Optimal. Leaf size=148 \[ \frac {2 b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 c^2 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4795, 4767, 8,
30} \begin {gather*} -\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 c^2 d}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 c^4 d}+\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4767
Rule 4795
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx &=-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac {2 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^2}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt {d-c^2 d x^2}}\\ &=\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 92, normalized size = 0.62 \begin {gather*} \frac {b c x \sqrt {1-c^2 x^2} \left (6+c^2 x^2\right )+3 a \left (-2+c^2 x^2+c^4 x^4\right )+3 b \left (-2+c^2 x^2+c^4 x^4\right ) \text {ArcSin}(c x)}{9 c^4 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.26, size = 407, normalized size = 2.75
method | result | size |
default | \(a \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (\arcsin \left (c x \right )+i\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(407\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 121, normalized size = 0.82 \begin {gather*} -\frac {1}{3} \, b {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} + \frac {{\left (c^{2} x^{3} + 6 \, x\right )} b}{9 \, c^{3} \sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.70, size = 120, normalized size = 0.81 \begin {gather*} -\frac {{\left (b c^{3} x^{3} + 6 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 3 \, {\left (a c^{4} x^{4} + a c^{2} x^{2} + {\left (b c^{4} x^{4} + b c^{2} x^{2} - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{9 \, {\left (c^{6} d x^{2} - c^{4} d\right )}} \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^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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