Optimal. Leaf size=268 \[ \frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac {35 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac {35 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{1024 c^2}+\frac {35 b^2 c^2 d^3 x^4}{3072}+\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}-\frac {175 b^2 d^3 x^2}{3072} \]
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Rubi [A] time = 0.25, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4677, 4649, 4647, 4641, 30, 14, 261} \[ \frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac {35 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac {35 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{1024 c^2}+\frac {35 b^2 c^2 d^3 x^4}{3072}+\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}-\frac {175 b^2 d^3 x^2}{3072} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 261
Rule 4641
Rule 4647
Rule 4649
Rule 4677
Rubi steps
\begin {align*} \int x \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac {\left (b d^3\right ) \int \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 c}\\ &=\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {1}{32} \left (b^2 d^3\right ) \int x \left (1-c^2 x^2\right )^3 \, dx+\frac {\left (7 b d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{32 c}\\ &=\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {1}{192} \left (7 b^2 d^3\right ) \int x \left (1-c^2 x^2\right )^2 \, dx+\frac {\left (35 b d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{192 c}\\ &=\frac {7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {1}{768} \left (35 b^2 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac {\left (35 b d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{256 c}\\ &=\frac {7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {35 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}-\frac {1}{768} \left (35 b^2 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac {1}{512} \left (35 b^2 d^3\right ) \int x \, dx+\frac {\left (35 b d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{512 c}\\ &=-\frac {175 b^2 d^3 x^2}{3072}+\frac {35 b^2 c^2 d^3 x^4}{3072}+\frac {7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {35 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{512 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{768 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{192 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{32 c}+\frac {35 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 257, normalized size = 0.96 \[ -\frac {d^3 \left (c x \left (1152 a^2 c x \left (c^6 x^6-4 c^4 x^4+6 c^2 x^2-4\right )+6 a b \sqrt {1-c^2 x^2} \left (48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right )+b^2 c x \left (-36 c^6 x^6+200 c^4 x^4-489 c^2 x^2+837\right )\right )+6 b \sin ^{-1}(c x) \left (3 a \left (128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right )+b c x \sqrt {1-c^2 x^2} \left (48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right )\right )+9 b^2 \left (128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right ) \sin ^{-1}(c x)^2\right )}{9216 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 354, normalized size = 1.32 \[ -\frac {36 \, {\left (32 \, a^{2} - b^{2}\right )} c^{8} d^{3} x^{8} - 8 \, {\left (576 \, a^{2} - 25 \, b^{2}\right )} c^{6} d^{3} x^{6} + 3 \, {\left (2304 \, a^{2} - 163 \, b^{2}\right )} c^{4} d^{3} x^{4} - 9 \, {\left (512 \, a^{2} - 93 \, b^{2}\right )} c^{2} d^{3} x^{2} + 9 \, {\left (128 \, b^{2} c^{8} d^{3} x^{8} - 512 \, b^{2} c^{6} d^{3} x^{6} + 768 \, b^{2} c^{4} d^{3} x^{4} - 512 \, b^{2} c^{2} d^{3} x^{2} + 93 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (128 \, a b c^{8} d^{3} x^{8} - 512 \, a b c^{6} d^{3} x^{6} + 768 \, a b c^{4} d^{3} x^{4} - 512 \, a b c^{2} d^{3} x^{2} + 93 \, a b d^{3}\right )} \arcsin \left (c x\right ) + 6 \, {\left (48 \, a b c^{7} d^{3} x^{7} - 200 \, a b c^{5} d^{3} x^{5} + 326 \, a b c^{3} d^{3} x^{3} - 279 \, a b c d^{3} x + {\left (48 \, b^{2} c^{7} d^{3} x^{7} - 200 \, b^{2} c^{5} d^{3} x^{5} + 326 \, b^{2} c^{3} d^{3} x^{3} - 279 \, b^{2} c d^{3} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{9216 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.48, size = 492, normalized size = 1.84 \[ -\frac {1}{8} \, a^{2} c^{6} d^{3} x^{8} + \frac {1}{2} \, a^{2} c^{4} d^{3} x^{6} - \frac {3}{4} \, a^{2} c^{2} d^{3} x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{32 