Optimal. Leaf size=219 \[ \frac {1284329 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3780}+\frac {4853}{105} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {5}{3} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {42696881 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900}+\frac {1284329 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900} \]
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Rubi [A]
time = 0.05, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164,
114, 120} \begin {gather*} \frac {1284329 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900}+\frac {42696881 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900}+\frac {(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}+\frac {5}{3} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {4853}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {1284329 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{3780} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2} \left (\frac {95}{2}+75 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {5}{3} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {1}{45} \int \frac {\left (-\frac {13425}{2}-\frac {20925 x}{2}\right ) \sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {93}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {5}{3} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {\int \frac {(3+5 x)^{3/2} \left (\frac {2862975}{4}+1091925 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1575}\\ &=\frac {4853}{105} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {5}{3} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\int \frac {\left (-\frac {187797825}{4}-\frac {288974025 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{23625}\\ &=\frac {1284329 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3780}+\frac {4853}{105} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {5}{3} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {\int \frac {\frac {12163900725}{8}+\frac {9606798225 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{212625}\\ &=\frac {1284329 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3780}+\frac {4853}{105} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {5}{3} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {14127619 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{37800}-\frac {42696881 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{18900}\\ &=\frac {1284329 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3780}+\frac {4853}{105} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {5}{3} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {42696881 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900}+\frac {1284329 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900}\\ \end {align*}
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Mathematica [A]
time = 8.08, size = 120, normalized size = 0.55 \begin {gather*} \frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-2283923+1258906 x+795150 x^2+392400 x^3+94500 x^4\right )-85393762 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+43010905 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{113400 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 153, normalized size = 0.70
method | result | size |
default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (42525000 x^{6}+42382857 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-85393762 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+230445000 x^{5}+598495500 x^{4}+1090375200 x^{3}-167061930 x^{2}-1075233030 x -411106140\right )}{3402000 x^{3}+2608200 x^{2}-793800 x -680400}\) | \(153\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {25 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2}+\frac {4885 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{84}+\frac {22555 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{168}+\frac {3532787 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15120}-\frac {54061781 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{158760 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {42696881 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{79380 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5929 \left (-30 x^{2}-38 x -12\right )}{32 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(296\) |
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 = 0.22, size = 50, normalized size = 0.23 \begin {gather*} \frac {{\left (94500 \, x^{4} + 392400 \, x^{3} + 795150 \, x^{2} + 1258906 \, x - 2283923\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3780 \, {\left (2 \, x - 1\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 (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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