Optimal. Leaf size=180 \[ -\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {28291441 \sqrt {5 x+3} \sqrt {1-2 x}}{1185408 (3 x+2)}+\frac {270463 \sqrt {5 x+3} \sqrt {1-2 x}}{84672 (3 x+2)^2}+\frac {7723 \sqrt {5 x+3} \sqrt {1-2 x}}{15120 (3 x+2)^3}+\frac {41 \sqrt {5 x+3} \sqrt {1-2 x}}{360 (3 x+2)^4}-\frac {11988317 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ -\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {28291441 \sqrt {5 x+3} \sqrt {1-2 x}}{1185408 (3 x+2)}+\frac {270463 \sqrt {5 x+3} \sqrt {1-2 x}}{84672 (3 x+2)^2}+\frac {7723 \sqrt {5 x+3} \sqrt {1-2 x}}{15120 (3 x+2)^3}+\frac {41 \sqrt {5 x+3} \sqrt {1-2 x}}{360 (3 x+2)^4}-\frac {11988317 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 93
Rule 97
Rule 149
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\left (-\frac {13}{2}-20 x\right ) \sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}-\frac {1}{180} \int \frac {-\frac {1361}{4}+455 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {7723 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}-\frac {\int \frac {-\frac {244825}{8}+38615 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{3780}\\ &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {7723 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {270463 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}-\frac {\int \frac {-\frac {29121535}{16}+\frac {6761575 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{52920}\\ &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {7723 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {270463 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {28291441 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {\int -\frac {1618422795}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{370440}\\ &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {7723 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {270463 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {28291441 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}+\frac {11988317 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {7723 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {270463 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {28291441 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}+\frac {11988317 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {7723 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {270463 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {28291441 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {11988317 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 140, normalized size = 0.78 \[ \frac {9007 \left (7 \sqrt {1-2 x} \sqrt {5 x+3} \left (3103 x^2+4366 x+1488\right )-3993 \sqrt {7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{921984 (3 x+2)^3}+\frac {153 (5 x+3)^{3/2} (1-2 x)^{5/2}}{392 (3 x+2)^4}+\frac {3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{35 (3 x+2)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.96, size = 131, normalized size = 0.73 \[ -\frac {179824755 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1273114845 \, x^{4} + 3451770150 \, x^{3} + 3511594796 \, x^{2} + 1588955864 \, x + 269759904\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9219840 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 3.06, size = 426, normalized size = 2.37 \[ \frac {11988317}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (27021 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 52500560 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 18029240320 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 2768103296000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {166086197760000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {664344791040000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{65856 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 298, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (43697415465 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+145658051550 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+17823607830 \sqrt {-10 x^{2}-x +3}\, x^{4}+194210735400 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+48324782100 \sqrt {-10 x^{2}-x +3}\, x^{3}+129473823600 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+49162327144 \sqrt {-10 x^{2}-x +3}\, x^{2}+43157941200 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+22245382096 \sqrt {-10 x^{2}-x +3}\, x +5754392160 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3776638656 \sqrt {-10 x^{2}-x +3}\right )}{9219840 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 198, normalized size = 1.10 \[ \frac {11988317}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {495385}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{5 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {239 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {8395 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2352 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {297231 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {3665849 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^6} \,d x \]
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]
________________________________________________________________________________________