Optimal. Leaf size=156 \[ -\frac {8 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{121 \sqrt {33}}-\frac {245 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 \sqrt {5 x+3}}+\frac {8 \sqrt {3 x+2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}+\frac {49 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ -\frac {245 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 \sqrt {5 x+3}}+\frac {8 \sqrt {3 x+2}}{363 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {8 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}+\frac {49 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {7 \sqrt {2+3 x}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1}{33} \int \frac {-\frac {19}{2}-9 x}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {7 \sqrt {2+3 x}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2 \int \frac {\frac {847}{4}+210 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2541}\\ &=\frac {7 \sqrt {2+3 x}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {245 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}-\frac {4 \int \frac {\frac {2163}{4}+\frac {5145 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{27951}\\ &=\frac {7 \sqrt {2+3 x}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {245 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {4}{121} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {49 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1331}\\ &=\frac {7 \sqrt {2+3 x}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {245 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {49 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}-\frac {8 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{121 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 99, normalized size = 0.63 \[ \frac {\sqrt {2} \left (181 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-49 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )-\frac {2 \sqrt {3 x+2} \left (490 x^2-402 x-345\right )}{(1-2 x)^{3/2} \sqrt {5 x+3}}}{3993} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, x - 9}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.03, size = 228, normalized size = 1.46 \[ -\frac {\left (2940 x^{3}-452 x^{2}-98 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+362 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-3678 x +49 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-181 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1380\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {3 x +2}}{3993 \left (15 x^{2}+19 x +6\right ) \left (2 x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________