Integrand size = 16, antiderivative size = 88 \[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=\frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arcsin (c x))}{3 d}-\frac {4 b \sqrt {d} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{9 c^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4723, 327, 335, 227} \[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=\frac {2 (d x)^{3/2} (a+b \arcsin (c x))}{3 d}-\frac {4 b \sqrt {d} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{9 c^{3/2}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d x}}{9 c} \]
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Rule 227
Rule 327
Rule 335
Rule 4723
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{3/2} (a+b \arcsin (c x))}{3 d}-\frac {(2 b c) \int \frac {(d x)^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{3 d} \\ & = \frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arcsin (c x))}{3 d}-\frac {(2 b d) \int \frac {1}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{9 c} \\ & = \frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arcsin (c x))}{3 d}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{9 c} \\ & = \frac {4 b \sqrt {d x} \sqrt {1-c^2 x^2}}{9 c}+\frac {2 (d x)^{3/2} (a+b \arcsin (c x))}{3 d}-\frac {4 b \sqrt {d} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{9 c^{3/2}} \\ \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 = 0.75 \[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=\frac {2 \sqrt {d x} \left (3 a c x+2 b \sqrt {1-c^2 x^2}+3 b c x \arcsin (c x)-2 b \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )\right )}{9 c} \]
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Time = 0.41 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \arcsin \left (c x \right )}{3}-\frac {2 c \left (-\frac {d^{2} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}+\frac {d^{2} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{3 c^{2} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{3 d}\right )}{d}\) | \(119\) |
default | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \arcsin \left (c x \right )}{3}-\frac {2 c \left (-\frac {d^{2} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}+\frac {d^{2} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{3 c^{2} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{3 d}\right )}{d}\) | \(119\) |
parts | \(\frac {2 a \left (d x \right )^{\frac {3}{2}}}{3 d}+\frac {2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \arcsin \left (c x \right )}{3}-\frac {2 c \left (-\frac {d^{2} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}+\frac {d^{2} \sqrt {-c x +1}\, \sqrt {c x +1}\, \operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{3 c^{2} \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{3 d}\right )}{d}\) | \(121\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.77 \[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=\frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right ) + {\left (3 \, b c^{3} x \arcsin \left (c x\right ) + 3 \, a c^{3} x + 2 \, \sqrt {-c^{2} x^{2} + 1} b c^{2}\right )} \sqrt {d x}\right )}}{9 \, c^{3}} \]
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Time = 3.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=a \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {3}{2}}}{3 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} \frac {\sqrt {d} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {c^{2} x^{2} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {9}{4}\right )} & \text {for}\: d > -\infty \wedge d < \infty \wedge d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {2 \left (d x\right )^{\frac {3}{2}}}{3 d} & \text {for}\: d \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \operatorname {asin}{\left (c x \right )} \]
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\[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=\int { \sqrt {d x} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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\[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=\int { \sqrt {d x} {\left (b \arcsin \left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {d x} (a+b \arcsin (c x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d\,x} \,d x \]
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