Optimal. Leaf size=151 \[ -\frac {13915 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {1265 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}-\frac {153065 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \]
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Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210}
\begin {gather*} -\frac {153065 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}}+\frac {115 \sqrt {1-2 x} (5 x+3)^{5/2}}{168 (3 x+2)^3}+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}-\frac {1265 \sqrt {1-2 x} (5 x+3)^{3/2}}{4704 (3 x+2)^2}-\frac {13915 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx &=\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115}{56} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {1265}{336} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {1265 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {13915 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{3136}\\ &=-\frac {13915 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {1265 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {153065 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904}\\ &=-\frac {13915 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {1265 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}+\frac {153065 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952}\\ &=-\frac {13915 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {1265 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}-\frac {153065 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 79, normalized size = 0.52 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (328464+1512052 x+2269240 x^2+1104135 x^3\right )}{(2+3 x)^4}-459195 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{460992} \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. \(249\) vs.
\(2(118)=236\).
time = 0.13, size = 250, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (1104135 x^{3}+2269240 x^{2}+1512052 x +328464\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{65856 \left (2+3 x \right )^{4} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {153065 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (37194795 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+99186120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+99186120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+15457890 x^{3} \sqrt {-10 x^{2}-x +3}+44082720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +31769360 x^{2} \sqrt {-10 x^{2}-x +3}+7347120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+21168728 x \sqrt {-10 x^{2}-x +3}+4598496 \sqrt {-10 x^{2}-x +3}\right )}{921984 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 157, normalized size = 1.04 \begin {gather*} \frac {153065}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {6325}{16464} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {95 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1176 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {3795 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {46805 \, \sqrt {-10 \, x^{2} - x + 3}}{65856 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 116, normalized size = 0.77 \begin {gather*} -\frac {459195 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1104135 \, x^{3} + 2269240 \, x^{2} + 1512052 \, x + 328464\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{921984 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 368 vs.
\(2 (118) = 236\).
time = 1.08, size = 368, normalized size = 2.44 \begin {gather*} \frac {30613}{614656} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {6655 \, \sqrt {10} {\left (69 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 70840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 15821120 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {1514688000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {6058752000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{32928 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \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 {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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