Optimal. Leaf size=450 \[ -\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b f^2 g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}+\frac{b g^3 x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.583574, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ -\frac{3 f^2 g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}+\frac{3 f g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b f^2 g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}+\frac{b g^3 x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4777
Rule 4763
Rule 4641
Rule 4677
Rule 8
Rule 4707
Rule 30
Rubi steps
\begin{align*} \int \frac{(f+g x)^3 \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)^3 \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^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{3 f^2 g x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{3 f g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{g^3 x^3 \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^3 \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 (3 f^2 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 (3 f 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{\left (g^3 \sqrt{1-c^2 x^2}\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{3 f^2 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{3 f 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^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \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 (3 b f^2 g \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{c \sqrt{d-c^2 d x^2}}+\frac{\left (3 f 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 (3 b f g^2 \sqrt{1-c^2 x^2}\right ) \int x \, dx}{2 c \sqrt{d-c^2 d x^2}}+\frac{\left (2 g^3 \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}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b g^3 \sqrt{1-c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b f^2 g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}+\frac{b g^3 x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 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{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f 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^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \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{3 f 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{\left (2 b g^3 \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b f^2 g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}+\frac{b g^3 x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 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{2 g^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f 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^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{f^3 \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{3 f 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*}
Mathematica [A] time = 1.06612, size = 343, normalized size = 0.76 \[ \frac{-\sqrt{d} g \left (c^2 x^2-1\right ) \left (-12 a \sqrt{1-c^2 x^2} \left (c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )+4 g^2\right )+8 b c x \left (c^2 \left (27 f^2+g^2 x^2\right )+6 g^2\right )-27 b c f g \cos \left (2 \sin ^{-1}(c x)\right )\right )-36 a c f \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (2 c^2 f^2+3 g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-18 b c \sqrt{d} f \left (c^2 x^2-1\right ) \left (2 c^2 f^2+3 g^2\right ) \sin ^{-1}(c x)^2+6 b \sqrt{d} g \left (c^2 x^2-1\right ) \sin ^{-1}(c x) \left (4 \sqrt{1-c^2 x^2} \left (c^2 \left (9 f^2+g^2 x^2\right )+2 g^2\right )+9 c f g \sin \left (2 \sin ^{-1}(c x)\right )\right )}{72 c^4 \sqrt{d} \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.546, size = 845, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} +{\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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