Integrand size = 16, antiderivative size = 120 \[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {20 b d^2 \sqrt {d x} \sqrt {1-c^2 x^2}}{147 c^3}+\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} (a+b \arcsin (c x))}{7 d}-\frac {20 b d^{5/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{147 c^{7/2}} \]
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Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4723, 327, 335, 227} \[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {2 (d x)^{7/2} (a+b \arcsin (c x))}{7 d}-\frac {20 b d^{5/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{147 c^{7/2}}+\frac {4 b \sqrt {1-c^2 x^2} (d x)^{5/2}}{49 c}+\frac {20 b d^2 \sqrt {1-c^2 x^2} \sqrt {d x}}{147 c^3} \]
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Rule 227
Rule 327
Rule 335
Rule 4723
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{7/2} (a+b \arcsin (c x))}{7 d}-\frac {(2 b c) \int \frac {(d x)^{7/2}}{\sqrt {1-c^2 x^2}} \, dx}{7 d} \\ & = \frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} (a+b \arcsin (c x))}{7 d}-\frac {(10 b d) \int \frac {(d x)^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{49 c} \\ & = \frac {20 b d^2 \sqrt {d x} \sqrt {1-c^2 x^2}}{147 c^3}+\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} (a+b \arcsin (c x))}{7 d}-\frac {\left (10 b d^3\right ) \int \frac {1}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{147 c^3} \\ & = \frac {20 b d^2 \sqrt {d x} \sqrt {1-c^2 x^2}}{147 c^3}+\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} (a+b \arcsin (c x))}{7 d}-\frac {\left (20 b d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{147 c^3} \\ & = \frac {20 b d^2 \sqrt {d x} \sqrt {1-c^2 x^2}}{147 c^3}+\frac {4 b (d x)^{5/2} \sqrt {1-c^2 x^2}}{49 c}+\frac {2 (d x)^{7/2} (a+b \arcsin (c x))}{7 d}-\frac {20 b d^{5/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{147 c^{7/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {2 d^2 \sqrt {d x} \left (21 a c^3 x^3+10 b \sqrt {1-c^2 x^2}+6 b c^2 x^2 \sqrt {1-c^2 x^2}+21 b c^3 x^3 \arcsin (c x)-10 b \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )\right )}{147 c^3} \]
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Time = 1.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {2 a \left (d x \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d x \right )^{\frac {7}{2}} \arcsin \left (c x \right )}{7}-\frac {2 c \left (-\frac {d^{2} \left (d x \right )^{\frac {5}{2}} \sqrt {-c^{2} x^{2}+1}}{7 c^{2}}-\frac {5 d^{4} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{21 c^{4}}+\frac {5 d^{4} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{21 c^{4} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{7 d}\right )}{d}\) | \(144\) |
default | \(\frac {\frac {2 a \left (d x \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d x \right )^{\frac {7}{2}} \arcsin \left (c x \right )}{7}-\frac {2 c \left (-\frac {d^{2} \left (d x \right )^{\frac {5}{2}} \sqrt {-c^{2} x^{2}+1}}{7 c^{2}}-\frac {5 d^{4} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{21 c^{4}}+\frac {5 d^{4} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{21 c^{4} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{7 d}\right )}{d}\) | \(144\) |
parts | \(\frac {2 a \left (d x \right )^{\frac {7}{2}}}{7 d}+\frac {2 b \left (\frac {\left (d x \right )^{\frac {7}{2}} \arcsin \left (c x \right )}{7}-\frac {2 c \left (-\frac {d^{2} \left (d x \right )^{\frac {5}{2}} \sqrt {-c^{2} x^{2}+1}}{7 c^{2}}-\frac {5 d^{4} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{21 c^{4}}+\frac {5 d^{4} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{21 c^{4} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{7 d}\right )}{d}\) | \(146\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82 \[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=\frac {2 \, {\left (10 \, \sqrt {-c^{2} d} b d^{2} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right ) + {\left (21 \, b c^{5} d^{2} x^{3} \arcsin \left (c x\right ) + 21 \, a c^{5} d^{2} x^{3} + 2 \, {\left (3 \, b c^{4} d^{2} x^{2} + 5 \, b c^{2} d^{2}\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {d x}\right )}}{147 \, c^{5}} \]
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Time = 68.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71 \[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=a \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {7}{2}}}{7 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} \frac {d^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {c^{2} x^{2} e^{2 i \pi }} \right )}}{7 \Gamma \left (\frac {13}{4}\right )} & \text {for}\: d > -\infty \wedge d < \infty \wedge d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {7}{2}}}{7 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \operatorname {asin}{\left (c x \right )} \]
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\[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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\[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=\int { \left (d x\right )^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (d x)^{5/2} (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d\,x\right )}^{5/2} \,d x \]
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