Optimal. Leaf size=341 \[ \frac{3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^{7/2} \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{1617 c^{11}}+\frac{b \left (1-c^2 x^2\right )^{5/2} \left (495 c^4 d^2 e+77 c^6 d^3+770 c^2 d e^2+350 e^3\right )}{1925 c^{11}}-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (1485 c^4 d^2 e+462 c^6 d^3+1540 c^2 d e^2+525 e^3\right )}{3465 c^{11}}+\frac{b \sqrt{1-c^2 x^2} \left (495 c^4 d^2 e+231 c^6 d^3+385 c^2 d e^2+105 e^3\right )}{1155 c^{11}}+\frac{b e^2 \left (1-c^2 x^2\right )^{9/2} \left (11 c^2 d+15 e\right )}{297 c^{11}}-\frac{b e^3 \left (1-c^2 x^2\right )^{11/2}}{121 c^{11}} \]
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Rubi [A] time = 0.434835, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {270, 4731, 12, 1799, 1620} \[ \frac{3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^{7/2} \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{1617 c^{11}}+\frac{b \left (1-c^2 x^2\right )^{5/2} \left (495 c^4 d^2 e+77 c^6 d^3+770 c^2 d e^2+350 e^3\right )}{1925 c^{11}}-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (1485 c^4 d^2 e+462 c^6 d^3+1540 c^2 d e^2+525 e^3\right )}{3465 c^{11}}+\frac{b \sqrt{1-c^2 x^2} \left (495 c^4 d^2 e+231 c^6 d^3+385 c^2 d e^2+105 e^3\right )}{1155 c^{11}}+\frac{b e^2 \left (1-c^2 x^2\right )^{9/2} \left (11 c^2 d+15 e\right )}{297 c^{11}}-\frac{b e^3 \left (1-c^2 x^2\right )^{11/2}}{121 c^{11}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 4731
Rule 12
Rule 1799
Rule 1620
Rubi steps
\begin{align*} \int x^4 \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{1155 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac{(b c) \int \frac{x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{\sqrt{1-c^2 x^2}} \, dx}{1155}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac{(b c) \operatorname{Subst}\left (\int \frac{x^2 \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2310}\\ &=\frac{1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac{(b c) \operatorname{Subst}\left (\int \left (\frac{231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3}{c^{10} \sqrt{1-c^2 x}}+\frac{\left (-462 c^6 d^3-1485 c^4 d^2 e-1540 c^2 d e^2-525 e^3\right ) \sqrt{1-c^2 x}}{c^{10}}+\frac{3 \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (1-c^2 x\right )^{3/2}}{c^{10}}-\frac{5 e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (1-c^2 x\right )^{5/2}}{c^{10}}+\frac{35 e^2 \left (11 c^2 d+15 e\right ) \left (1-c^2 x\right )^{7/2}}{c^{10}}-\frac{105 e^3 \left (1-c^2 x\right )^{9/2}}{c^{10}}\right ) \, dx,x,x^2\right )}{2310}\\ &=\frac{b \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right ) \sqrt{1-c^2 x^2}}{1155 c^{11}}-\frac{b \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) \left (1-c^2 x^2\right )^{3/2}}{3465 c^{11}}+\frac{b \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (1-c^2 x^2\right )^{5/2}}{1925 c^{11}}-\frac{b e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (1-c^2 x^2\right )^{7/2}}{1617 c^{11}}+\frac{b e^2 \left (11 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{9/2}}{297 c^{11}}-\frac{b e^3 \left (1-c^2 x^2\right )^{11/2}}{121 c^{11}}+\frac{1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.261638, size = 271, normalized size = 0.79 \[ \frac{3465 a x^5 \left (495 d^2 e x^2+231 d^3+385 d e^2 x^4+105 e^3 x^6\right )+\frac{b \sqrt{1-c^2 x^2} \left (c^{10} x^4 \left (245025 d^2 e x^2+160083 d^3+148225 d e^2 x^4+33075 e^3 x^6\right )+2 c^8 \left (147015 d^2 e x^4+106722 d^3 x^2+84700 d e^2 x^6+18375 e^3 x^8\right )+24 c^6 \left (16335 d^2 e x^2+17787 d^3+8470 d e^2 x^4+1750 e^3 x^6\right )+80 c^4 e \left (9801 d^2+3388 d e x^2+630 e^2 x^4\right )+4480 c^2 e^2 \left (121 d+15 e x^2\right )+134400 e^3\right )}{c^{11}}+3465 b x^5 \sin ^{-1}(c x) \left (495 d^2 e x^2+231 d^3+385 d e^2 x^4+105 e^3 x^6\right )}{4002075} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 497, normalized size = 1.5 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{11}{x}^{11}}{11}}+{\frac{{c}^{11}d{e}^{2}{x}^{9}}{3}}+{\frac{3\,{c}^{11}{d}^{2}e{x}^{7}}{7}}+{\frac{{c}^{11}{x}^{5}{d}^{3}}{5}} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{\arcsin \left ( cx \right ){e}^{3}{c}^{11}{x}^{11}}{11}}+{\frac{\arcsin \left ( cx \right ){c}^{11}d{e}^{2}{x}^{9}}{3}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{11}{d}^{2}e{x}^{7}}{7}}+{\frac{\arcsin \left ( cx \right ){c}^{11}{x}^{5}{d}^{3}}{5}}-{\frac{{e}^{3}}{11} \left ( -{\frac{{c}^{10}{x}^{10}}{11}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{10\,{c}^{8}{x}^{8}}{99}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{80\,{c}^{6}{x}^{6}}{693}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{32\,{c}^{4}{x}^{4}}{231}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{128\,{c}^{2}{x}^{2}}{693}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{256}{693}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{{c}^{2}d{e}^{2}}{3} \left ( -{\frac{{c}^{8}{x}^{8}}{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8\,{c}^{6}{x}^{6}}{63}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{16\,{c}^{4}{x}^{4}}{105}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{64\,{c}^{2}{x}^{2}}{315}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{128}{315}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{3\,{c}^{4}{d}^{2}e}{7} \left ( -{\frac{{c}^{6}{x}^{6}}{7}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{6\,{c}^{4}{x}^{4}}{35}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8\,{c}^{2}{x}^{2}}{35}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{16}{35}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{{d}^{3}{c}^{6}}{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 ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49536, size = 628, normalized size = 1.84 \begin{align*} \frac{1}{11} \, a e^{3} x^{11} + \frac{1}{3} \, a d e^{2} x^{9} + \frac{3}{7} \, a d^{2} e x^{7} + \frac{1}{5} \, a d^{3} x^{5} + \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 d^{3} + \frac{3}{245} \,{\left (35 \, x^{7} \arcsin \left (c x\right ) +{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d^{2} e + \frac{1}{945} \,{\left (315 \, x^{9} \arcsin \left (c x\right ) +{\left (\frac{35 \, \sqrt{-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac{40 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac{64 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b d e^{2} + \frac{1}{7623} \,{\left (693 \, x^{11} \arcsin \left (c x\right ) +{\left (\frac{63 \, \sqrt{-c^{2} x^{2} + 1} x^{10}}{c^{2}} + \frac{70 \, \sqrt{-c^{2} x^{2} + 1} x^{8}}{c^{4}} + \frac{80 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{6}} + \frac{96 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{8}} + \frac{128 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{10}} + \frac{256 \, \sqrt{-c^{2} x^{2} + 1}}{c^{12}}\right )} c\right )} b e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06498, size = 846, normalized size = 2.48 \begin{align*} \frac{363825 \, a c^{11} e^{3} x^{11} + 1334025 \, a c^{11} d e^{2} x^{9} + 1715175 \, a c^{11} d^{2} e x^{7} + 800415 \, a c^{11} d^{3} x^{5} + 3465 \,{\left (105 \, b c^{11} e^{3} x^{11} + 385 \, b c^{11} d e^{2} x^{9} + 495 \, b c^{11} d^{2} e x^{7} + 231 \, b c^{11} d^{3} x^{5}\right )} \arcsin \left (c x\right ) +{\left (33075 \, b c^{10} e^{3} x^{10} + 426888 \, b c^{6} d^{3} + 1225 \,{\left (121 \, b c^{10} d e^{2} + 30 \, b c^{8} e^{3}\right )} x^{8} + 784080 \, b c^{4} d^{2} e + 25 \,{\left (9801 \, b c^{10} d^{2} e + 6776 \, b c^{8} d e^{2} + 1680 \, b c^{6} e^{3}\right )} x^{6} + 542080 \, b c^{2} d e^{2} + 3 \,{\left (53361 \, b c^{10} d^{3} + 98010 \, b c^{8} d^{2} e + 67760 \, b c^{6} d e^{2} + 16800 \, b c^{4} e^{3}\right )} x^{4} + 134400 \, b e^{3} + 4 \,{\left (53361 \, b c^{8} d^{3} + 98010 \, b c^{6} d^{2} e + 67760 \, b c^{4} d e^{2} + 16800 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{4002075 \, c^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 65.6259, size = 631, normalized size = 1.85 \begin{align*} \begin{cases} \frac{a d^{3} x^{5}}{5} + \frac{3 a d^{2} e x^{7}}{7} + \frac{a d e^{2} x^{9}}{3} + \frac{a e^{3} x^{11}}{11} + \frac{b d^{3} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{3 b d^{2} e x^{7} \operatorname{asin}{\left (c x \right )}}{7} + \frac{b d e^{2} x^{9} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e^{3} x^{11} \operatorname{asin}{\left (c x \right )}}{11} + \frac{b d^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{3 b d^{2} e x^{6} \sqrt{- c^{2} x^{2} + 1}}{49 c} + \frac{b d e^{2} x^{8} \sqrt{- c^{2} x^{2} + 1}}{27 c} + \frac{b e^{3} x^{10} \sqrt{- c^{2} x^{2} + 1}}{121 c} + \frac{4 b d^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{3}} + \frac{18 b d^{2} e x^{4} \sqrt{- c^{2} x^{2} + 1}}{245 c^{3}} + \frac{8 b d e^{2} x^{6} \sqrt{- c^{2} x^{2} + 1}}{189 c^{3}} + \frac{10 b e^{3} x^{8} \sqrt{- c^{2} x^{2} + 1}}{1089 c^{3}} + \frac{8 b d^{3} \sqrt{- c^{2} x^{2} + 1}}{75 c^{5}} + \frac{24 b d^{2} e x^{2} \sqrt{- c^{2} x^{2} + 1}}{245 c^{5}} + \frac{16 b d e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{315 c^{5}} + \frac{80 b e^{3} x^{6} \sqrt{- c^{2} x^{2} + 1}}{7623 c^{5}} + \frac{48 b d^{2} e \sqrt{- c^{2} x^{2} + 1}}{245 c^{7}} + \frac{64 b d e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{945 c^{7}} + \frac{32 b e^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{2541 c^{7}} + \frac{128 b d e^{2} \sqrt{- c^{2} x^{2} + 1}}{945 c^{9}} + \frac{128 b e^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{7623 c^{9}} + \frac{256 b e^{3} \sqrt{- c^{2} x^{2} + 1}}{7623 c^{11}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{3} x^{5}}{5} + \frac{3 d^{2} e x^{7}}{7} + \frac{d e^{2} x^{9}}{3} + \frac{e^{3} x^{11}}{11}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36202, size = 1253, normalized size = 3.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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