Optimal. Leaf size=119 \[ -\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+24 b^4 x \]
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Rubi [A] time = 0.16, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4803, 4619, 4677, 8} \[ -\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+24 b^4 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 4619
Rule 4677
Rule 4803
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {\left (24 b^3\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {\left (24 b^4\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=24 b^4 x-\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 115, normalized size = 0.97 \[ \frac {-12 b^2 \left (2 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2-2 b^2 (c+d x)\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4+4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 233, normalized size = 1.96 \[ \frac {{\left (b^{4} d x + b^{4} c\right )} \arcsin \left (d x + c\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{4} - 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x + 6 \, {\left ({\left (a^{2} b^{2} - 2 \, b^{4}\right )} d x + {\left (a^{2} b^{2} - 2 \, b^{4}\right )} c\right )} \arcsin \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{3} b - 6 \, a b^{3}\right )} d x + {\left (a^{3} b - 6 \, a b^{3}\right )} c\right )} \arcsin \left (d x + c\right ) + 4 \, {\left (b^{4} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{3} \arcsin \left (d x + c\right )^{2} + a^{3} b - 6 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 329, normalized size = 2.76 \[ \frac {{\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{4}}{d} + \frac {4 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right )^{3}}{d} + \frac {6 \, {\left (d x + c\right )} a^{2} b^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {12 \, {\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{2}}{d} + \frac {12 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3} \arcsin \left (d x + c\right )^{2}}{d} + \frac {4 \, {\left (d x + c\right )} a^{3} b \arcsin \left (d x + c\right )}{d} - \frac {24 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )}{d} + \frac {12 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{2} \arcsin \left (d x + c\right )}{d} - \frac {24 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{4}}{d} - \frac {12 \, {\left (d x + c\right )} a^{2} b^{2}}{d} + \frac {24 \, {\left (d x + c\right )} b^{4}}{d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{3} b}{d} - \frac {24 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 255, normalized size = 2.14 \[ \frac {a^{4} \left (d x +c \right )+b^{4} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{4}+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a \,b^{3} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{3}+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+6 a^{2} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a^{3} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b^{4} x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{4} + a^{4} x + \frac {4 \, {\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} a^{3} b}{d} + \int \frac {2 \, {\left (2 \, \sqrt {d x + c + 1} \sqrt {-d x - c + 1} b^{4} d x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 2 \, {\left (a b^{3} d^{2} x^{2} + 2 \, a b^{3} c d x + a b^{3} c^{2} - a b^{3}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 3 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{2} b^{2} c d x + a^{2} b^{2} c^{2} - a^{2} b^{2}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 229, normalized size = 1.92 \[ a^4\,x+\frac {b^4\,\left (c+d\,x\right )\,\left ({\mathrm {asin}\left (c+d\,x\right )}^4-12\,{\mathrm {asin}\left (c+d\,x\right )}^2+24\right )}{d}-\frac {b^4\,\left (24\,\mathrm {asin}\left (c+d\,x\right )-4\,{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}+\frac {6\,a^2\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {4\,a^3\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {4\,a\,b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}-\frac {4\,a\,b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.54, size = 444, normalized size = 3.73 \[ \begin {cases} a^{4} x + \frac {4 a^{3} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a^{3} b x \operatorname {asin}{\left (c + d x \right )} + \frac {4 a^{3} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {6 a^{2} b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 12 a^{2} b^{2} x + \frac {12 a^{2} b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {4 a b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 24 a b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {12 a b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {b^{4} c \operatorname {asin}^{4}{\left (c + d x \right )}}{d} - \frac {12 b^{4} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + b^{4} x \operatorname {asin}^{4}{\left (c + d x \right )} - 12 b^{4} x \operatorname {asin}^{2}{\left (c + d x \right )} + 24 b^{4} x + \frac {4 b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {24 b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\relax (c )}\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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