Optimal. Leaf size=222 \[ -\frac {48478}{729} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}+\frac {136028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}-\frac {48478 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645} \]
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Rubi [A]
time = 0.05, antiderivative size = 222, 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 {48478 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}+\frac {136028 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}-\frac {6464 \sqrt {1-2 x} (5 x+3)^{5/2}}{81 \sqrt {3 x+2}}+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 (3 x+2)^{3/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {48478}{729} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \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} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {2}{15} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {4}{135} \int \frac {\left (-295-\frac {4925 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}+\frac {8}{405} \int \frac {\left (\frac {118045}{4}-\frac {206925 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}-\frac {8 \int \frac {\left (\frac {137475}{2}-\frac {1817925 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6075}\\ &=-\frac {48478}{729} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}+\frac {8 \int \frac {-\frac {2121825}{8}-\frac {2550525 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{54675}\\ &=-\frac {48478}{729} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}-\frac {136028 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3645}+\frac {266629 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3645}\\ &=-\frac {48478}{729} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11036}{81} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 (2+3 x)^{3/2}}-\frac {6464 \sqrt {1-2 x} (3+5 x)^{5/2}}{81 \sqrt {2+3 x}}+\frac {136028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}-\frac {48478 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3645}\\ \end {align*}
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Mathematica [A]
time = 9.09, size = 109, normalized size = 0.49 \begin {gather*} \frac {\frac {6 \sqrt {1-2 x} \sqrt {3+5 x} \left (-158237-517257 x-461043 x^2-45090 x^3+24300 x^4\right )}{(2+3 x)^{5/2}}+\sqrt {2} \left (-136028 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+935915 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{10935} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 318, normalized size = 1.43
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {40 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81}-\frac {1388 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729}-\frac {28291 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15309 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {136028 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{15309 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{32805 \left (\frac {2}{3}+x \right )^{3}}+\frac {5698 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{32805 \left (\frac {2}{3}+x \right )^{2}}-\frac {68014 \left (-30 x^{2}-3 x +9\right )}{10935 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(299\) |
default | \(-\frac {\left (7198983 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1224252 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+9598644 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1632336 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1458000 x^{6}+3199548 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+544112 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2559600 x^{5}+28370520 x^{4}+32990058 x^{3}+4298988 x^{2}-8361204 x -2848266\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{10935 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(318\) |
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 = 0.25, size = 60, normalized size = 0.27 \begin {gather*} \frac {2 \, {\left (24300 \, x^{4} - 45090 \, x^{3} - 461043 \, x^{2} - 517257 \, x - 158237\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3645 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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 = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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