Optimal. Leaf size=160 \[ \frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {1736 \sqrt {1-2 x} \sqrt {3+5 x}}{135 (2+3 x)^{3/2}}+\frac {16564 \sqrt {1-2 x} \sqrt {3+5 x}}{135 \sqrt {2+3 x}}-\frac {16564}{135} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {496}{135} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157,
164, 114, 120} \begin {gather*} -\frac {496}{135} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {16564}{135} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}+\frac {16564 \sqrt {5 x+3} \sqrt {1-2 x}}{135 \sqrt {3 x+2}}+\frac {1736 \sqrt {5 x+3} \sqrt {1-2 x}}{135 (3 x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {2}{15} \int \frac {(128-25 x) \sqrt {1-2 x}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {1736 \sqrt {1-2 x} \sqrt {3+5 x}}{135 (2+3 x)^{3/2}}-\frac {4}{135} \int \frac {-\frac {6869}{2}+2095 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {1736 \sqrt {1-2 x} \sqrt {3+5 x}}{135 (2+3 x)^{3/2}}+\frac {16564 \sqrt {1-2 x} \sqrt {3+5 x}}{135 \sqrt {2+3 x}}-\frac {8}{945} \int \frac {-\frac {91735}{2}-\frac {144935 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {1736 \sqrt {1-2 x} \sqrt {3+5 x}}{135 (2+3 x)^{3/2}}+\frac {16564 \sqrt {1-2 x} \sqrt {3+5 x}}{135 \sqrt {2+3 x}}+\frac {2728}{135} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {16564}{135} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {1736 \sqrt {1-2 x} \sqrt {3+5 x}}{135 (2+3 x)^{3/2}}+\frac {16564 \sqrt {1-2 x} \sqrt {3+5 x}}{135 \sqrt {2+3 x}}-\frac {16564}{135} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {496}{135} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 8.06, size = 101, normalized size = 0.63 \begin {gather*} \frac {4}{405} \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (34927+101862 x+74538 x^2\right )}{2 (2+3 x)^{5/2}}+\sqrt {2} \left (4141 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2095 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \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. \(307\) vs.
\(2(116)=232\).
time = 0.10, size = 308, normalized size = 1.92
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1215 \left (\frac {2}{3}+x \right )^{3}}+\frac {1652 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1215 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {33128}{27} x^{2}-\frac {16564}{135} x +\frac {16564}{45}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {10484 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16564 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(249\) |
default | \(-\frac {2 \left (36828 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-74538 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+49104 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-99384 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+16368 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-33128 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2236140 x^{4}-3279474 x^{3}-682554 x^{2}+811977 x +314343\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{405 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(308\) |
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 = 0.24, size = 50, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (74538 \, x^{2} + 101862 \, x + 34927\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{135 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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