Optimal. Leaf size=308 \[ \frac{1}{2} x^2 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} x^3 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} x^4 (d i+e h) \left (a+b \sin ^{-1}(c x)\right )+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e i x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (225 c^2 x \left (8 c^2 (d g+e f)+3 (d i+e h)\right )+32 \left (50 c^2 (d h+e g)+225 c^4 d f+24 e i\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (8 c^2 (d g+e f)+3 (d i+e h)\right )}{32 c^4}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 (d h+e g)+12 e i\right )}{225 c^3}+\frac{b x^3 \sqrt{1-c^2 x^2} (d i+e h)}{16 c}+\frac{b e i x^4 \sqrt{1-c^2 x^2}}{25 c} \]
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Rubi [A] time = 0.948063, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {4749, 12, 1809, 780, 216} \[ \frac{1}{2} x^2 (d g+e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} x^3 (d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} x^4 (d i+e h) \left (a+b \sin ^{-1}(c x)\right )+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e i x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \sqrt{1-c^2 x^2} \left (225 c^2 x \left (8 c^2 (d g+e f)+3 (d i+e h)\right )+32 \left (50 c^2 (d h+e g)+225 c^4 d f+24 e i\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (8 c^2 (d g+e f)+3 (d i+e h)\right )}{32 c^4}+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 (d h+e g)+12 e i\right )}{225 c^3}+\frac{b x^3 \sqrt{1-c^2 x^2} (d i+e h)}{16 c}+\frac{b e i x^4 \sqrt{1-c^2 x^2}}{25 c} \]
Antiderivative was successfully verified.
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Rule 4749
Rule 12
Rule 1809
Rule 780
Rule 216
Rubi steps
\begin{align*} \int (d+e x) \left (f+g x+h x^2+108 x^3\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (60 d f+30 (e f+d g) x+20 (e g+d h) x^2+15 (108 d+e h) x^3+1296 e x^4\right )}{60 \sqrt{1-c^2 x^2}} \, dx\\ &=d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{60} (b c) \int \frac{x \left (60 d f+30 (e f+d g) x+20 (e g+d h) x^2+15 (108 d+e h) x^3+1296 e x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{108 b e x^4 \sqrt{1-c^2 x^2}}{25 c}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-300 c^2 d f-150 c^2 (e f+d g) x-4 \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2-75 c^2 (108 d+e h) x^3\right )}{\sqrt{1-c^2 x^2}} \, dx}{300 c}\\ &=\frac{b (108 d+e h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{108 b e x^4 \sqrt{1-c^2 x^2}}{25 c}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (1200 c^4 d f+75 c^2 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x+16 c^2 \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{1200 c^3}\\ &=\frac{b \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b (108 d+e h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{108 b e x^4 \sqrt{1-c^2 x^2}}{25 c}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-16 c^2 \left (2592 e+225 c^4 d f+50 c^2 (e g+d h)\right )-225 c^4 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x\right )}{\sqrt{1-c^2 x^2}} \, dx}{3600 c^5}\\ &=\frac{b \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b (108 d+e h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{108 b e x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b \left (32 \left (2592 e+225 c^4 d f+50 c^2 (e g+d h)\right )+225 c^2 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac{b \left (e \left (1296+25 c^2 g\right )+25 c^2 d h\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b (108 d+e h) x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{108 b e x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{b \left (32 \left (2592 e+225 c^4 d f+50 c^2 (e g+d h)\right )+225 c^2 \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}-\frac{b \left (324 d+8 c^2 e f+8 c^2 d g+3 e h\right ) \sin ^{-1}(c x)}{32 c^4}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} (e g+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} (108 d+e h) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{108}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.465432, size = 253, normalized size = 0.82 \[ \frac{120 a c^5 x (5 d (12 f+x (6 g+x (4 h+3 i x)))+e x (30 f+x (20 g+3 x (5 h+4 i x))))+b \sqrt{1-c^2 x^2} \left (2 c^4 (25 d (144 f+x (36 g+x (16 h+9 i x)))+e x (900 f+x (400 g+9 x (25 h+16 i x))))+c^2 \left (25 d (64 h+27 i x)+e \left (1600 g+675 h x+384 i x^2\right )\right )+768 e i\right )+15 b c \sin ^{-1}(c x) \left (8 c^4 x (5 d (12 f+x (6 g+x (4 h+3 i x)))+e x (30 f+x (20 g+3 x (5 h+4 i x))))-120 c^2 (d g+e f)-45 (d i+e h)\right )}{7200 c^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 428, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{4}} \left ({\frac{ei{c}^{5}{x}^{5}}{5}}+{\frac{ \left ( dci+ech \right ){c}^{4}{x}^{4}}{4}}+{\frac{ \left ( d{c}^{2}h+e{c}^{2}g \right ){c}^{3}{x}^{3}}{3}}+{\frac{ \left ( d{c}^{3}g+ef{c}^{3} \right ){c}^{2}{x}^{2}}{2}}+{c}^{5}fdx \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ) ei{c}^{5}{x}^{5}}{5}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{4}di}{4}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{4}eh}{4}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{3}dh}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{3}eg}{3}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{2}dg}{2}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{2}ef}{2}}+\arcsin \left ( cx \right ){c}^{5}fdx-{\frac{ei}{5} \left ( -{\frac{{c}^{4}{x}^{4}}{5}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{2}{x}^{2}}{15}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8}{15}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{15\,dci+15\,ech}{60} \left ( -{\frac{{c}^{3}{x}^{3}}{4}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,cx}{8}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,\arcsin \left ( cx \right ) }{8}} \right ) }-{\frac{20\,d{c}^{2}h+20\,e{c}^{2}g}{60} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{30\,d{c}^{3}g+30\,ef{c}^{3}}{60} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) }+{c}^{4}df\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49323, size = 726, normalized size = 2.36 \begin{align*} \frac{1}{5} \, a e i x^{5} + \frac{1}{4} \, a e h x^{4} + \frac{1}{4} \, a d i x^{4} + \frac{1}{3} \, a e g x^{3} + \frac{1}{3} \, a d h x^{3} + \frac{1}{2} \, a e f x^{2} + \frac{1}{2} \, a d g x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b e f + \frac{1}{4} \,{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d g + \frac{1}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e g + \frac{1}{9} \,{\left (3 \, x^{3} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d h + \frac{1}{32} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b e h + \frac{1}{32} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d i + \frac{1}{75} \,{\left (15 \, x^{5} \arcsin \left (c x\right ) +{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e i + a d f x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d f}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.80817, size = 846, normalized size = 2.75 \begin{align*} \frac{1440 \, a c^{5} e i x^{5} + 7200 \, a c^{5} d f x + 1800 \,{\left (a c^{5} e h + a c^{5} d i\right )} x^{4} + 2400 \,{\left (a c^{5} e g + a c^{5} d h\right )} x^{3} + 3600 \,{\left (a c^{5} e f + a c^{5} d g\right )} x^{2} + 15 \,{\left (96 \, b c^{5} e i x^{5} + 480 \, b c^{5} d f x - 120 \, b c^{3} e f - 120 \, b c^{3} d g + 120 \,{\left (b c^{5} e h + b c^{5} d i\right )} x^{4} - 45 \, b c e h - 45 \, b c d i + 160 \,{\left (b c^{5} e g + b c^{5} d h\right )} x^{3} + 240 \,{\left (b c^{5} e f + b c^{5} d g\right )} x^{2}\right )} \arcsin \left (c x\right ) +{\left (288 \, b c^{4} e i x^{4} + 7200 \, b c^{4} d f + 1600 \, b c^{2} e g + 1600 \, b c^{2} d h + 450 \,{\left (b c^{4} e h + b c^{4} d i\right )} x^{3} + 768 \, b e i + 32 \,{\left (25 \, b c^{4} e g + 25 \, b c^{4} d h + 12 \, b c^{2} e i\right )} x^{2} + 225 \,{\left (8 \, b c^{4} e f + 8 \, b c^{4} d g + 3 \, b c^{2} e h + 3 \, b c^{2} d i\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{7200 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.03738, size = 658, normalized size = 2.14 \begin{align*} \begin{cases} a d f x + \frac{a d g x^{2}}{2} + \frac{a d h x^{3}}{3} + \frac{a d i x^{4}}{4} + \frac{a e f x^{2}}{2} + \frac{a e g x^{3}}{3} + \frac{a e h x^{4}}{4} + \frac{a e i x^{5}}{5} + b d f x \operatorname{asin}{\left (c x \right )} + \frac{b d g x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b d h x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b d i x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b e f x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b e g x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e h x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b e i x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{b d f \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{b d g x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{b d h x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b d i x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} + \frac{b e f x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{b e g x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b e h x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} + \frac{b e i x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} - \frac{b d g \operatorname{asin}{\left (c x \right )}}{4 c^{2}} - \frac{b e f \operatorname{asin}{\left (c x \right )}}{4 c^{2}} + \frac{2 b d h \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{3 b d i x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} + \frac{2 b e g \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{3 b e h x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} + \frac{4 b e i x^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{3}} - \frac{3 b d i \operatorname{asin}{\left (c x \right )}}{32 c^{4}} - \frac{3 b e h \operatorname{asin}{\left (c x \right )}}{32 c^{4}} + \frac{8 b e i \sqrt{- c^{2} x^{2} + 1}}{75 c^{5}} & \text{for}\: c \neq 0 \\a \left (d f x + \frac{d g x^{2}}{2} + \frac{d h x^{3}}{3} + \frac{d i x^{4}}{4} + \frac{e f x^{2}}{2} + \frac{e g x^{3}}{3} + \frac{e h x^{4}}{4} + \frac{e i x^{5}}{5}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19007, size = 1041, normalized size = 3.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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