Optimal. Leaf size=382 \[ -\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{32 b^2 d^2 \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{12 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2} \]
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Rubi [A] time = 0.292748, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4677, 194, 4645, 12, 1799, 1850} \[ -\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{32 b^2 d^2 \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{12 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 194
Rule 4645
Rule 12
Rule 1799
Rule 1850
Rubi steps
\begin{align*} \int x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}+\frac{\left (2 b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{7 c \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt{1-c^2 x^2}} \, dx}{7 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac{\left (2 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt{1-c^2 x^2}} \, dx}{245 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac{\left (b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}-\frac{\left (b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{16}{\sqrt{1-c^2 x}}+8 \sqrt{1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{245 \sqrt{1-c^2 x^2}}\\ &=\frac{32 b^2 d^2 \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{735 c^2}+\frac{12 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{7 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )^2}{7 c^2 d}\\ \end{align*}
Mathematica [A] time = 0.324334, size = 216, normalized size = 0.57 \[ -\frac{d^2 \sqrt{d-c^2 d x^2} \left (3675 a^2 \left (1-c^2 x^2\right )^{7/2}+210 a b c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+210 b \sin ^{-1}(c x) \left (35 a \left (1-c^2 x^2\right )^{7/2}+b c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )\right )+2 b^2 \left (75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right ) \sqrt{1-c^2 x^2}+3675 b^2 \left (1-c^2 x^2\right )^{7/2} \sin ^{-1}(c x)^2\right )}{25725 c^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.402, size = 1888, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6254, size = 379, normalized size = 0.99 \begin{align*} -\frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} b^{2} \arcsin \left (c x\right )^{2}}{7 \, c^{2} d} - \frac{2 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} a b \arcsin \left (c x\right )}{7 \, c^{2} d} - \frac{2}{25725} \, b^{2}{\left (\frac{75 \, \sqrt{-c^{2} x^{2} + 1} c^{4} d^{\frac{7}{2}} x^{6} - 351 \, \sqrt{-c^{2} x^{2} + 1} c^{2} d^{\frac{7}{2}} x^{4} + 757 \, \sqrt{-c^{2} x^{2} + 1} d^{\frac{7}{2}} x^{2} - \frac{2161 \, \sqrt{-c^{2} x^{2} + 1} d^{\frac{7}{2}}}{c^{2}}}{d} + \frac{105 \,{\left (5 \, c^{6} d^{\frac{7}{2}} x^{7} - 21 \, c^{4} d^{\frac{7}{2}} x^{5} + 35 \, c^{2} d^{\frac{7}{2}} x^{3} - 35 \, d^{\frac{7}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} a^{2}}{7 \, c^{2} d} - \frac{2 \,{\left (5 \, c^{6} d^{\frac{7}{2}} x^{7} - 21 \, c^{4} d^{\frac{7}{2}} x^{5} + 35 \, c^{2} d^{\frac{7}{2}} x^{3} - 35 \, d^{\frac{7}{2}} x\right )} a b}{245 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97908, size = 888, normalized size = 2.32 \begin{align*} \frac{210 \,{\left (5 \, a b c^{7} d^{2} x^{7} - 21 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3} - 35 \, a b c d^{2} x +{\left (5 \, b^{2} c^{7} d^{2} x^{7} - 21 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3} - 35 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} +{\left (75 \,{\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{8} d^{2} x^{8} - 12 \,{\left (1225 \, a^{2} - 71 \, b^{2}\right )} c^{6} d^{2} x^{6} + 2 \,{\left (11025 \, a^{2} - 1108 \, b^{2}\right )} c^{4} d^{2} x^{4} - 4 \,{\left (3675 \, a^{2} - 1459 \, b^{2}\right )} c^{2} d^{2} x^{2} +{\left (3675 \, a^{2} - 4322 \, b^{2}\right )} d^{2} + 3675 \,{\left (b^{2} c^{8} d^{2} x^{8} - 4 \, b^{2} c^{6} d^{2} x^{6} + 6 \, b^{2} c^{4} d^{2} x^{4} - 4 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 7350 \,{\left (a b c^{8} d^{2} x^{8} - 4 \, a b c^{6} d^{2} x^{6} + 6 \, a b c^{4} d^{2} x^{4} - 4 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{25725 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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