Optimal. Leaf size=270 \[ \frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b f g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.43, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4777, 4763, 4641, 4677, 8, 4707, 30} \[ \frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b f g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4641
Rule 4677
Rule 4707
Rule 4763
Rule 4777
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {f^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {2 f g x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (f^2 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (2 f g \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {\left (2 b f g \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{c \sqrt {d-c^2 d x^2}}+\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b g^2 \sqrt {1-c^2 x^2}\right ) \int x \, dx}{2 c \sqrt {d-c^2 d x^2}}\\ &=\frac {2 b f g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {b g^2 x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}-\frac {2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}}-\frac {g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt {d-c^2 d x^2}}+\frac {f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 266, normalized size = 0.99 \[ \frac {\sqrt {d} g \left (c^2 x^2-1\right ) \left (4 c \left (a \sqrt {1-c^2 x^2} (4 f+g x)-4 b c f x\right )+b g \cos \left (2 \sin ^{-1}(c x)\right )\right )-4 a \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (2 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-2 b \sqrt {d} \left (c^2 x^2-1\right ) \left (2 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^2+2 b \sqrt {d} g \left (c^2 x^2-1\right ) \sin ^{-1}(c x) \left (8 c f \sqrt {1-c^2 x^2}+g \sin \left (2 \sin ^{-1}(c x)\right )\right )}{8 c^3 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} + {\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \arcsin \left (c x\right )\right )}}{c^{2} d x^{2} - d}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.87, size = 548, normalized size = 2.03 \[ -\frac {a \,g^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2 c^{2} d}+\frac {a \,g^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{2} \sqrt {c^{2} d}}-\frac {2 a f g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a \,f^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f^{2}}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} g^{2}}{4 c^{3} d \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \sqrt {-c^{2} x^{2}+1}\, x}{c d \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \arcsin \left (c x \right ) x^{2}}{d \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f g \arcsin \left (c x \right )}{c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, g^{2}}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arcsin \left (c x \right ) x}{8 c^{2} d \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \cos \left (3 \arcsin \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g^{2} \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{8 c^{3} d \left (c^{2} x^{2}-1\right )} \]
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 \[ -\frac {1}{2} \, a g^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x}{c^{2} d} - \frac {\arcsin \left (c x\right )}{c^{3} \sqrt {d}}\right )} + \frac {b f^{2} \arcsin \left (c x\right )^{2}}{2 \, c \sqrt {d}} + \frac {b g^{2} \int \frac {x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{\sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x}}{\sqrt {d}} + \frac {2 \, b f g x}{c \sqrt {d}} + \frac {a f^{2} \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} b f g \arcsin \left (c x\right )}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a f g}{c^{2} d} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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