Optimal. Leaf size=222 \[ \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {310208 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \]
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Rubi [A]
time = 0.05, antiderivative size = 222, 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 = {100, 157, 164,
114, 120} \begin {gather*} \frac {310208 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {10312712 \sqrt {1-2 x} \sqrt {3 x+2}}{1029 \sqrt {5 x+3}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {12276 \sqrt {1-2 x}}{245 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {176 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}} \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 {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx &=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {154-231 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {4}{735} \int \frac {\frac {33649}{2}-23100 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \int \frac {1272579-\frac {2900205 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{15435}\\ &=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {217164255}{4}-33589710 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{108045}\\ &=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}-\frac {32 \int \frac {\frac {2827810755}{4}+\frac {4466693385 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1188495}\\ &=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}-\frac {1706144 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}-\frac {10312712 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1715}\\ &=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {310208 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}\\ \end {align*}
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Mathematica [A]
time = 7.54, size = 110, normalized size = 0.50 \begin {gather*} \frac {2 \left (-\frac {3 \sqrt {1-2 x} \left (130497191+793777840 x+1809835578 x^2+1833255216 x^3+696108060 x^4\right )}{(2+3 x)^{7/2} \sqrt {3+5 x}}-4 \sqrt {2} \left (1289089 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-649285 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{5145} \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. \(400\) vs.
\(2(162)=324\).
time = 0.10, size = 401, normalized size = 1.81
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {550 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {878 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945 \left (\frac {2}{3}+x \right )^{3}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{4}}-\frac {7482962 \left (-30 x^{2}-3 x +9\right )}{5145 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {67558 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2205 \left (\frac {2}{3}+x \right )^{2}}-\frac {6528856 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {10312712 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(301\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (139221612 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-69098832 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+278443224 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-138197664 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+185628816 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-92131776 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+41250848 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-20473728 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+4176648360 x^{5}+8911207116 x^{4}+5359247820 x^{3}-666839694 x^{2}-1598350374 x -391491573\right )}{5145 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\) | \(401\) |
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 = 70, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (696108060 \, x^{4} + 1833255216 \, x^{3} + 1809835578 \, x^{2} + 793777840 \, x + 130497191\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1715 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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 (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{9/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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