Optimal. Leaf size=408 \[ \frac {b c (2-m) m \sqrt {1-c^2 x^2} x^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{3 d^2 \left (m^2+3 m+2\right ) \sqrt {d-c^2 d x^2}}-\frac {(2-m) m \sqrt {1-c^2 x^2} x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 (m+1) \sqrt {d-c^2 d x^2}}+\frac {(2-m) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c (2-m) \sqrt {1-c^2 x^2} x^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \, _2F_1\left (2,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.45, antiderivative size = 408, 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 = {4705, 4713, 4711, 364} \[ \frac {b c (2-m) m \sqrt {1-c^2 x^2} x^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{3 d^2 \left (m^2+3 m+2\right ) \sqrt {d-c^2 d x^2}}-\frac {(2-m) m \sqrt {1-c^2 x^2} x^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 (m+1) \sqrt {d-c^2 d x^2}}+\frac {(2-m) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c (2-m) \sqrt {1-c^2 x^2} x^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} x^{m+2} \, _2F_1\left (2,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{3 d^2 (m+2) \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4705
Rule 4711
Rule 4713
Rubi steps
\begin {align*} \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(2-m) \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {x^{1+m}}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(2-m) x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {((2-m) m) \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx}{3 d^2}-\frac {\left (b c (2-m) \sqrt {1-c^2 x^2}\right ) \int \frac {x^{1+m}}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(2-m) x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c (2-m) x^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {b c x^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {\left ((2-m) m \sqrt {1-c^2 x^2}\right ) \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(2-m) x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {(2-m) m x^{1+m} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{3 d^2 (1+m) \sqrt {d-c^2 d x^2}}-\frac {b c (2-m) x^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {b c x^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{3 d^2 (2+m) \sqrt {d-c^2 d x^2}}+\frac {b c (2-m) m x^{2+m} \sqrt {1-c^2 x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{3 d^2 \left (2+3 m+m^2\right ) \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 279, normalized size = 0.68 \[ \frac {x^{m+1} \left ((2-m) \left (d-c^2 d x^2\right ) \left (-m \sqrt {1-c^2 x^2} \left ((m+2) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-b c x \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )\right )+(m+1) (m+2) \left (a+b \sin ^{-1}(c x)\right )-b c (m+1) x \sqrt {1-c^2 x^2} \, _2F_1\left (1,\frac {m}{2}+1;\frac {m}{2}+2;c^2 x^2\right )\right )+d (m+1) (m+2) \left (a+b \sin ^{-1}(c x)\right )-b c d (m+1) x \left (1-c^2 x^2\right )^{3/2} \, _2F_1\left (2,\frac {m}{2}+1;\frac {m}{2}+2;c^2 x^2\right )\right )}{3 d^2 (m+1) (m+2) \left (d-c^2 d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.81, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a +b \arcsin \left (c x \right )\right )}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^m\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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