Optimal. Leaf size=222 \[ \frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {301304}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {33232}{35} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164,
114, 120} \begin {gather*} -\frac {33232}{35} \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {301304}{35} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {301304 \sqrt {1-2 x} \sqrt {3 x+2}}{21 \sqrt {5 x+3}}-\frac {16616 \sqrt {1-2 x} \sqrt {3 x+2}}{7 (5 x+3)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {156-235 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4}{315} \int \frac {\frac {33999}{2}-23450 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {8 \int \frac {1277955-\frac {2936955 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{2205}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}-\frac {16 \int \frac {\frac {209379555}{4}-\frac {64771245 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{72765}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {32 \int \frac {681610545+\frac {4306575735 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{800415}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {49848}{35} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {301304}{35} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {301304}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {33232}{35} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 7.74, size = 109, normalized size = 0.49 \begin {gather*} \frac {2}{105} \left (\frac {\sqrt {1-2 x} \left (17157169+107221804 x+251053266 x^2+261029520 x^3+101690100 x^4\right )}{(2+3 x)^{5/2} (3+5 x)^{3/2}}+4 \sqrt {2} \left (37663 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-18970 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(397\) vs.
\(2(162)=324\).
time = 0.10, size = 398, normalized size = 1.79
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3 \left (x +\frac {3}{5}\right )^{2}}+\frac {-37100 x^{2}-\frac {18550}{3} x +\frac {37100}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45 \left (\frac {2}{3}+x \right )^{3}}+\frac {-\frac {342908}{7} x^{2}-\frac {171454}{35} x +\frac {514362}{35}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {956 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45 \left (\frac {2}{3}+x \right )^{2}}+\frac {63584 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{49 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {301304 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(301\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (3364740 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-6779340 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+6505164 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-13106724 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+4187232 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-8436512 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+897264 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1807824 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-203380200 x^{5}-420368940 x^{4}-241077012 x^{3}+36609658 x^{2}+72907466 x +17157169\right )}{105 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(398\) |
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 = 0.23, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (101690100 \, x^{4} + 261029520 \, x^{3} + 251053266 \, x^{2} + 107221804 \, x + 17157169\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{105 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________