Optimal. Leaf size=400 \[ \frac {2 b x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}+\frac {16 a b x \sqrt {1-c^2 x^2}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}+\frac {8 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt {d-c^2 d x^2}}-\frac {76 b^2 \left (1-c^2 x^2\right )^2}{675 c^6 \sqrt {d-c^2 d x^2}}+\frac {298 b^2 \left (1-c^2 x^2\right )}{225 c^6 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.58, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {4707, 4677, 4619, 261, 4627, 266, 43} \[ \frac {16 a b x \sqrt {1-c^2 x^2}}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {2 b x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac {8 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}+\frac {2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt {d-c^2 d x^2}}-\frac {76 b^2 \left (1-c^2 x^2\right )^2}{675 c^6 \sqrt {d-c^2 d x^2}}+\frac {298 b^2 \left (1-c^2 x^2\right )}{225 c^6 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 4619
Rule 4627
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac {4 \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{5 c^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \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 )^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}+\frac {8 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx}{15 c^4}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{25 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt {d-c^2 d x^2}}\\ &=\frac {8 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{25 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{45 c^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1-c^2 x^2}}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {8 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1-c^2 x}}-\frac {2 \sqrt {1-c^2 x}}{c^4}+\frac {\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{25 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{45 c^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1-c^2 x^2}}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{25 c^6 \sqrt {d-c^2 d x^2}}-\frac {4 b^2 \left (1-c^2 x^2\right )^2}{75 c^6 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {8 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}-\frac {\left (16 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{15 c^4 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{45 c^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1-c^2 x^2}}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {298 b^2 \left (1-c^2 x^2\right )}{225 c^6 \sqrt {d-c^2 d x^2}}-\frac {76 b^2 \left (1-c^2 x^2\right )^2}{675 c^6 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {8 b x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{45 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 230, normalized size = 0.58 \[ \frac {225 a^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )+30 a b c x \sqrt {1-c^2 x^2} \left (9 c^4 x^4+20 c^2 x^2+120\right )+30 b \sin ^{-1}(c x) \left (15 a \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )+b c x \sqrt {1-c^2 x^2} \left (9 c^4 x^4+20 c^2 x^2+120\right )\right )-2 b^2 \left (27 c^6 x^6+109 c^4 x^4+1936 c^2 x^2-2072\right )+225 b^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right ) \sin ^{-1}(c x)^2}{3375 c^6 \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 276, normalized size = 0.69 \[ -\frac {30 \, {\left (9 \, a b c^{5} x^{5} + 20 \, a b c^{3} x^{3} + 120 \, a b c x + {\left (9 \, b^{2} c^{5} x^{5} + 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} x^{6} + {\left (225 \, a^{2} - 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} - 968 \, b^{2}\right )} c^{2} x^{2} + 225 \, {\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, b^{2}\right )} \arcsin \left (c x\right )^{2} - 1800 \, a^{2} + 4144 \, b^{2} + 450 \, {\left (3 \, a b c^{6} x^{6} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \]
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 = 0.79, size = 1020, normalized size = 2.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 365, normalized size = 0.91 \[ -\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \arcsin \left (c x\right )^{2} - \frac {2}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \arcsin \left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {-c^{2} x^{2} + 1} c^{2} x^{4} + 136 \, \sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{4} \sqrt {d}} + \frac {15 \, {\left (9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x\right )} \arcsin \left (c x\right )}{c^{5} \sqrt {d}}\right )} + \frac {2 \, {\left (9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x\right )} a b}{225 \, c^{5} \sqrt {d}} \]
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 \[ \int \frac {x^5\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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 \[ \int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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