Optimal. Leaf size=189 \[ -\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt {3+5 x}}+\frac {8874 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}-\frac {572}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {9206 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {7738 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625} \]
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Rubi [A]
time = 0.04, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {7738 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}+\frac {9206 \sqrt {33} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac {178 (3 x+2)^{3/2} (1-2 x)^{3/2}}{75 \sqrt {5 x+3}}-\frac {572}{625} (3 x+2)^{3/2} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {8874 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{3125} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (-\frac {11}{2}-24 x\right ) (1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (-\frac {291}{2}-\frac {1287 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt {3+5 x}}-\frac {572}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (\frac {5175}{2}-\frac {119799 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{5625}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt {3+5 x}}+\frac {8874 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}-\frac {572}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {8 \int \frac {\frac {217593}{8}+\frac {372843 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{84375}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt {3+5 x}}+\frac {8874 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}-\frac {572}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {27618 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{15625}+\frac {42559 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15625}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {178 (1-2 x)^{3/2} (2+3 x)^{3/2}}{75 \sqrt {3+5 x}}+\frac {8874 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}-\frac {572}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {9206 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {7738 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}\\ \end {align*}
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Mathematica [A]
time = 7.56, size = 107, normalized size = 0.57 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (-25421-48650 x-9450 x^2+4500 x^3\right )}{(3+5 x)^{3/2}}-27618 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+155295 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{46875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 225, normalized size = 1.19
method | result | size |
default | \(-\frac {\left (638385 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+138090 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+383031 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+82854 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-270000 x^{5}+522000 x^{4}+3103500 x^{3}+1822760 x^{2}-718790 x -508420\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{46875 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(225\) |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {24 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}-\frac {396 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3125}-\frac {8059 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{65625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9206 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46875 \left (x +\frac {3}{5}\right )^{2}}-\frac {2596 \left (-30 x^{2}-5 x +10\right )}{9375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(275\) |
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 = 0.18, size = 50, normalized size = 0.26 \begin {gather*} \frac {2 \, {\left (4500 \, x^{3} - 9450 \, x^{2} - 48650 \, x - 25421\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9375 \, {\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}\,{\left (3\,x+2\right )}^{3/2}}{{\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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