Optimal. Leaf size=183 \[ \frac {2 b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {1-c^2 x^2}}-\frac {b c x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{5 c^4 d^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {272, 45, 4779,
12} \begin {gather*} \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{5 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{3 c^4 d}-\frac {b c x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {b x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {1-c^2 x^2}}+\frac {2 b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 4779
Rubi steps
\begin {align*} \int x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^3 \sqrt {d-c^2 d x^2} \, dx\\ &=-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \text {Subst}\left (\int x \sqrt {d-c^2 d x} \, dx,x,x^2\right )\\ &=\frac {2 b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {1-c^2 x^2}}-\frac {b c x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \text {Subst}\left (\int \left (\frac {\sqrt {d-c^2 d x}}{c^2}-\frac {\left (d-c^2 d x\right )^{3/2}}{c^2 d}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b x \sqrt {d-c^2 d x^2}}{15 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^3 \sqrt {d-c^2 d x^2}}{45 c \sqrt {1-c^2 x^2}}-\frac {b c x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 134, normalized size = 0.73 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (15 a \sqrt {1-c^2 x^2} \left (-2-c^2 x^2+3 c^4 x^4\right )+b \left (30 c x+5 c^3 x^3-9 c^5 x^5\right )+15 b \sqrt {1-c^2 x^2} \left (-2-c^2 x^2+3 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{225 c^4 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.23, size = 544, normalized size = 2.97
method | result | size |
default | \(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{5 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{15 d \,c^{4}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}-16 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+13 c^{2} x^{2}+20 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-5 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (i+5 \arcsin \left (c x \right )\right )}{800 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (\arcsin \left (c x \right )+i\right )}{16 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{16 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{288 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (17 i+15 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{3600 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i x^{2} c^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (2 i+15 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right )}{900 c^{4} \left (c^{2} x^{2}-1\right )}\right )\) | \(544\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 138, normalized size = 0.75 \begin {gather*} -\frac {1}{15} \, b {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \arcsin \left (c x\right ) - \frac {1}{15} \, a {\left (\frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} - \frac {{\left (9 \, c^{4} \sqrt {d} x^{5} - 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} b}{225 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.91, size = 150, normalized size = 0.82 \begin {gather*} \frac {{\left (9 \, b c^{5} x^{5} - 5 \, b c^{3} x^{3} - 30 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 15 \, {\left (3 \, a c^{6} x^{6} - 4 \, a c^{4} x^{4} - a c^{2} x^{2} + {\left (3 \, b c^{6} x^{6} - 4 \, 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}}{225 \, {\left (c^{6} x^{2} - c^{4}\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 x^{3} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\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 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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