Optimal. Leaf size=123 \[ -\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{36} b c d x^5 \sqrt{1-c^2 x^2}+\frac{b d x^3 \sqrt{1-c^2 x^2}}{36 c}+\frac{b d x \sqrt{1-c^2 x^2}}{24 c^3}-\frac{b d \sin ^{-1}(c x)}{24 c^4} \]
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Rubi [A] time = 0.0959112, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {14, 4687, 12, 459, 321, 216} \[ -\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{36} b c d x^5 \sqrt{1-c^2 x^2}+\frac{b d x^3 \sqrt{1-c^2 x^2}}{36 c}+\frac{b d x \sqrt{1-c^2 x^2}}{24 c^3}-\frac{b d \sin ^{-1}(c x)}{24 c^4} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4687
Rule 12
Rule 459
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x^3 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{d x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{12} (b c d) \int \frac{x^4 \left (3-2 c^2 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{1}{36} b c d x^5 \sqrt{1-c^2 x^2}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{9} (b c d) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d x^3 \sqrt{1-c^2 x^2}}{36 c}-\frac{1}{36} b c d x^5 \sqrt{1-c^2 x^2}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{(b d) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{12 c}\\ &=\frac{b d x \sqrt{1-c^2 x^2}}{24 c^3}+\frac{b d x^3 \sqrt{1-c^2 x^2}}{36 c}-\frac{1}{36} b c d x^5 \sqrt{1-c^2 x^2}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{(b d) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{24 c^3}\\ &=\frac{b d x \sqrt{1-c^2 x^2}}{24 c^3}+\frac{b d x^3 \sqrt{1-c^2 x^2}}{36 c}-\frac{1}{36} b c d x^5 \sqrt{1-c^2 x^2}-\frac{b d \sin ^{-1}(c x)}{24 c^4}+\frac{1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0903986, size = 89, normalized size = 0.72 \[ \frac{d \left (-6 a c^4 x^4 \left (2 c^2 x^2-3\right )+b c x \sqrt{1-c^2 x^2} \left (-2 c^4 x^4+2 c^2 x^2+3\right )-3 b \left (4 c^6 x^6-6 c^4 x^4+1\right ) \sin ^{-1}(c x)\right )}{72 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 118, normalized size = 1. \begin{align*}{\frac{1}{{c}^{4}} \left ( -da \left ({\frac{{c}^{6}{x}^{6}}{6}}-{\frac{{c}^{4}{x}^{4}}{4}} \right ) -db \left ({\frac{\arcsin \left ( cx \right ){c}^{6}{x}^{6}}{6}}-{\frac{{c}^{4}{x}^{4}\arcsin \left ( cx \right ) }{4}}+{\frac{{c}^{5}{x}^{5}}{36}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{c}^{3}{x}^{3}}{36}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{cx}{24}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{24}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52922, size = 261, normalized size = 2.12 \begin{align*} -\frac{1}{6} \, a c^{2} d x^{6} + \frac{1}{4} \, a d x^{4} - \frac{1}{288} \,{\left (48 \, x^{6} \arcsin \left (c x\right ) +{\left (\frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{6}} - \frac{15 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b c^{2} d + \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 d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0412, size = 221, normalized size = 1.8 \begin{align*} -\frac{12 \, a c^{6} d x^{6} - 18 \, a c^{4} d x^{4} + 3 \,{\left (4 \, b c^{6} d x^{6} - 6 \, b c^{4} d x^{4} + b d\right )} \arcsin \left (c x\right ) +{\left (2 \, b c^{5} d x^{5} - 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt{-c^{2} x^{2} + 1}}{72 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.7657, size = 138, normalized size = 1.12 \begin{align*} \begin{cases} - \frac{a c^{2} d x^{6}}{6} + \frac{a d x^{4}}{4} - \frac{b c^{2} d x^{6} \operatorname{asin}{\left (c x \right )}}{6} - \frac{b c d x^{5} \sqrt{- c^{2} x^{2} + 1}}{36} + \frac{b d x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b d x^{3} \sqrt{- c^{2} x^{2} + 1}}{36 c} + \frac{b d x \sqrt{- c^{2} x^{2} + 1}}{24 c^{3}} - \frac{b d \operatorname{asin}{\left (c x \right )}}{24 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37521, size = 220, normalized size = 1.79 \begin{align*} -\frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d x}{36 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b d \arcsin \left (c x\right )}{6 \, c^{4}} + \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d x}{36 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a d}{6 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right )}{4 \, c^{4}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d x}{24 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a d}{4 \, c^{4}} + \frac{b d \arcsin \left (c x\right )}{24 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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