Optimal. Leaf size=502 \[ \frac {5 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}+\frac {5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5 b (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}-\frac {5 b c x^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {1}{6} x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {65 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac {245 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}+\frac {115 b^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac {1}{108} b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2} \]
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Rubi [A] time = 0.57, antiderivative size = 502, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \frac {5 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}+\frac {5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {5 b (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}-\frac {5 b c x^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {1}{6} x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {65 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac {245 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}+\frac {115 b^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac {1}{108} b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2} \]
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)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left ((d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{5/2}}\\ &=\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{6 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \left (1-c^2 x^2\right )^{5/2}}\\ &=\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{12 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac {65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac {5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{144 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{64 \left (1-c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 c^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac {245 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac {65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac {5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{288 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 \left (1-c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac {245 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac {65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}+\frac {115 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac {5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 3.13, size = 450, normalized size = 0.90 \[ \frac {d^2 e^2 \left (\sqrt {c d x+d} \sqrt {e-c e x} \left (9504 a^2 c x \sqrt {1-c^2 x^2}+2304 a^2 c^5 x^5 \sqrt {1-c^2 x^2}-7488 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+3240 a b \cos \left (2 \sin ^{-1}(c x)\right )+324 a b \cos \left (4 \sin ^{-1}(c x)\right )+24 a b \cos \left (6 \sin ^{-1}(c x)\right )-1620 b^2 \sin \left (2 \sin ^{-1}(c x)\right )-81 b^2 \sin \left (4 \sin ^{-1}(c x)\right )-4 b^2 \sin \left (6 \sin ^{-1}(c x)\right )\right )-4320 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+72 b \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (60 a+45 b \sin \left (2 \sin ^{-1}(c x)\right )+9 b \sin \left (4 \sin ^{-1}(c x)\right )+b \sin \left (6 \sin ^{-1}(c x)\right )\right )+12 b \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (540 a \sin \left (2 \sin ^{-1}(c x)\right )+108 a \sin \left (4 \sin ^{-1}(c x)\right )+12 a \sin \left (6 \sin ^{-1}(c x)\right )+270 b \cos \left (2 \sin ^{-1}(c x)\right )+27 b \cos \left (4 \sin ^{-1}(c x)\right )+2 b \cos \left (6 \sin ^{-1}(c x)\right )\right )+1440 b^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3\right )}{13824 c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{4} d^{2} e^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (b^{2} c^{4} d^{2} e^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} e^{2} x^{2} + b^{2} d^{2} e^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} e^{2} x^{4} - 2 \, a b c^{2} d^{2} e^{2} x^{2} + a b d^{2} e^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}, x\right ) \]
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 = 0.31, size = 0, normalized size = 0.00 \[ \int \left (c d x +d \right )^{\frac {5}{2}} \left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]
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 \[ \frac {1}{48} \, {\left (15 \, \sqrt {-c^{2} d e x^{2} + d e} d^{2} e^{2} x + \frac {15 \, d^{3} e^{3} \arcsin \left (c x\right )}{\sqrt {d e} c} + 10 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} d e x + 8 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {5}{2}} x\right )} a^{2} + \sqrt {d} \sqrt {e} \int {\left ({\left (b^{2} c^{4} d^{2} e^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} e^{2} x^{2} + b^{2} d^{2} e^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{4} d^{2} e^{2} x^{4} - 2 \, a b c^{2} d^{2} e^{2} x^{2} + a b d^{2} e^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}\,{\left (e-c\,e\,x\right )}^{5/2} \,d x \]
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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