Optimal. Leaf size=670 \[ -\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{c^2}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}-\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^4}+\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}+\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{3 b f g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}+\frac{b c g^3 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{b g^3 x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}-\frac{2 b g^3 x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.705186, antiderivative size = 670, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {4778, 4764, 4648, 4642, 30, 4678, 4698, 4708, 266, 43, 4690, 12} \[ -\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{c^2}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}-\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^4}+\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}+\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{3 b f g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}+\frac{b c g^3 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{b g^3 x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}-\frac{2 b g^3 x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4778
Rule 4764
Rule 4648
Rule 4642
Rule 30
Rule 4678
Rule 4698
Rule 4708
Rule 266
Rule 43
Rule 4690
Rule 12
Rubi steps
\begin{align*} \int (f+g x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{\sqrt{d-c^2 d x^2} \int (f+g x)^3 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{d-c^2 d x^2} \int \left (f^3 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )+3 f^2 g x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )+3 f g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )+g^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (f^3 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 f^2 g \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (3 f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (g^3 \sqrt{d-c^2 d x^2}\right ) \int x^3 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{c^2}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^4}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}+\frac{\left (f^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (b c f^3 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b f^2 g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{c \sqrt{1-c^2 x^2}}+\frac{\left (3 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c f g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (b c g^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}+\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}+\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{c^2}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^4}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{\left (3 f g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (3 b f g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt{1-c^2 x^2}}+\frac{\left (b g^3 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3 \sqrt{1-c^2 x^2}}\\ &=-\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}-\frac{2 b g^3 x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}}+\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{3 b f g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}+\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{b g^3 x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}+\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{b c g^3 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{f^2 g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{c^2}-\frac{g^3 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^4}+\frac{g^3 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^4}-\frac{f^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.20368, size = 442, normalized size = 0.66 \[ \frac{240 a \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (6 c^4 x \left (20 f^2 g x+10 f^3+15 f g^2 x^2+4 g^3 x^3\right )-c^2 g \left (120 f^2+45 f g x+8 g^2 x^2\right )-16 g^3\right )-3600 a c \sqrt{d} f \sqrt{1-c^2 x^2} \left (4 c^2 f^2+3 g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-2400 b c^2 f^2 g \sqrt{d-c^2 d x^2} \left (12 \left (1-c^2 x^2\right )^{3/2} \cos ^{-1}(c x)+9 c x-\cos \left (3 \cos ^{-1}(c x)\right )\right )+3600 b c^3 f^3 \sqrt{d-c^2 d x^2} \left (\cos \left (2 \cos ^{-1}(c x)\right )+2 \cos ^{-1}(c x) \left (\sin \left (2 \cos ^{-1}(c x)\right )-\cos ^{-1}(c x)\right )\right )+675 b c f g^2 \sqrt{d-c^2 d x^2} \left (-8 \cos ^{-1}(c x)^2+\cos \left (4 \cos ^{-1}(c x)\right )+4 \cos ^{-1}(c x) \sin \left (4 \cos ^{-1}(c x)\right )\right )-8 b g^3 \sqrt{d-c^2 d x^2} \left (16 c x \left (-9 c^4 x^4+5 c^2 x^2+30\right )+15 \cos ^{-1}(c x) \left (30 \sqrt{1-c^2 x^2}-5 \sin \left (3 \cos ^{-1}(c x)\right )-3 \sin \left (5 \cos ^{-1}(c x)\right )\right )\right )}{28800 c^4 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.848, size = 1285, 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 ({\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 )} \arccos \left (c x\right )\right )} \sqrt{-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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sqrt{-c^{2} d x^{2} + d}{\left (g x + f\right )}^{3}{\left (b \arccos \left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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