Optimal. Leaf size=191 \[ \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {496 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {582 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401} \]
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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 = {100, 157, 164,
114, 120} \begin {gather*} -\frac {582 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}+\frac {496 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {496 \sqrt {1-2 x} \sqrt {5 x+3}}{2401 \sqrt {3 x+2}}-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{343 (3 x+2)^{3/2}}+\frac {58 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {1}{21} \int \frac {-\frac {169}{2}-150 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {2 \int \frac {\frac {16203}{4}+\frac {14355 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {10593}{2}+\frac {44055 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{33957}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {8 \int \frac {-\frac {60885}{8}-30690 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{237699}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}-\frac {496 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2401}+\frac {873 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2401}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {496 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {582 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}\\ \end {align*}
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Mathematica [A]
time = 8.63, size = 104, normalized size = 0.54 \begin {gather*} \frac {-\frac {2 \sqrt {3+5 x} \left (-885-4616 x+762 x^2+8928 x^3\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}}+\sqrt {2} \left (-496 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+3115 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{7203} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs.
\(2(139)=278\).
time = 0.10, size = 305, normalized size = 1.60
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {23}{2646}+\frac {31 x}{2646}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (-\frac {121}{43218}+\frac {248 x}{7203}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {205 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{16807 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2480 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{50421 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
default | \(-\frac {\left (15714 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2976 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2619 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+496 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5238 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-992 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+89280 x^{4}+61188 x^{3}-41588 x^{2}-36546 x -5310\right ) \sqrt {1-2 x}}{7203 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(305\) |
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 = 60, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (8928 \, x^{3} + 762 \, x^{2} - 4616 \, x - 885\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{7203 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \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 (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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