Optimal. Leaf size=230 \[ -\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {c d x+d} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt {c d x+d} \sqrt {e-c e x}} \]
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Rubi [A] time = 0.44, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4673, 4763, 4641, 4677, 4619, 261} \[ -\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {c d x+d} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt {c d x+d} \sqrt {e-c e x}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4641
Rule 4673
Rule 4677
Rule 4763
Rubi steps
\begin {align*} \int \frac {\sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d+c d x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(e-c e x) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {e \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c e x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\left (e \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (c e \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b e \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 e \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 c e \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [A] time = 1.28, size = 296, normalized size = 1.29 \[ \frac {3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a^2 \sqrt {1-c^2 x^2}-2 a b c x-2 b^2 \sqrt {1-c^2 x^2}\right )-3 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+3 b \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (a+b \sqrt {1-c^2 x^2}\right )-6 b \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (b c x-a \sqrt {1-c^2 x^2}\right )+b^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3}{3 c d \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt {-c e x + e}}{\sqrt {c d x + d}}, x\right ) \]
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 = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c e x +e}}{\sqrt {c d x +d}}\, dx \]
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 \[ a^{2} {\left (\frac {e \arcsin \left (c x\right )}{c d \sqrt {\frac {e}{d}}} + \frac {\sqrt {-c^{2} d e x^{2} + d e}}{c d}\right )} + \sqrt {d} \sqrt {e} \int \frac {{\left (b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c d x + d}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {e-c\,e\,x}}{\sqrt {d+c\,d\,x}} \,d x \]
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 \[ \int \frac {\sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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