Integrand size = 23, antiderivative size = 81 \[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}} \]
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Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4889, 4723, 326, 324, 435} \[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}} \]
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Rule 324
Rule 326
Rule 435
Rule 4723
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \arcsin (x)}{\sqrt {e x}} \, dx,x,c+d x\right )}{d} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))}{d e}-\frac {(2 b) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))}{d e}-\frac {\left (2 b \sqrt {e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e \sqrt {c+d x}} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))}{d e}+\frac {\left (4 b \sqrt {e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1-2 x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )}{d e \sqrt {c+d x}} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \arcsin (c+d x))}{d e}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {c+d x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=-\frac {2 \sqrt {e (c+d x)} \left (-3 (a+b \arcsin (c+d x))+2 b (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},(c+d x)^2\right )\right )}{3 d e} \]
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Result contains complex when optimal does not.
Time = 1.77 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.84
method | result | size |
derivativedivides | \(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \arcsin \left (\frac {d e x +c e}{e}\right )+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{\sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{d e}\) | \(149\) |
default | \(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \arcsin \left (\frac {d e x +c e}{e}\right )+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{\sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{d e}\) | \(149\) |
parts | \(\frac {2 a \sqrt {d e x +c e}}{d e}+\frac {2 b \left (\sqrt {d e x +c e}\, \arcsin \left (\frac {d e x +c e}{e}\right )+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{\sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{e d}\) | \(155\) |
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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 = 0.84 \[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {-d^{3} e} b {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) - \sqrt {d e x + c e} {\left (b d \arcsin \left (d x + c\right ) + a d\right )}\right )}}{d^{2} e} \]
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Exception generated. \[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: TypeError} \]
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Exception generated. \[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{\sqrt {d e x + c e}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{\sqrt {c e+d e x}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{\sqrt {c\,e+d\,e\,x}} \,d x \]
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