Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {16}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {494}{135} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {214}{135} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 129, 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, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=-\frac {214}{135} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {494}{135} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}-\frac {16}{27} \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x} \]
[In]
[Out]
Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {13}{2}-20 x\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {16}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{135} \int \frac {-\frac {305}{4}-\frac {1235 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {16}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {494}{135} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {1177}{135} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {16}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {494}{135} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {214}{135} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=\frac {2}{405} \left (-\frac {15 \sqrt {1-2 x} \sqrt {3+5 x} (25+6 x)}{\sqrt {2+3 x}}-247 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+140 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \]
[In]
[Out]
Time = 1.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (132 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-247 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-900 x^{3}-3840 x^{2}-105 x +1125\right )}{405 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(140\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {14 \left (-30 x^{2}-3 x +9\right )}{27 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {122 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2835 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {988 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2835 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(214\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=-\frac {2 \, {\left (675 \, {\left (6 \, x + 25\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 1468 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 11115 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{18225 \, {\left (3 \, x + 2\right )}} \]
[In]
[Out]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]
[In]
[Out]