Optimal. Leaf size=115 \[ -\frac {185}{126} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac {125}{54} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {173 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{189 \sqrt {7}} \]
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Rubi [A]
time = 0.03, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 159, 163,
56, 222, 95, 210} \begin {gather*} \frac {125}{54} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {173 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{189 \sqrt {7}}+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)}-\frac {185}{126} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 100
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^2} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}-\frac {1}{21} \int \frac {\left (-\frac {189}{2}-185 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {185}{126} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac {1}{126} \int \frac {1516+\frac {4375 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {185}{126} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac {173}{378} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {625}{108} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {185}{126} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac {173}{189} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1}{54} \left (125 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {185}{126} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac {125}{54} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {173 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{189 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 103, normalized size = 0.90 \begin {gather*} \frac {-\frac {42 \sqrt {1-2 x} \left (1056+3335 x+2625 x^2\right )}{(2+3 x) \sqrt {3+5 x}}-6125 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-692 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5292} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 146, normalized size = 1.27
method | result | size |
risch | \(\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (352+525 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{126 \left (2+3 x \right ) \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {125 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{216}+\frac {173 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{2646}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(132\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (18375 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +2076 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +12250 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1384 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-44100 x \sqrt {-10 x^{2}-x +3}-29568 \sqrt {-10 x^{2}-x +3}\right )}{10584 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 75, normalized size = 0.65 \begin {gather*} \frac {125}{216} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {173}{2646} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {25}{18} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {\sqrt {-10 \, x^{2} - x + 3}}{63 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 127, normalized size = 1.10 \begin {gather*} -\frac {6125 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 692 \, \sqrt {7} {\left (3 \, x + 2\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 ) + 84 \, {\left (525 \, x + 352\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{10584 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{2}}\, dx \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. 279 vs.
\(2 (83) = 166\).
time = 2.33, size = 279, normalized size = 2.43 \begin {gather*} \frac {173}{26460} \, \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 {125}{216} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {5}{18} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {22 \, \sqrt {10} {\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 )}}{63 \, {\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 )}} \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 )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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