Optimal. Leaf size=301 \[ -\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^3}+\frac{2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^2}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{1-c^2 x^2}}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{1-c^2 x^2}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{1-c^2 x^2}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.237526, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {266, 43, 4691, 12, 1153} \[ -\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^3}+\frac{2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^2}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{1-c^2 x^2}}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{1-c^2 x^2}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{1-c^2 x^2}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{1-c^2 x^2}}+\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 4691
Rule 12
Rule 1153
Rubi steps
\begin{align*} \int x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right )}{315 c^6} \, dx}{\sqrt{1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^5 \left (d-c^2 d x^2\right )^{3/2} \, dx\\ &=-\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right ) \, dx}{315 c^5 \sqrt{1-c^2 x^2}}+\frac{1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname{Subst}\left (\int x^2 \left (d-c^2 d x\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+50 c^6 x^6-35 c^8 x^8\right ) \, dx}{315 c^5 \sqrt{1-c^2 x^2}}+\frac{1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname{Subst}\left (\int \left (\frac{\left (d-c^2 d x\right )^{3/2}}{c^4}-\frac{2 \left (d-c^2 d x\right )^{5/2}}{c^4 d}+\frac{\left (d-c^2 d x\right )^{7/2}}{c^4 d^2}\right ) \, dx,x,x^2\right )\\ &=\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{1-c^2 x^2}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{1-c^2 x^2}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{1-c^2 x^2}}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d}+\frac{2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^2}-\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^3}\\ \end{align*}
Mathematica [A] time = 0.15775, size = 150, normalized size = 0.5 \[ \frac{d \sqrt{d-c^2 d x^2} \left (-315 a \left (35 c^4 x^4+20 c^2 x^2+8\right ) \left (1-c^2 x^2\right )^{5/2}+b c x \left (1225 c^8 x^8-2250 c^6 x^6+189 c^4 x^4+420 c^2 x^2+2520\right )-315 b \left (35 c^4 x^4+20 c^2 x^2+8\right ) \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)\right )}{99225 c^6 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.41, size = 1327, normalized size = 4.4 \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 = 1.91501, size = 513, normalized size = 1.7 \begin{align*} -\frac{{\left (1225 \, b c^{9} d x^{9} - 2250 \, b c^{7} d x^{7} + 189 \, b c^{5} d x^{5} + 420 \, b c^{3} d x^{3} + 2520 \, b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 315 \,{\left (35 \, a c^{10} d x^{10} - 85 \, a c^{8} d x^{8} + 53 \, a c^{6} d x^{6} + a c^{4} d x^{4} + 4 \, a c^{2} d x^{2} - 8 \, a d +{\left (35 \, b c^{10} d x^{10} - 85 \, b c^{8} d x^{8} + 53 \, b c^{6} d x^{6} + b c^{4} d x^{4} + 4 \, b c^{2} d x^{2} - 8 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{99225 \,{\left (c^{8} x^{2} - c^{6}\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{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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