Optimal. Leaf size=142 \[ \frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 155, 148,
56, 222} \begin {gather*} \frac {2997 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}}+\frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {357 (3 x+2)^3}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {5281 \sqrt {1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{8784600 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 100
Rule 148
Rule 155
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^3 \left (141+\frac {507 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-6537-\frac {48861 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-\frac {1453983}{4}-\frac {5027355 x}{8}\right ) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}}\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 74, normalized size = 0.52 \begin {gather*} -\frac {168318961+19593966 x-1260430251 x^2-1247811640 x^3+213465780 x^4}{8784600 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2997 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{200 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 182, normalized size = 1.28
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \left (13163723100 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}+2632744620 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-4269315600 x^{4} \sqrt {-10 x^{2}-x +3}-7766596629 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+24956232800 x^{3} \sqrt {-10 x^{2}-x +3}-789823386 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +25208605020 x^{2} \sqrt {-10 x^{2}-x +3}+1184735079 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-391879320 x \sqrt {-10 x^{2}-x +3}-3366379220 \sqrt {-10 x^{2}-x +3}\right )}{175692000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 197, normalized size = 1.39 \begin {gather*} -\frac {243 \, x^{4}}{10 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {999}{5856400} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {2997}{4000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {360639}{2928200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {5842159 \, x}{878460 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3429 \, x^{2}}{25 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {947293}{21961500 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3016649 \, x}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1851167}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 121, normalized size = 0.85 \begin {gather*} -\frac {131637231 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (213465780 \, x^{4} - 1247811640 \, x^{3} - 1260430251 \, x^{2} + 19593966 \, x + 168318961\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{175692000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{5}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 191, normalized size = 1.35 \begin {gather*} -\frac {1}{439230000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {4092 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {2997}{2000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (10673289 \, \sqrt {5} {\left (5 \, x + 3\right )} - 440040554 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7233942969 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{5490375000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1023 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{27451875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \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 (3\,x+2\right )}^5}{{\left (1-2\,x\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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