Optimal. Leaf size=249 \[ -\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}-\frac {584888452 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}-\frac {13235368 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157,
164, 114, 120} \begin {gather*} -\frac {13235368 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}-\frac {584888452 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}+\frac {36980 \sqrt {1-2 x} (5 x+3)^{5/2}}{18711 (3 x+2)^{7/2}}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {55772 \sqrt {1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{5/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {5 x+3}}{57760857 \sqrt {3 x+2}}-\frac {17089252 \sqrt {1-2 x} \sqrt {5 x+3}}{8251551 (3 x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2} \left (-\frac {3065}{2}+\frac {25 x}{2}\right )}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {8 \int \frac {\left (\frac {147745}{4}-23025 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{18711}\\ &=-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {16 \int \frac {\left (\frac {14799465}{8}-\frac {4921575 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{1964655}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {32 \int \frac {\frac {469321365}{16}-\frac {49085475 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{123773265}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {64 \int \frac {\frac {3426487275}{8}+\frac {10966658475 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{866412855}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {6617684 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5250987}+\frac {584888452 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{57760857}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}-\frac {584888452 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}-\frac {13235368 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 9.41, size = 112, normalized size = 0.45 \begin {gather*} \frac {\frac {48 \sqrt {2-4 x} \sqrt {3+5 x} \left (9770732477+72620507583 x+215597947743 x^2+320012032635 x^3+237923150688 x^4+71063946918 x^5\right )}{(2+3 x)^{11/2}}+9358215232 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5864078080 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{1386260568 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs.
\(2(185)=370\).
time = 0.10, size = 587, normalized size = 2.36
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {11914 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5845851 \left (\frac {2}{3}+x \right )^{5}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1948617 \left (\frac {2}{3}+x \right )^{6}}+\frac {2101378 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{31827411 \left (\frac {2}{3}+x \right )^{3}}-\frac {339634 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{13640319 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {5848884520}{57760857} x^{2}-\frac {584888452}{57760857} x +\frac {584888452}{19253619}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {3680168 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{74263959 \left (\frac {2}{3}+x \right )^{2}}+\frac {1827459880 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1212977997 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2924442260 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1212977997 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(332\) |
default | \(\frac {2 \left (71063946918 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-26533603998 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+236879823060 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-88445346660 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+315839764080 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-117927128880 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+210559842720 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-78618085920 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2131918407540 x^{7}+70186614240 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-26206028640 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+7350886361394 x^{6}+9358215232 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3494137152 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9674554908852 x^{5}+5286666174003 x^{4}-54699222996 x^{3}-1429398032628 x^{2}-624272370816 x -87936592293\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{173282571 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(587\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.17, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (71063946918 \, x^{5} + 237923150688 \, x^{4} + 320012032635 \, x^{3} + 215597947743 \, x^{2} + 72620507583 \, x + 9770732477\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{57760857 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________