Optimal. Leaf size=220 \[ -\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4801 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {25643 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {24369 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159,
164, 114, 120} \begin {gather*} -\frac {24369 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {25643 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {6 \sqrt {1-2 x} (3 x+2)^{7/2}}{\sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac {622}{175} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {3872 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{4375}+\frac {4801 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{21875} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
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 {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {9}{2}-30 x\right ) \sqrt {1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (\frac {1545}{2}-\frac {4665 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {4 \int \frac {(2+3 x)^{3/2} \left (-\frac {15705}{4}+14520 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2625}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {4 \int \frac {\left (\frac {29625}{2}-\frac {216045 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{65625}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4801 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {4 \int \frac {-\frac {2042685}{8}-\frac {1153935 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{984375}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4801 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {25643 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{109375}+\frac {73107 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{218750}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4801 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {25643 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {24369 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 7.90, size = 112, normalized size = 0.51 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (52067+216050 x+174525 x^2-189000 x^3-202500 x^4\right )}{(3+5 x)^{3/2}}+51286 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+168035 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{656250} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 230, normalized size = 1.05
method | result | size |
default | \(-\frac {\left (1096605 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-256430 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+12150000 x^{6}+657963 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-153858 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+13365000 x^{5}-12631500 x^{4}-18488250 x^{3}-1794020 x^{2}+3800330 x +1041340\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{656250 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(230\) |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {108 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}+\frac {144 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {487 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {15131 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{306250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {25643 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{459375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{234375 \left (x +\frac {3}{5}\right )^{2}}-\frac {644 \left (-30 x^{2}-5 x +10\right )}{46875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(297\) |
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 = 0.14, size = 55, normalized size = 0.25 \begin {gather*} -\frac {{\left (202500 \, x^{4} + 189000 \, x^{3} - 174525 \, x^{2} - 216050 \, x - 52067\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{65625 \, {\left (25 \, x^{2} + 30 \, x + 9\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]
________________________________________________________________________________________