Optimal. Leaf size=127 \[ -\frac {22 (1-2 x)^{3/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {572 \sqrt {1-2 x} \sqrt {2+3 x}}{25 \sqrt {3+5 x}}-\frac {584}{125} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {68}{125} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 155, 164,
114, 120} \begin {gather*} -\frac {68}{125} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {584}{125} \sqrt {33} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {22 \sqrt {3 x+2} (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac {572 \sqrt {3 x+2} \sqrt {1-2 x}}{25 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} (3+5 x)^{5/2}} \, dx &=-\frac {22 (1-2 x)^{3/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {2}{15} \int \frac {\sqrt {1-2 x} (102+27 x)}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {22 (1-2 x)^{3/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {572 \sqrt {1-2 x} \sqrt {2+3 x}}{25 \sqrt {3+5 x}}-\frac {4}{75} \int \frac {-\frac {1689}{2}-1314 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {22 (1-2 x)^{3/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {572 \sqrt {1-2 x} \sqrt {2+3 x}}{25 \sqrt {3+5 x}}+\frac {374}{125} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {1752}{125} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {22 (1-2 x)^{3/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {572 \sqrt {1-2 x} \sqrt {2+3 x}}{25 \sqrt {3+5 x}}-\frac {584}{125} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {68}{125} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 7.31, size = 97, normalized size = 0.76 \begin {gather*} \frac {2}{375} \left (\frac {55 \sqrt {1-2 x} \sqrt {2+3 x} (229+400 x)}{(3+5 x)^{3/2}}+876 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-315 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs.
\(2(93)=186\).
time = 0.10, size = 215, normalized size = 1.69
method | result | size |
default | \(-\frac {2 \left (2805 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4380 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1683 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2628 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-132000 x^{3}-97570 x^{2}+31405 x +25190\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{375 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {1126 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{525 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {584 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{175 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1875 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {704}{5} x^{2}-\frac {352}{15} x +\frac {704}{15}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(225\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.24, size = 40, normalized size = 0.31 \begin {gather*} \frac {22 \, {\left (400 \, x + 229\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{75 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}}{\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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