Optimal. Leaf size=125 \[ -\frac {b}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {ArcSin}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4771, 272, 45}
\begin {gather*} \frac {x^3 (a+b \text {ArcSin}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b}{6 c^3 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 )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4771
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \left (-1+c^2 x\right )^2}+\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 103, normalized size = 0.82 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (2 a c^3 x^3-b \sqrt {1-c^2 x^2}+2 b c^3 x^3 \text {ArcSin}(c x)-b \left (1-c^2 x^2\right )^{3/2} \log \left (-1+c^2 x^2\right )\right )}{6 c^3 d^3 \left (-1+c^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 1242, normalized size = 9.94
method | result | size |
default | \(a \left (\frac {x}{2 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}}{2 c^{2}}\right )+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-c^{2} x^{2}+1\right ) x^{3}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \left (-c^{2} x^{2}+1\right ) x^{5}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} \arcsin \left (c x \right ) x^{7}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{6}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} x^{5}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \sqrt {-c^{2} x^{2}+1}\, x^{4}}{2 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} \arcsin \left (c x \right ) x^{5}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c^{3}}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{3 c^{3} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4}}{d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x^{2}}{2 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{3}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} x^{7}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right )}+\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2}}{3 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{6 d^{3} \left (3 c^{8} x^{8}-9 c^{6} x^{6}+10 c^{4} x^{4}-5 c^{2} x^{2}+1\right ) c^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{3} d^{3} \left (c^{2} x^{2}-1\right )}\) | \(1242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 153, normalized size = 1.22 \begin {gather*} \frac {1}{6} \, b c {\left (\frac {1}{c^{6} d^{\frac {5}{2}} x^{2} - c^{4} d^{\frac {5}{2}}} - \frac {\log \left (c x + 1\right )}{c^{4} d^{\frac {5}{2}}} - \frac {\log \left (c x - 1\right )}{c^{4} d^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {x}{\sqrt {-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac {x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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