Optimal. Leaf size=362 \[ \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac {3 x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {3 b c x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{32} b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2} \]
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Rubi [A] time = 0.43, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {4673, 4649, 4647, 4641, 4627, 321, 216, 4677, 195} \[ \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac {3 x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {3 b c x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{32} b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 321
Rule 4627
Rule 4641
Rule 4647
Rule 4649
Rule 4673
Rule 4677
Rubi steps
\begin {align*} \int (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (3 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {\left (3 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b c (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{32 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 c^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 2.06, size = 373, normalized size = 1.03 \[ \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (160 a^2 c x \sqrt {1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+64 a b \cos \left (2 \sin ^{-1}(c x)\right )+4 a b \cos \left (4 \sin ^{-1}(c x)\right )-32 b^2 \sin \left (2 \sin ^{-1}(c x)\right )-b^2 \sin \left (4 \sin ^{-1}(c x)\right )\right )-96 a^2 d^{3/2} e^{3/2} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+8 b d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (12 a+8 b \sin \left (2 \sin ^{-1}(c x)\right )+b \sin \left (4 \sin ^{-1}(c x)\right )\right )+4 b d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (4 a \left (8 \sin \left (2 \sin ^{-1}(c x)\right )+\sin \left (4 \sin ^{-1}(c x)\right )\right )+16 b \cos \left (2 \sin ^{-1}(c x)\right )+b \cos \left (4 \sin ^{-1}(c x)\right )\right )+32 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3}{256 c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d e x^{2} - a^{2} d e + {\left (b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d e x^{2} - a b d e\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}, x\right ) \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]
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 \[ \frac {1}{8} \, {\left (3 \, \sqrt {-c^{2} d e x^{2} + d e} d e x + \frac {3 \, d^{2} e^{2} \arcsin \left (c x\right )}{\sqrt {d e} c} + 2 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} x\right )} a^{2} + \sqrt {d} \sqrt {e} \int -{\left ({\left (b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d e x^{2} - a b d e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \]
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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