Integrand size = 23, antiderivative size = 102 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 (a+b \arcsin (c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{15 d e^{7/2}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 102, 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, 331, 335, 227} \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=-\frac {2 (a+b \arcsin (c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{15 d e^{7/2}}-\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}} \]
[In]
[Out]
Rule 227
Rule 331
Rule 335
Rule 4723
Rule 4889
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \arcsin (x)}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 (a+b \arcsin (c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{(e x)^{5/2} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{5 d e} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 (a+b \arcsin (c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1-x^2}} \, dx,x,c+d x\right )}{15 d e^3} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 (a+b \arcsin (c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{15 d e^4} \\ & = -\frac {4 b \sqrt {1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 (a+b \arcsin (c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{15 d e^{7/2}} \\ \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 = 0.58 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=\frac {-6 (a+b \arcsin (c+d x))-4 b (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},(c+d x)^2\right )}{15 d e (e (c+d x))^{5/2}} \]
[In]
[Out]
Time = 2.10 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.66
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{15 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{15 e^{2} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(169\) |
default | \(\frac {-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{15 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{15 e^{2} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(169\) |
parts | \(-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}} d e}+\frac {2 b \left (-\frac {\arcsin \left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {-\frac {2 \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{15 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1-\frac {d e x +c e}{e}}\, \sqrt {1+\frac {d e x +c e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {1}{e}}, i\right )}{15 e^{2} \sqrt {\frac {1}{e}}\, \sqrt {-\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{e d}\) | \(174\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.72 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (3 \, b d^{2} \arcsin \left (d x + c\right ) + 3 \, a d^{2} + 2 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d e x + c e}\right )}}{15 \, {\left (d^{6} e^{4} x^{3} + 3 \, c d^{5} e^{4} x^{2} + 3 \, c^{2} d^{4} e^{4} x + c^{3} d^{3} e^{4}\right )}} \]
[In]
[Out]
\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {7}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \]
[In]
[Out]