Optimal. Leaf size=219 \[ \frac {1284329 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{18900}+\frac {(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}+\frac {5}{3} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {4853}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {1284329 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{3780}+\frac {42696881 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900} \]
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Rubi [A] time = 0.07, 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 = {97, 154, 158, 113, 119} \[ \frac {(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}+\frac {5}{3} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {93}{14} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {4853}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {1284329 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{3780}+\frac {1284329 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900}+\frac {42696881 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{18900} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 154
Rule 158
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 = 0.30, size = 120, normalized size = 0.55 \[ \frac {43010905 \sqrt {2-4 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-30 \sqrt {3 x+2} \sqrt {5 x+3} \left (94500 x^4+392400 x^3+795150 x^2+1258906 x-2283923\right )-85393762 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{113400 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 155, normalized size = 0.71 \[ -\frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-42525000 x^{6}-230445000 x^{5}-598495500 x^{4}-1090375200 x^{3}+167061930 x^{2}+1075233030 x -85393762 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+43010905 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+411106140\right )}{113400 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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