Optimal. Leaf size=195 \[ -\frac {10385 \sqrt {1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {2075}{72} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {48625 \sqrt {1-2 x} \sqrt {5 x+3}}{1944}-\frac {21935 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{5832 \sqrt {7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {10385 \sqrt {1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {2075}{72} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {48625 \sqrt {1-2 x} \sqrt {5 x+3}}{1944}-\frac {21935 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{5832 \sqrt {7}} \]
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)^{5/2}}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (-\frac {2005}{4}-2050 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {1}{162} \int \frac {\left (\frac {109865}{8}-56025 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {\int \frac {\left (\frac {19665}{2}-291750 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{1944}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {\int \frac {-468735-1316100 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{11664}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {408665 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{11664}-\frac {109675 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2916}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {408665 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{5832}-\frac {\left (21935 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1458}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {21935 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5832 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.32, size = 144, normalized size = 0.74 \[ \frac {21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (32400 x^4-93420 x^3-420531 x^2-391014 x-107984\right )-408665 \sqrt {14 x-7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+307090 \sqrt {10-20 x} (3 x+2)^3 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{40824 \sqrt {2 x-1} (3 x+2)^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.13, size = 172, normalized size = 0.88 \[ \frac {307090 \, \sqrt {5} \sqrt {2} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 408665 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (32400 \, x^{4} - 93420 \, x^{3} - 420531 \, x^{2} - 391014 \, x - 107984\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{81648 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 5.14, size = 409, normalized size = 2.10 \[ \frac {81733}{163296} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{486} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 329 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {21935}{5832} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (2803 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1982400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {411208000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {1644832000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{324 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 287, normalized size = 1.47 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-1360800 \sqrt {-10 x^{2}-x +3}\, x^{4}+8291430 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-11033955 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3923640 \sqrt {-10 x^{2}-x +3}\, x^{3}+16582860 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-22067910 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+17662302 \sqrt {-10 x^{2}-x +3}\, x^{2}+11055240 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-14711940 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+16422588 \sqrt {-10 x^{2}-x +3}\, x +2456720 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3269320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4535328 \sqrt {-10 x^{2}-x +3}\right )}{81648 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.44, size = 190, normalized size = 0.97 \[ -\frac {185}{882} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {16075}{1764} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {189865}{31752} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {6347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3528 \, {\left (3 \, x + 2\right )}} + \frac {41225}{2268} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21935}{5832} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {408665}{81648} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {191965}{13608} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,d x \]
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]
________________________________________________________________________________________