Optimal. Leaf size=400 \[ \frac{16 a b x \sqrt{1-c^2 x^2}}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{2 b x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{8 b x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}+\frac{2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt{d-c^2 d x^2}}-\frac{76 b^2 \left (1-c^2 x^2\right )^2}{675 c^6 \sqrt{d-c^2 d x^2}}+\frac{298 b^2 \left (1-c^2 x^2\right )}{225 c^6 \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.583019, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {4707, 4677, 4619, 261, 4627, 266, 43} \[ \frac{16 a b x \sqrt{1-c^2 x^2}}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{2 b x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{8 b x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}+\frac{2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt{d-c^2 d x^2}}-\frac{76 b^2 \left (1-c^2 x^2\right )^2}{675 c^6 \sqrt{d-c^2 d x^2}}+\frac{298 b^2 \left (1-c^2 x^2\right )}{225 c^6 \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4707
Rule 4677
Rule 4619
Rule 261
Rule 4627
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx &=-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{4 \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx}{5 c^2}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac{8 \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx}{15 c^4}-\frac{\left (2 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^5}{\sqrt{1-c^2 x^2}} \, dx}{25 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b \sqrt{1-c^2 x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt{d-c^2 d x^2}}\\ &=\frac{8 b x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}+\frac{2 b x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{25 \sqrt{d-c^2 d x^2}}+\frac{\left (16 b \sqrt{1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{\left (8 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2}} \, dx}{45 c^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{1-c^2 x^2}}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{8 b x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}+\frac{2 b x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^4 \sqrt{1-c^2 x}}-\frac{2 \sqrt{1-c^2 x}}{c^4}+\frac{\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{25 \sqrt{d-c^2 d x^2}}+\frac{\left (16 b^2 \sqrt{1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{45 c^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{1-c^2 x^2}}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left (1-c^2 x^2\right )}{25 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 b^2 \left (1-c^2 x^2\right )^2}{75 c^6 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{8 b x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}+\frac{2 b x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{\left (16 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \sqrt{1-c^2 x}}-\frac{\sqrt{1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{45 c^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{1-c^2 x^2}}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{298 b^2 \left (1-c^2 x^2\right )}{225 c^6 \sqrt{d-c^2 d x^2}}-\frac{76 b^2 \left (1-c^2 x^2\right )^2}{675 c^6 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{8 b x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}+\frac{2 b x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}\\ \end{align*}
Mathematica [A] time = 0.168043, size = 230, normalized size = 0.57 \[ \frac{225 a^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )+30 a b c x \sqrt{1-c^2 x^2} \left (9 c^4 x^4+20 c^2 x^2+120\right )+30 b \sin ^{-1}(c x) \left (15 a \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )+b c x \sqrt{1-c^2 x^2} \left (9 c^4 x^4+20 c^2 x^2+120\right )\right )-2 b^2 \left (27 c^6 x^6+109 c^4 x^4+1936 c^2 x^2-2072\right )+225 b^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right ) \sin ^{-1}(c x)^2}{3375 c^6 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.597, size = 1304, normalized size = 3.3 \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 [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.
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Fricas [A] time = 2.00462, size = 628, normalized size = 1.57 \begin{align*} -\frac{30 \,{\left (9 \, a b c^{5} x^{5} + 20 \, a b c^{3} x^{3} + 120 \, a b c x +{\left (9 \, b^{2} c^{5} x^{5} + 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} +{\left (27 \,{\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} x^{6} +{\left (225 \, a^{2} - 218 \, b^{2}\right )} c^{4} x^{4} + 4 \,{\left (225 \, a^{2} - 968 \, b^{2}\right )} c^{2} x^{2} + 225 \,{\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, b^{2}\right )} \arcsin \left (c x\right )^{2} - 1800 \, a^{2} + 4144 \, b^{2} + 450 \,{\left (3 \, a b c^{6} x^{6} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, a b\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{3375 \,{\left (c^{8} d x^{2} - c^{6} d\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 \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{5}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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