Optimal. Leaf size=218 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {1473539 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750 \sqrt {33}}-\frac {31288 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375 \sqrt {33}} \]
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Rubi [A]
time = 0.06, antiderivative size = 218, 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 {31288 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375 \sqrt {33}}-\frac {1473539 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac {118 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 \sqrt {5 x+3}}+\frac {958 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{1925}+\frac {5153 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{48125}-\frac {12601 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{240625} \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 {\sqrt {1-2 x} (2+3 x)^{9/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {23}{2}-30 x\right ) (2+3 x)^{7/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}+\frac {4}{825} \int \frac {\left (\frac {4875}{4}-\frac {7185 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {4 \int \frac {(2+3 x)^{3/2} \left (-\frac {32295}{4}+\frac {77295 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{28875}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}+\frac {4 \int \frac {\sqrt {2+3 x} \left (\frac {1297125}{8}+\frac {567045 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{721875}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {4 \int \frac {-\frac {42883065}{8}-\frac {66309255 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10828125}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}+\frac {15644 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{109375}+\frac {1473539 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2406250}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{9/2}}{15 (3+5 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 \sqrt {3+5 x}}-\frac {12601 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{240625}+\frac {5153 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{48125}+\frac {958 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{1925}-\frac {1473539 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{218750 \sqrt {33}}-\frac {31288 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 4.88, size = 112, normalized size = 0.51 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (54083+1854575 x+6882975 x^2+8575875 x^3+3341250 x^4\right )}{(3+5 x)^{3/2}}+1473539 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-441035 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{7218750} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 230, normalized size = 1.06
method | result | size |
default | \(-\frac {\left (5162520 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7367695 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-200475000 x^{6}+3097512 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4420617 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-547965000 x^{5}-431912250 x^{4}-8586750 x^{3}+115868770 x^{2}+36550670 x +1081660\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{7218750 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(230\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {162 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}+\frac {1107 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {243 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21875}+\frac {952957 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{10106250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1473539 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{10106250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{234375 \left (x +\frac {3}{5}\right )^{2}}-\frac {854 \left (-30 x^{2}-5 x +10\right )}{515625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(286\) |
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 = 0.12, size = 55, normalized size = 0.25 \begin {gather*} \frac {{\left (3341250 \, x^{4} + 8575875 \, x^{3} + 6882975 \, x^{2} + 1854575 \, x + 54083\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{721875 \, {\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(-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 {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{9/2}}{{\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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