Optimal. Leaf size=202 \[ -\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2 d}-\frac{b c^5 d^2 x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}-\frac{b c d^2 x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}+\frac{b d^2 x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.0907841, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4677, 194} \[ -\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2 d}-\frac{b c^5 d^2 x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}-\frac{b c d^2 x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}+\frac{b d^2 x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 194
Rubi steps
\begin{align*} \int x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^2 d}+\frac{\left (b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \, dx}{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 )}{7 c^2 d}+\frac{\left (b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-3 c^2 x^2+3 c^4 x^4-c^6 x^6\right ) \, dx}{7 c \sqrt{1-c^2 x^2}}\\ &=\frac{b d^2 x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}-\frac{b c d^2 x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^7 \sqrt{d-c^2 d x^2}}{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 )}{7 c^2 d}\\ \end{align*}
Mathematica [A] time = 0.0694342, size = 93, normalized size = 0.46 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (\left (c^2 x^2-1\right )^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{b c \left (-\frac{1}{7} c^6 x^7+\frac{3 c^4 x^5}{5}-c^2 x^3+x\right )}{\sqrt{1-c^2 x^2}}\right )}{7 c^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.223, size = 921, normalized size = 4.6 \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 = 1.6537, size = 132, normalized size = 0.65 \begin{align*} -\frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} b \arcsin \left (c x\right )}{7 \, c^{2} d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{7}{2}} a}{7 \, c^{2} d} - \frac{{\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 )} b}{245 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25337, size = 448, normalized size = 2.22 \begin{align*} \frac{{\left (5 \, b c^{7} d^{2} x^{7} - 21 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3} - 35 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 35 \,{\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{245 \,{\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 )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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