Optimal. Leaf size=221 \[ -\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^6 d^2}+\frac {a+b \sin ^{-1}(c x)}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{c^6 d^2 \sqrt {1-c^2 x^2}}-\frac {5 b x \sqrt {d-c^2 d x^2}}{3 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b x^3 \sqrt {d-c^2 d x^2}}{9 c^3 d^2 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.29, antiderivative size = 229, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {4703, 4707, 4677, 8, 30, 302, 206} \[ \frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^2}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c^3 d \sqrt {d-c^2 d x^2}}-\frac {5 b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{c^6 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 206
Rule 302
Rule 4677
Rule 4703
Rule 4707
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx}{c^2 d}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {x^4}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2}-\frac {8 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^4 d}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int x^2 \, dx}{3 c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {b x \sqrt {1-c^2 x^2}}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{3 c^5 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {5 b x \sqrt {1-c^2 x^2}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{c^6 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 166, normalized size = 0.75 \[ \frac {\sqrt {d-c^2 d x^2} \left (\sqrt {-c^2} \left (3 a \left (c^4 x^4+4 c^2 x^2-8\right )+b c x \sqrt {1-c^2 x^2} \left (c^2 x^2+15\right )+3 b \left (c^4 x^4+4 c^2 x^2-8\right ) \sin ^{-1}(c x)\right )-9 i b c \sqrt {1-c^2 x^2} F\left (\left .i \sinh ^{-1}\left (\sqrt {-c^2} x\right )\right |1\right )\right )}{9 c^6 \sqrt {-c^2} d^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 441, normalized size = 2.00 \[ \left [\frac {9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 4 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 12 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \arcsin \left (c x\right ) - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{36 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}, -\frac {9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 6 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \arcsin \left (c x\right ) - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{18 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 419, normalized size = 1.90 \[ -\frac {a \,x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {4 a \,x^{2}}{3 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8 a}{3 c^{6} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{3 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {31 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{18 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {65 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{24 d^{2} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{72 d^{2} c^{6} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a {\left (\frac {x^{4}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} + \frac {4 \, x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{4} d} - \frac {8}{\sqrt {-c^{2} d x^{2} + d} c^{6} d}\right )} - \frac {\frac {1}{30} \, {\left (\sqrt {c x + 1} \sqrt {-c x + 1} c^{6} d^{2} {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 25 \, c^{2} x^{3} - 45 \, x\right )}}{c^{5} d^{2}} + \frac {45 \, \log \left (c x + 1\right )}{c^{6} d^{2}} - \frac {45 \, \log \left (c x - 1\right )}{c^{6} d^{2}}\right )} + 30 \, {\left (c^{4} x^{4} + 4 \, c^{2} x^{2} - 8\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} b}{3 \, \sqrt {c x + 1} \sqrt {-c x + 1} c^{6} d^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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