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} \arcsin \left (c x\right )^{2}}{8 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{32 \, c} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{192 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{192 \, c} + \frac {35 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} x \arcsin \left (c x\right )}{768 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3}}{256 \, c^{2}} + \frac {35 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3} x}{768 \, c} + \frac {35 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{512 \, c} - \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3}}{1152 \, c^{2}} + \frac {35 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{512 \, c} + \frac {35 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3}}{3072 \, c^{2}} + \frac {35 \, b^{2} d^{3} \arcsin \left (c x\right )^{2}}{1024 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} d^{3}}{2 \, c^{2}} - \frac {35 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3}}{1024 \, c^{2}} + \frac {35 \, a b d^{3} \arcsin \left (c x\right )}{512 \, c^{2}} - \frac {7175 \, b^{2} d^{3}}{294912 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 358, normalized size = 1.34 \[ \frac {-d^{3} a^{2} \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{2} c^{6} x^{6}+\frac {3}{4} c^{4} x^{4}-\frac {1}{2} c^{2} x^{2}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {35 \arcsin \left (c x \right )^{2}}{1024}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}-\frac {35}{1024}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}+\frac {93 \arcsin \left (c x \right )}{1024}\right )}{c^{2}} \]
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 \[ -\frac {1}{8} \, a^{2} c^{6} d^{3} x^{8} + \frac {1}{2} \, a^{2} c^{4} d^{3} x^{6} - \frac {1}{1536} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} a b c^{6} d^{3} - \frac {3}{4} \, a^{2} c^{2} d^{3} x^{4} + \frac {1}{48} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} a b c^{4} d^{3} - \frac {3}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} a b c^{2} d^{3} + \frac {1}{2} \, a^{2} d^{3} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b d^{3} - \frac {1}{8} \, {\left (b^{2} c^{6} d^{3} x^{8} - 4 \, b^{2} c^{4} d^{3} x^{6} + 6 \, b^{2} c^{2} d^{3} x^{4} - 4 \, b^{2} d^{3} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} - \int \frac {{\left (b^{2} c^{7} d^{3} x^{8} - 4 \, b^{2} c^{5} d^{3} x^{6} + 6 \, b^{2} c^{3} d^{3} x^{4} - 4 \, b^{2} c d^{3} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{4 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.71, size = 573, normalized size = 2.14 \[ \begin {cases} - \frac {a^{2} c^{6} d^{3} x^{8}}{8} + \frac {a^{2} c^{4} d^{3} x^{6}}{2} - \frac {3 a^{2} c^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{2}}{2} - \frac {a b c^{6} d^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{4} - \frac {a b c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{32} + a b c^{4} d^{3} x^{6} \operatorname {asin}{\left (c x \right )} + \frac {25 a b c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{192} - \frac {3 a b c^{2} d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{2} - \frac {163 a b c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{768} + a b d^{3} x^{2} \operatorname {asin}{\left (c x \right )} + \frac {93 a b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{512 c} - \frac {93 a b d^{3} \operatorname {asin}{\left (c x \right )}}{512 c^{2}} - \frac {b^{2} c^{6} d^{3} x^{8} \operatorname {asin}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{6} d^{3} x^{8}}{256} - \frac {b^{2} c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{32} + \frac {b^{2} c^{4} d^{3} x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {25 b^{2} c^{4} d^{3} x^{6}}{1152} + \frac {25 b^{2} c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{192} - \frac {3 b^{2} c^{2} d^{3} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} + \frac {163 b^{2} c^{2} d^{3} x^{4}}{3072} - \frac {163 b^{2} c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{768} + \frac {b^{2} d^{3} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {93 b^{2} d^{3} x^{2}}{1024} + \frac {93 b^{2} d^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{512 c} - \frac {93 b^{2} d^{3} \operatorname {asin}^{2}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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