Optimal. Leaf size=396 \[ -\frac {b c f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {2 b g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}-\frac {2 b c g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 f x \sqrt {d-c^2 d x^2}+\frac {b^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {4 b^2 g \sqrt {d-c^2 d x^2}}{9 c^2}+\frac {2 b^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2} \]
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Rubi [A] time = 0.50, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {4777, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43} \[ -\frac {b c f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}-\frac {2 b c g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {2 b g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}-\frac {1}{4} b^2 f x \sqrt {d-c^2 d x^2}+\frac {b^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {4 b^2 g \sqrt {d-c^2 d x^2}}{9 c^2}+\frac {2 b^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 216
Rule 321
Rule 444
Rule 4627
Rule 4641
Rule 4645
Rule 4647
Rule 4677
Rule 4763
Rule 4777
Rubi steps
\begin {align*} \int (f+g x) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\sqrt {d-c^2 d x^2} \int (f+g x) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {d-c^2 d x^2} \int \left (f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+g x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c f \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 c \sqrt {1-c^2 x^2}}\\ &=\frac {2 b g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {2 b c g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 f x \sqrt {d-c^2 d x^2}+\frac {2 b g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {2 b c g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 f x \sqrt {d-c^2 d x^2}+\frac {b^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {2 b g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {2 b c g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}-\frac {\left (b^2 g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}\\ &=\frac {4 b^2 g \sqrt {d-c^2 d x^2}}{9 c^2}-\frac {1}{4} b^2 f x \sqrt {d-c^2 d x^2}+\frac {2 b^2 g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {b^2 f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {2 b g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {2 b c g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2}+\frac {f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 224, normalized size = 0.57 \[ \frac {\sqrt {d-c^2 d x^2} \left (-27 b^2 c f \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)\right )+8 b^2 g \left (-3 c^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+9 c x \left (a+b \sin ^{-1}(c x)\right )-b \sqrt {1-c^2 x^2} \left (c^2 x^2-7\right )\right )+54 b c^2 f x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-36 b g \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+18 c f \left (a+b \sin ^{-1}(c x)\right )^3\right )}{108 b c^2 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} {\left (a^{2} g x + a^{2} f + {\left (b^{2} g x + b^{2} f\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g x + a b f\right )} \arcsin \left (c x\right )\right )}, x\right ) \]
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 [B] time = 0.79, size = 1139, normalized size = 2.88 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, {\left (\sqrt {-c^{2} d x^{2} + d} x + \frac {\sqrt {d} \arcsin \left (c x\right )}{c}\right )} a^{2} f - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2} g}{3 \, c^{2} d} + \sqrt {d} \int {\left ({\left (b^{2} g x + b^{2} f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b g x + a b f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
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 \[ \int \left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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