Optimal. Leaf size=398 \[ \frac {e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {e^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b c e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {4 b e^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {c d x+d} \sqrt {e-c e x}}-\frac {4 b^2 e^2 \left (1-c^2 x^2\right )}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b^2 e^2 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {b^2 e^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}} \]
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Rubi [A] time = 0.58, antiderivative size = 398, 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, 4773, 3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac {e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {e^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b c e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {4 b e^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {c d x+d} \sqrt {e-c e x}}-\frac {4 b^2 e^2 \left (1-c^2 x^2\right )}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b^2 e^2 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {b^2 e^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3317
Rule 4673
Rule 4773
Rubi steps
\begin {align*} \int \frac {(e-c e x)^{3/2} \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)^2 \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} \operatorname {Subst}\left (\int (a+b x)^2 (c e-c e \sin (x))^2 \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \left (c^2 e^2 (a+b x)^2-2 c^2 e^2 (a+b x)^2 \sin (x)+c^2 e^2 (a+b x)^2 \sin ^2(x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {e^2 \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 (e^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 e^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {b c e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 e^2 \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^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e^2 \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 (e^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (4 b e^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (b^2 e^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {b^2 e^2 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {4 b e^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b c e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 e^2 \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^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (b^2 e^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (4 b^2 e^2 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {4 b^2 e^2 \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b^2 e^2 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {b^2 e^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {4 b e^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b c e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 e^2 \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^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [A] time = 2.39, size = 358, normalized size = 0.90 \[ \frac {e \sqrt {c d x+d} \sqrt {e-c e x} \left (-4 \left (a^2 (c x-4) \sqrt {1-c^2 x^2}+8 a b c x+8 b^2 \sqrt {1-c^2 x^2}\right )-2 a b \cos \left (2 \sin ^{-1}(c x)\right )+b^2 \sin \left (2 \sin ^{-1}(c x)\right )\right )-12 a^2 \sqrt {d} e^{3/2} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+2 b e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (6 a+8 b \sqrt {1-c^2 x^2}-b \sin \left (2 \sin ^{-1}(c x)\right )\right )-2 b e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (4 a (c x-4) \sqrt {1-c^2 x^2}+16 b c x+b \cos \left (2 \sin ^{-1}(c x)\right )\right )+4 b^2 e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3}{8 c d \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} c e x - a^{2} e + {\left (b^{2} c e x - b^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c e x - a b e\right )} \arcsin \left (c x\right )\right )} \sqrt {-c e x + e}}{\sqrt {c d x + d}}, x\right ) \]
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 \[ \int \frac {\left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (\frac {\sqrt {-c^{2} d e x^{2} + d e} e x}{d} - \frac {3 \, e^{2} \arcsin \left (c x\right )}{\sqrt {d e} c} - \frac {4 \, \sqrt {-c^{2} d e x^{2} + d e} e}{c d}\right )} a^{2} - \sqrt {d} \sqrt {e} \int \frac {{\left ({\left (b^{2} c e x - b^{2} e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c e x - a b e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c d x + d}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{3/2}}{\sqrt {d+c\,d\,x}} \,d x \]
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 \[ \int \frac {\left (- e \left (c x - 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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