3}{16 c}+\frac{7 b d \sqrt{1-c^2 x^2} (d+e x)^2}{48 c} \]
Antiderivative was successfully verified.
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Rule 4743
Rule 743
Rule 833
Rule 780
Rule 216
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{4 e}-\frac{(b c) \int \frac{(d+e x)^4}{\sqrt{1-c^2 x^2}} \, dx}{4 e}\\ &=\frac{b (d+e x)^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{4 e}+\frac{b \int \frac{(d+e x)^2 \left (-4 c^2 d^2-3 e^2-7 c^2 d e x\right )}{\sqrt{1-c^2 x^2}} \, dx}{16 c e}\\ &=\frac{7 b d (d+e x)^2 \sqrt{1-c^2 x^2}}{48 c}+\frac{b (d+e x)^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{4 e}-\frac{b \int \frac{(d+e x) \left (c^2 d \left (12 c^2 d^2+23 e^2\right )+c^2 e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt{1-c^2 x^2}} \, dx}{48 c^3 e}\\ &=\frac{7 b d (d+e x)^2 \sqrt{1-c^2 x^2}}{48 c}+\frac{b (d+e x)^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right ) \sqrt{1-c^2 x^2}}{96 c^3}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{4 e}-\frac{\left (b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3 e}\\ &=\frac{7 b d (d+e x)^2 \sqrt{1-c^2 x^2}}{48 c}+\frac{b (d+e x)^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right ) \sqrt{1-c^2 x^2}}{96 c^3}-\frac{b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \sin ^{-1}(c x)}{32 c^4 e}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{4 e}\\ \end{align*}
Mathematica [A] time = 0.1484, size = 165, normalized size = 0.92 \[ \frac{24 a c^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+b c \sqrt{1-c^2 x^2} \left (c^2 \left (72 d^2 e x+96 d^3+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )+3 b \sin ^{-1}(c x) \left (8 c^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )}{96 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 265, normalized size = 1.5 \begin{align*}{\frac{1}{c} \left ({\frac{ \left ( ecx+dc \right ) ^{4}a}{4\,{c}^{3}e}}+{\frac{b}{{c}^{3}} \left ({\frac{{e}^{3}\arcsin \left ( cx \right ){c}^{4}{x}^{4}}{4}}+{e}^{2}\arcsin \left ( cx \right ){c}^{4}{x}^{3}d+{\frac{3\,e\arcsin \left ( cx \right ){c}^{4}{x}^{2}{d}^{2}}{2}}+\arcsin \left ( cx \right ){c}^{4}x{d}^{3}+{\frac{{c}^{4}{d}^{4}\arcsin \left ( cx \right ) }{4\,e}}-{\frac{1}{4\,e} \left ({e}^{4} \left ( -{\frac{{c}^{3}{x}^{3}}{4}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,cx}{8}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,\arcsin \left ( cx \right ) }{8}} \right ) +4\,dc{e}^{3} \left ( -1/3\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}-2/3\,\sqrt{-{c}^{2}{x}^{2}+1} \right ) +6\,{c}^{2}{d}^{2}{e}^{2} \left ( -1/2\,cx\sqrt{-{c}^{2}{x}^{2}+1}+1/2\,\arcsin \left ( cx \right ) \right ) -4\,{c}^{3}{d}^{3}e\sqrt{-{c}^{2}{x}^{2}+1}+{c}^{4}{d}^{4}\arcsin \left ( cx \right ) \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47708, size = 344, normalized size = 1.92 \begin{align*} \frac{1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac{3}{2} \, a d^{2} e x^{2} + \frac{3}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d^{2} e + \frac{1}{3} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac{1}{32} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b e^{3} + a d^{3} x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d^{3}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.47397, size = 447, normalized size = 2.5 \begin{align*} \frac{24 \, a c^{4} e^{3} x^{4} + 96 \, a c^{4} d e^{2} x^{3} + 144 \, a c^{4} d^{2} e x^{2} + 96 \, a c^{4} d^{3} x + 3 \,{\left (8 \, b c^{4} e^{3} x^{4} + 32 \, b c^{4} d e^{2} x^{3} + 48 \, b c^{4} d^{2} e x^{2} + 32 \, b c^{4} d^{3} x - 24 \, b c^{2} d^{2} e - 3 \, b e^{3}\right )} \arcsin \left (c x\right ) +{\left (6 \, b c^{3} e^{3} x^{3} + 32 \, b c^{3} d e^{2} x^{2} + 96 \, b c^{3} d^{3} + 64 \, b c d e^{2} + 9 \,{\left (8 \, b c^{3} d^{2} e + b c e^{3}\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{96 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.85828, size = 316, normalized size = 1.77 \begin{align*} \begin{cases} a d^{3} x + \frac{3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac{a e^{3} x^{4}}{4} + b d^{3} x \operatorname{asin}{\left (c x \right )} + \frac{3 b d^{2} e x^{2} \operatorname{asin}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname{asin}{\left (c x \right )} + \frac{b e^{3} x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b d^{3} \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{3 b d^{2} e x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{b d e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{3 c} + \frac{b e^{3} x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} - \frac{3 b d^{2} e \operatorname{asin}{\left (c x \right )}}{4 c^{2}} + \frac{2 b d e^{2} \sqrt{- c^{2} x^{2} + 1}}{3 c^{3}} + \frac{3 b e^{3} x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} - \frac{3 b e^{3} \operatorname{asin}{\left (c x \right )}}{32 c^{4}} & \text{for}\: c \neq 0 \\a \left (d^{3} x + \frac{3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac{e^{3} x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24641, size = 456, normalized size = 2.55 \begin{align*} b d^{3} x \arcsin \left (c x\right ) + a d x^{3} e^{2} + a d^{3} x + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} b d^{2} x e}{4 \, c} + \frac{{\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) e^{2}}{c^{2}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )} b d^{2} \arcsin \left (c x\right ) e}{2 \, c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d^{3}}{c} + \frac{b d x \arcsin \left (c x\right ) e^{2}}{c^{2}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )} a d^{2} e}{2 \, c^{2}} + \frac{3 \, b d^{2} \arcsin \left (c x\right ) e}{4 \, c^{2}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b x e^{3}}{16 \, c^{3}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d e^{2}}{3 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b \arcsin \left (c x\right ) e^{3}}{4 \, c^{4}} + \frac{5 \, \sqrt{-c^{2} x^{2} + 1} b x e^{3}}{32 \, c^{3}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d e^{2}}{c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a e^{3}}{4 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )} b \arcsin \left (c x\right ) e^{3}}{2 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )} a e^{3}}{2 \, c^{4}} + \frac{5 \, b \arcsin \left (c x\right ) e^{3}}{32 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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