Optimal. Leaf size=106 \[ \frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d}+\frac {b e^4 \left (1-(c+d x)^2\right )^{5/2}}{25 d}-\frac {2 b e^4 \left (1-(c+d x)^2\right )^{3/2}}{15 d}+\frac {b e^4 \sqrt {1-(c+d x)^2}}{5 d} \]
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Rubi [A] time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4805, 12, 4627, 266, 43} \[ \frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d}+\frac {b e^4 \left (1-(c+d x)^2\right )^{5/2}}{25 d}-\frac {2 b e^4 \left (1-(c+d x)^2\right )^{3/2}}{15 d}+\frac {b e^4 \sqrt {1-(c+d x)^2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 4627
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^4 \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int e^4 x^4 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int x^4 \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d}-\frac {\left (b e^4\right ) \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d}-\frac {\left (b e^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x}} \, dx,x,(c+d x)^2\right )}{10 d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d}-\frac {\left (b e^4\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-2 \sqrt {1-x}+(1-x)^{3/2}\right ) \, dx,x,(c+d x)^2\right )}{10 d}\\ &=\frac {b e^4 \sqrt {1-(c+d x)^2}}{5 d}-\frac {2 b e^4 \left (1-(c+d x)^2\right )^{3/2}}{15 d}+\frac {b e^4 \left (1-(c+d x)^2\right )^{5/2}}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 77, normalized size = 0.73 \[ \frac {e^4 \left (\frac {1}{5} (c+d x)^5 \left (a+b \sin ^{-1}(c+d x)\right )-\frac {1}{75} b \sqrt {1-(c+d x)^2} \left (-3 \left ((c+d x)^2-1\right )^2+10 \left (1-(c+d x)^2\right )-15\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 262, normalized size = 2.47 \[ \frac {15 \, a d^{5} e^{4} x^{5} + 75 \, a c d^{4} e^{4} x^{4} + 150 \, a c^{2} d^{3} e^{4} x^{3} + 150 \, a c^{3} d^{2} e^{4} x^{2} + 75 \, a c^{4} d e^{4} x + 15 \, {\left (b d^{5} e^{4} x^{5} + 5 \, b c d^{4} e^{4} x^{4} + 10 \, b c^{2} d^{3} e^{4} x^{3} + 10 \, b c^{3} d^{2} e^{4} x^{2} + 5 \, b c^{4} d e^{4} x + b c^{5} e^{4}\right )} \arcsin \left (d x + c\right ) + {\left (3 \, b d^{4} e^{4} x^{4} + 12 \, b c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d e^{4} x + {\left (3 \, b c^{4} + 4 \, b c^{2} + 8 \, b\right )} e^{4}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{75 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 167, normalized size = 1.58 \[ \frac {{\left (d x + c\right )}^{5} a e^{4}}{5 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} {\left (d x + c\right )} b \arcsin \left (d x + c\right ) e^{4}}{5 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} {\left (d x + c\right )} b \arcsin \left (d x + c\right ) e^{4}}{5 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} \sqrt {-{\left (d x + c\right )}^{2} + 1} b e^{4}}{25 \, d} + \frac {{\left (d x + c\right )} b \arcsin \left (d x + c\right ) e^{4}}{5 \, d} - \frac {2 \, {\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} b e^{4}}{15 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} b e^{4}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 99, normalized size = 0.93 \[ \frac {\frac {e^{4} \left (d x +c \right )^{5} a}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \arcsin \left (d x +c \right )}{5}+\frac {\left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}}{75}+\frac {8 \sqrt {1-\left (d x +c \right )^{2}}}{75}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 1280, normalized size = 12.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^4\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.14, size = 527, normalized size = 4.97 \[ \begin {cases} a c^{4} e^{4} x + 2 a c^{3} d e^{4} x^{2} + 2 a c^{2} d^{2} e^{4} x^{3} + a c d^{3} e^{4} x^{4} + \frac {a d^{4} e^{4} x^{5}}{5} + \frac {b c^{5} e^{4} \operatorname {asin}{\left (c + d x \right )}}{5 d} + b c^{4} e^{4} x \operatorname {asin}{\left (c + d x \right )} + \frac {b c^{4} e^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{25 d} + 2 b c^{3} d e^{4} x^{2} \operatorname {asin}{\left (c + d x \right )} + \frac {4 b c^{3} e^{4} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{25} + 2 b c^{2} d^{2} e^{4} x^{3} \operatorname {asin}{\left (c + d x \right )} + \frac {6 b c^{2} d e^{4} x^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{25} + \frac {4 b c^{2} e^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{75 d} + b c d^{3} e^{4} x^{4} \operatorname {asin}{\left (c + d x \right )} + \frac {4 b c d^{2} e^{4} x^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{25} + \frac {8 b c e^{4} x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{75} + \frac {b d^{4} e^{4} x^{5} \operatorname {asin}{\left (c + d x \right )}}{5} + \frac {b d^{3} e^{4} x^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{25} + \frac {4 b d e^{4} x^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{75} + \frac {8 b e^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{75 d} & \text {for}\: d \neq 0 \\c^{4} e^{4} x \left (a + b \operatorname {asin}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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