Optimal. Leaf size=164 \[ -\frac{60 b^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac{20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+120 a b^4 x+\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac{120 b^5 \sqrt{1-(c+d x)^2}}{d}+\frac{120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d} \]
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Rubi [A] time = 0.208758, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4803, 4619, 4677, 261} \[ -\frac{60 b^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac{20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+120 a b^4 x+\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac{120 b^5 \sqrt{1-(c+d x)^2}}{d}+\frac{120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4803
Rule 4619
Rule 4677
Rule 261
Rubi steps
\begin{align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^5 \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{x \left (a+b \sin ^{-1}(x)\right )^4}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}-\frac{\left (20 b^2\right ) \operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=-\frac{20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac{\left (60 b^3\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \sin ^{-1}(x)\right )^2}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{60 b^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac{20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac{\left (120 b^4\right ) \operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=120 a b^4 x-\frac{60 b^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac{20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}+\frac{\left (120 b^5\right ) \operatorname{Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=120 a b^4 x+\frac{120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d}-\frac{60 b^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac{20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}-\frac{\left (120 b^5\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=120 a b^4 x+\frac{120 b^5 \sqrt{1-(c+d x)^2}}{d}+\frac{120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d}-\frac{60 b^3 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac{20 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac{5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5}{d}\\ \end{align*}
Mathematica [A] time = 0.199896, size = 150, normalized size = 0.91 \[ \frac{-20 b^2 \left (-6 b^2 \left (a (c+d x)+b \sqrt{1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^3+3 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^2\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^5+5 b \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 367, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ({a}^{5} \left ( dx+c \right ) +{b}^{5} \left ( \left ( \arcsin \left ( dx+c \right ) \right ) ^{5} \left ( dx+c \right ) +5\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{4}\sqrt{1- \left ( dx+c \right ) ^{2}}-20\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3} \left ( dx+c \right ) -60\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\sqrt{1- \left ( dx+c \right ) ^{2}}+120\,\sqrt{1- \left ( dx+c \right ) ^{2}}+120\, \left ( dx+c \right ) \arcsin \left ( dx+c \right ) \right ) +5\,a{b}^{4} \left ( \left ( dx+c \right ) \left ( \arcsin \left ( dx+c \right ) \right ) ^{4}+4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}\sqrt{1- \left ( dx+c \right ) ^{2}}-12\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) +24\,dx+24\,c-24\,\sqrt{1- \left ( dx+c \right ) ^{2}}\arcsin \left ( dx+c \right ) \right ) +10\,{a}^{2}{b}^{3} \left ( \left ( \arcsin \left ( dx+c \right ) \right ) ^{3} \left ( dx+c \right ) +3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\sqrt{1- \left ( dx+c \right ) ^{2}}-6\,\sqrt{1- \left ( dx+c \right ) ^{2}}-6\, \left ( dx+c \right ) \arcsin \left ( dx+c \right ) \right ) +10\,{a}^{3}{b}^{2} \left ( \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) -2\,dx-2\,c+2\,\sqrt{1- \left ( dx+c \right ) ^{2}}\arcsin \left ( dx+c \right ) \right ) +5\,{a}^{4}b \left ( \left ( dx+c \right ) \arcsin \left ( dx+c \right ) +\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) \right ) } \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 [B] time = 2.39977, size = 749, normalized size = 4.57 \begin{align*} \frac{{\left (b^{5} d x + b^{5} c\right )} \arcsin \left (d x + c\right )^{5} + 5 \,{\left (a b^{4} d x + a b^{4} c\right )} \arcsin \left (d x + c\right )^{4} + 10 \,{\left ({\left (a^{2} b^{3} - 2 \, b^{5}\right )} d x +{\left (a^{2} b^{3} - 2 \, b^{5}\right )} c\right )} \arcsin \left (d x + c\right )^{3} +{\left (a^{5} - 20 \, a^{3} b^{2} + 120 \, a b^{4}\right )} d x + 10 \,{\left ({\left (a^{3} b^{2} - 6 \, a b^{4}\right )} d x +{\left (a^{3} b^{2} - 6 \, a b^{4}\right )} c\right )} \arcsin \left (d x + c\right )^{2} + 5 \,{\left ({\left (a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5}\right )} d x +{\left (a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5}\right )} c\right )} \arcsin \left (d x + c\right ) + 5 \,{\left (b^{5} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{4} \arcsin \left (d x + c\right )^{3} + a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5} + 6 \,{\left (a^{2} b^{3} - 2 \, b^{5}\right )} \arcsin \left (d x + c\right )^{2} + 4 \,{\left (a^{3} b^{2} - 6 \, a b^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.19545, size = 663, normalized size = 4.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21896, size = 651, normalized size = 3.97 \begin{align*} \frac{{\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )^{5}}{d} + \frac{5 \,{\left (d x + c\right )} a b^{4} \arcsin \left (d x + c\right )^{4}}{d} + \frac{5 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} b^{5} \arcsin \left (d x + c\right )^{4}}{d} + \frac{10 \,{\left (d x + c\right )} a^{2} b^{3} \arcsin \left (d x + c\right )^{3}}{d} - \frac{20 \,{\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )^{3}}{d} + \frac{20 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a b^{4} \arcsin \left (d x + c\right )^{3}}{d} + \frac{10 \,{\left (d x + c\right )} a^{3} b^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac{60 \,{\left (d x + c\right )} a b^{4} \arcsin \left (d x + c\right )^{2}}{d} + \frac{30 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a^{2} b^{3} \arcsin \left (d x + c\right )^{2}}{d} - \frac{60 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} b^{5} \arcsin \left (d x + c\right )^{2}}{d} + \frac{5 \,{\left (d x + c\right )} a^{4} b \arcsin \left (d x + c\right )}{d} - \frac{60 \,{\left (d x + c\right )} a^{2} b^{3} \arcsin \left (d x + c\right )}{d} + \frac{120 \,{\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )}{d} + \frac{20 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a^{3} b^{2} \arcsin \left (d x + c\right )}{d} - \frac{120 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a b^{4} \arcsin \left (d x + c\right )}{d} + \frac{{\left (d x + c\right )} a^{5}}{d} - \frac{20 \,{\left (d x + c\right )} a^{3} b^{2}}{d} + \frac{120 \,{\left (d x + c\right )} a b^{4}}{d} + \frac{5 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a^{4} b}{d} - \frac{60 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a^{2} b^{3}}{d} + \frac{120 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} b^{5}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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