Optimal. Leaf size=191 \[ -\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {1973 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {8366 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625} \]
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Rubi [A]
time = 0.05, antiderivative size = 191, 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 {8366 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}+\frac {1973 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {106 \sqrt {1-2 x} (3 x+2)^{5/2}}{25 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac {1558}{625} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {2264 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{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)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {3}{2}-24 x\right ) \sqrt {1-2 x} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (306-\frac {2337 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {4 \int \frac {\sqrt {2+3 x} \left (-\frac {2775}{4}+5094 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1875}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {4 \int \frac {\frac {5967}{2}-\frac {17757 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{28125}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {1973 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{15625}+\frac {12549 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15625}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {1973 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {8366 \sqrt {\frac {3}{11}} 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 = 7.60, size = 107, normalized size = 0.56 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (-106+2975 x+1050 x^2-6750 x^3\right )}{(3+5 x)^{3/2}}-1973 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+39620 \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 = 1.18
method | result | size |
default | \(-\frac {\left (188235 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+9865 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+112941 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+5919 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+405000 x^{5}+4500 x^{4}-324000 x^{3}-2390 x^{2}+60560 x -2120\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{46875 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(225\) |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {36 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}+\frac {244 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3125}+\frac {442 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1973 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{65625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46875 \left (x +\frac {3}{5}\right )^{2}}-\frac {446 \left (-30 x^{2}-5 x +10\right )}{9375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(275\) |
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 (6750 \, x^{3} - 1050 \, x^{2} - 2975 \, x + 106\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 )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}}{{\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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