Optimal. Leaf size=256 \[ -\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {1-c^2 x^2}}+\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {266, 43, 4691, 12} \[ -\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {1-c^2 x^2}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {1-c^2 x^2}}+\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 4691
Rubi steps
\begin {align*} \int x^5 \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 {-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^5 \sqrt {d-c^2 d x^2} \, dx\\ &=-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \, dx}{105 c^5 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname {Subst}\left (\int x^2 \sqrt {d-c^2 d x} \, dx,x,x^2\right )\\ &=\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {1-c^2 x^2}}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {d-c^2 d x}}{c^4}-\frac {2 \left (d-c^2 d x\right )^{3/2}}{c^4 d}+\frac {\left (d-c^2 d x\right )^{5/2}}{c^4 d^2}\right ) \, dx,x,x^2\right )\\ &=\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {1-c^2 x^2}}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \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^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 157, normalized size = 0.61 \[ \frac {\sqrt {d-c^2 d x^2} \left (105 a \sqrt {1-c^2 x^2} \left (15 c^6 x^6-3 c^4 x^4-4 c^2 x^2-8\right )+b c x \left (-225 c^6 x^6+63 c^4 x^4+140 c^2 x^2+840\right )+105 b \sqrt {1-c^2 x^2} \left (15 c^6 x^6-3 c^4 x^4-4 c^2 x^2-8\right ) \sin ^{-1}(c x)\right )}{11025 c^6 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 177, normalized size = 0.69 \[ \frac {{\left (225 \, b c^{7} x^{7} - 63 \, b c^{5} x^{5} - 140 \, b c^{3} x^{3} - 840 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 105 \, {\left (15 \, a c^{8} x^{8} - 18 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + {\left (15 \, b c^{8} x^{8} - 18 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \arcsin \left (c x\right ) + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{11025 \, {\left (c^{8} x^{2} - c^{6}\right )}} \]
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 = 0.52, size = 880, normalized size = 3.44 \[ a \left (-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{7 c^{2} d}+\frac {-\frac {4 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{35 c^{2} d}-\frac {8 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{105 d \,c^{4}}}{c^{2}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}-64 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}+104 c^{4} x^{4}+112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-25 c^{2} x^{2}-56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+7 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (i+7 \arcsin \left (c x \right )\right )}{6272 c^{6} \left (c^{2} x^{2}-1\right )}+\frac {3 \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 )}{3200 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {5 \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 (i+\arcsin \left (c x \right )\right )}{128 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {5 \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 )}{128 c^{6} \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 )}{1152 c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}+64 c^{8} x^{8}-112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-144 c^{6} x^{6}+56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+104 c^{4} x^{4}-7 i \sqrt {-c^{2} x^{2}+1}\, x c -25 c^{2} x^{2}+1\right ) \left (-i+7 \arcsin \left (c x \right )\right )}{6272 c^{6} \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 (-13 i+15 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{7200 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (-i+105 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right )}{14400 c^{6} \left (c^{2} x^{2}-1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 197, normalized size = 0.77 \[ -\frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} a - \frac {{\left (225 \, c^{6} \sqrt {d} x^{7} - 63 \, c^{4} \sqrt {d} x^{5} - 140 \, c^{2} \sqrt {d} x^{3} - 840 \, \sqrt {d} x\right )} b}{11025 \, c^{5}} \]
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 \[ \int x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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 \[ \int x^{5} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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