Optimal. Leaf size=154 \[ \frac{2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{b}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.078979, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4655, 4653, 260, 261} \[ \frac{2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{b}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4655
Rule 4653
Rule 260
Rule 261
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{2 \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b c \sqrt{1-c^2 x^2}\right ) \int \frac{x}{1-c^2 x^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{x \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{2 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.238211, size = 113, normalized size = 0.73 \[ -\frac{\sqrt{d-c^2 d x^2} \left (4 a c^3 x^3-6 a c x+b \sqrt{1-c^2 x^2}-2 b \left (1-c^2 x^2\right )^{3/2} \log \left (c^2 x^2-1\right )+2 b c x \left (2 c^2 x^2-3\right ) \sin ^{-1}(c x)\right )}{6 c d^3 \left (c^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.125, size = 1071, normalized size = 7. \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 = 1.76874, size = 190, normalized size = 1.23 \begin{align*} \frac{1}{6} \, b c{\left (\frac{1}{c^{4} d^{\frac{5}{2}} x^{2} - c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac{5}{2}}}\right )} + \frac{1}{3} \, b{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac{1}{3} \, a{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asin}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{5}{2}}}\, dx \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{b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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