Integrand size = 28, antiderivative size = 125 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}+\frac {62 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {8 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{25 \sqrt {33}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 157, 164, 114, 120} \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {8 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {62 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{165 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}} \]
[In]
[Out]
Rule 99
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {-\frac {1}{2}-6 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}-\frac {4}{165} \int \frac {\frac {69}{2}+\frac {93 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}-\frac {4}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {62}{275} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}+\frac {62 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {8 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {2}{825} \left (-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x} (104+155 x)}{(3+5 x)^{3/2}}-31 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+35 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(93)=186\).
Time = 1.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.75
method | result | size |
default | \(\frac {2 \left (165 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-155 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+99 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-93 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4650 x^{3}-3895 x^{2}+1030 x +1040\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{825 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(219\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{375 \left (x +\frac {3}{5}\right )^{2}}-\frac {62 \left (-30 x^{2}-5 x +10\right )}{825 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {92 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5775 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {124 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{5775 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(219\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=-\frac {450 \, {\left (155 \, x + 104\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1357 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2790 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{37125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
[In]
[Out]
\[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {3 x + 2}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
[In]
[Out]