Optimal. Leaf size=164 \[ -\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{36 (3 x+2)}+\frac {41}{18} (5 x+3)^{3/2} \sqrt {1-2 x}-\frac {1649}{108} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {6829 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{162 \sqrt {10}}-\frac {1945}{324} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ -\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{36 (3 x+2)}+\frac {41}{18} (5 x+3)^{3/2} \sqrt {1-2 x}-\frac {1649}{108} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {6829 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{162 \sqrt {10}}-\frac {1945}{324} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 54
Rule 93
Rule 97
Rule 149
Rule 154
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}-\frac {1}{18} \int \frac {\left (-\frac {1335}{4}-1230 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx\\ &=\frac {41}{18} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}-\frac {1}{540} \int \frac {\left (\frac {2115}{2}-49470 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {1649}{108} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {41}{18} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}+\frac {\int \frac {-68505-204870 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{3240}\\ &=-\frac {1649}{108} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {41}{18} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}+\frac {13615}{648} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx-\frac {6829}{324} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1649}{108} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {41}{18} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}+\frac {13615}{324} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {6829 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{162 \sqrt {5}}\\ &=-\frac {1649}{108} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {41}{18} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}-\frac {6829 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{162 \sqrt {10}}-\frac {1945}{324} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.27, size = 139, normalized size = 0.85 \[ \frac {15 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (360 x^3-1230 x^2-3471 x-1628\right )-9725 \sqrt {14 x-7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+6829 \sqrt {10-20 x} (3 x+2)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1620 \sqrt {2 x-1} (3 x+2)^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.99, size = 146, normalized size = 0.89 \[ -\frac {9725 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 6829 \, \sqrt {10} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (360 \, x^{3} - 1230 \, x^{2} - 3471 \, x - 1628\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3240 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.83, size = 351, normalized size = 2.14 \[ \frac {389}{1296} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{270} \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 107 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {6829}{3240} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {77 \, \sqrt {10} {\left (41 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {17640 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {70560 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{54 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 225, normalized size = 1.37 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-10800 \sqrt {-10 x^{2}-x +3}\, x^{3}+61461 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-87525 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+36900 \sqrt {-10 x^{2}-x +3}\, x^{2}+81948 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-116700 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+104130 \sqrt {-10 x^{2}-x +3}\, x +27316 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-38900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+48840 \sqrt {-10 x^{2}-x +3}\right )}{3240 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.30, size = 130, normalized size = 0.79 \[ \frac {5}{9} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {205}{18} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {6829}{3240} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1945}{648} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {911}{108} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {5 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^3} \,d x \]
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]
________________________________________________________________________________________