Integrand size = 23, antiderivative size = 122 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{3 d e^3 \sqrt {c+d x}} \]
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Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4889, 4723, 331, 326, 324, 435} \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {4 b \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {-c-d x+1}}{\sqrt {2}}\right )\right |2\right )}{3 d e^3 \sqrt {c+d x}}-\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}} \]
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Rule 324
Rule 326
Rule 331
Rule 435
Rule 4723
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \arcsin (x)}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{(e x)^{3/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}-\frac {(2 b) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{3 d e^3} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}-\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 )}{3 d e^3 \sqrt {c+d x}} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\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 )}{3 d e^3 \sqrt {c+d x}} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \arcsin (c+d x))}{3 d e (e (c+d x))^{3/2}}+\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 )}{3 d e^3 \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 = 56, normalized size of antiderivative = 0.46 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \left (a+b \arcsin (c+d x)+2 b (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},(c+d x)^2\right )\right )}{3 d e (e (c+d x))^{3/2}} \]
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Result contains complex when optimal does not.
Time = 1.72 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.56
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{3 e \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(190\) |
default | \(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{3 e \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(190\) |
parts | \(-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}} d e}+\frac {2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )\right )}{3 e \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{e d}\) | \(195\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + \sqrt {d e x + c e} {\left (b d \arcsin \left (d x + c\right ) + a d + 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )}\right )}}{3 \, {\left (d^{4} e^{3} x^{2} + 2 \, c d^{3} e^{3} x + c^{2} d^{2} e^{3}\right )}} \]
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\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]
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