Optimal. Leaf size=191 \[ \frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {481339 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}+\frac {124724 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875} \]
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Rubi [A]
time = 0.05, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {124724 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}-\frac {481339 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{3 \sqrt {3 x+2}}-\frac {32}{63} \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {2108 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{1575}+\frac {124724 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{14175} \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 {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {4}{315} \int \frac {\left (-\frac {1335}{4}-\frac {7905 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {8 \int \frac {\left (\frac {326745}{8}-\frac {467715 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{23625}\\ &=\frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {8 \int \frac {-\frac {2274105}{8}-\frac {7220085 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{212625}\\ &=\frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {481339 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{70875}-\frac {685982 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{70875}\\ &=\frac {124724 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{14175}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1575}-\frac {32}{63} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {481339 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}+\frac {124724 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{70875}\\ \end {align*}
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Mathematica [A]
time = 7.62, size = 107, normalized size = 0.56 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (32033+14727 x-21690 x^2+13500 x^3\right )}{\sqrt {2+3 x}}+481339 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2539285 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{212625} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 148, normalized size = 0.77
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (2057946 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+481339 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+4050000 x^{5}-6102000 x^{4}+2552400 x^{3}+12003810 x^{2}-364440 x -2882970\right )}{6378750 x^{3}+4890375 x^{2}-1488375 x -1275750}\) | \(148\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {40 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63}-\frac {1364 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945}+\frac {23458 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14175}+\frac {151607 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{297675 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {481339 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{297675 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {980}{81} x^{2}-\frac {98}{81} x +\frac {98}{27}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
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 = 0.18, size = 38, normalized size = 0.20 \begin {gather*} \frac {2 \, {\left (13500 \, x^{3} - 21690 \, x^{2} + 14727 \, x + 32033\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14175 \, \sqrt {3 \, x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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