Optimal. Leaf size=189 \[ \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {36 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {26 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {5636 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}-\frac {5636 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {4364 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005 \sqrt {33}} \]
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Rubi [A]
time = 0.04, antiderivative size = 189, 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 = {101, 157, 164,
114, 120} \begin {gather*} -\frac {4364 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005 \sqrt {33}}-\frac {5636 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}+\frac {5636 \sqrt {1-2 x} \sqrt {5 x+3}}{12005 \sqrt {3 x+2}}-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 (3 x+2)^{3/2}}-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{245 (3 x+2)^{5/2}}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 101
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {2}{7} \int \frac {-22-\frac {75 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {36 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {4}{245} \int \frac {-\frac {347}{4}-135 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {36 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {26 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {1539}{4}-\frac {195 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{5145}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {36 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {26 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {5636 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}-\frac {16 \int \frac {-\frac {28635}{8}-\frac {21135 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{36015}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {36 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {26 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {5636 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}+\frac {2182 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12005}+\frac {5636 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{12005}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {36 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {26 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {5636 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}-\frac {5636 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {4364 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.66, size = 104, normalized size = 0.55 \begin {gather*} \frac {2 \left (-\frac {3 \sqrt {3+5 x} \left (-11923-13127 x+41724 x^2+50724 x^3\right )}{\sqrt {1-2 x} (2+3 x)^{5/2}}+\sqrt {2} \left (2818 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+455 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{36015} \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.
\(2(139)=278\).
time = 0.10, size = 308, normalized size = 1.63
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {8 \left (-30 x^{2}-38 x -12\right )}{2401 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {3818 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{50421 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5636 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{50421 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2205 \left (\frac {2}{3}+x \right )^{3}}+\frac {34 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15435 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {11512}{2401} x^{2}-\frac {5756}{12005} x +\frac {17268}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(277\) |
default | \(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (29457 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-25362 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+39276 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-33816 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+13092 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-11272 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-760860 x^{4}-1082376 x^{3}-178611 x^{2}+296988 x +107307\right )}{36015 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(308\) |
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 = 0.22, size = 60, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (50724 \, x^{3} + 41724 \, x^{2} - 13127 \, x - 11923\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{12005 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\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 = -0.01 \begin {gather*} \int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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