Integrand size = 16, antiderivative size = 124 \[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=\frac {4 b (d x)^{3/2} \sqrt {1-c^2 x^2}}{25 c}+\frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}-\frac {12 b d^{3/2} E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{25 c^{5/2}}+\frac {12 b d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{25 c^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4723, 327, 335, 313, 227, 1213, 435} \[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=\frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}+\frac {12 b d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{25 c^{5/2}}-\frac {12 b d^{3/2} E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{25 c^{5/2}}+\frac {4 b \sqrt {1-c^2 x^2} (d x)^{3/2}}{25 c} \]
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Rule 227
Rule 313
Rule 327
Rule 335
Rule 435
Rule 1213
Rule 4723
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}-\frac {(2 b c) \int \frac {(d x)^{5/2}}{\sqrt {1-c^2 x^2}} \, dx}{5 d} \\ & = \frac {4 b (d x)^{3/2} \sqrt {1-c^2 x^2}}{25 c}+\frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}-\frac {(6 b d) \int \frac {\sqrt {d x}}{\sqrt {1-c^2 x^2}} \, dx}{25 c} \\ & = \frac {4 b (d x)^{3/2} \sqrt {1-c^2 x^2}}{25 c}+\frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}-\frac {(12 b) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{25 c} \\ & = \frac {4 b (d x)^{3/2} \sqrt {1-c^2 x^2}}{25 c}+\frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}+\frac {(12 b d) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{25 c^2}-\frac {(12 b d) \text {Subst}\left (\int \frac {1+\frac {c x^2}{d}}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{25 c^2} \\ & = \frac {4 b (d x)^{3/2} \sqrt {1-c^2 x^2}}{25 c}+\frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}+\frac {12 b d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{25 c^{5/2}}-\frac {(12 b d) \text {Subst}\left (\int \frac {\sqrt {1+\frac {c x^2}{d}}}{\sqrt {1-\frac {c x^2}{d}}} \, dx,x,\sqrt {d x}\right )}{25 c^2} \\ & = \frac {4 b (d x)^{3/2} \sqrt {1-c^2 x^2}}{25 c}+\frac {2 (d x)^{5/2} (a+b \arcsin (c x))}{5 d}-\frac {12 b d^{3/2} E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{25 c^{5/2}}+\frac {12 b d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{25 c^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.53 \[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=\frac {2 (d x)^{3/2} \left (5 a c x+2 b \sqrt {1-c^2 x^2}+5 b c x \arcsin (c x)-2 b \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )}{25 c} \]
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Time = 0.64 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d x \right )^{\frac {5}{2}} \arcsin \left (c x \right )}{5}-\frac {2 c \left (-\frac {d^{2} \left (d x \right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}{5 c^{2}}-\frac {3 d^{3} \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{5 c^{3} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{5 d}\right )}{d}\) | \(138\) |
default | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (d x \right )^{\frac {5}{2}} \arcsin \left (c x \right )}{5}-\frac {2 c \left (-\frac {d^{2} \left (d x \right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}{5 c^{2}}-\frac {3 d^{3} \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{5 c^{3} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{5 d}\right )}{d}\) | \(138\) |
parts | \(\frac {2 a \left (d x \right )^{\frac {5}{2}}}{5 d}+\frac {2 b \left (\frac {\left (d x \right )^{\frac {5}{2}} \arcsin \left (c x \right )}{5}-\frac {2 c \left (-\frac {d^{2} \left (d x \right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}{5 c^{2}}-\frac {3 d^{3} \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{5 c^{3} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{5 d}\right )}{d}\) | \(140\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69 \[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=-\frac {2 \, {\left (6 \, \sqrt {-c^{2} d} b d {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) - {\left (5 \, b c^{3} d x^{2} \arcsin \left (c x\right ) + 5 \, a c^{3} d x^{2} + 2 \, \sqrt {-c^{2} x^{2} + 1} b c^{2} d x\right )} \sqrt {d x}\right )}}{25 \, c^{3}} \]
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Time = 11.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69 \[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=a \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {5}{2}}}{5 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} \frac {d^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {c^{2} x^{2} e^{2 i \pi }} \right )}}{5 \Gamma \left (\frac {11}{4}\right )} & \text {for}\: d > -\infty \wedge d < \infty \wedge d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {5}{2}}}{5 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \operatorname {asin}{\left (c x \right )} \]
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\[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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\[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (d x)^{3/2} (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d\,x\right )}^{3/2} \,d x \]
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