Optimal. Leaf size=151 \[ -\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 95, 210}
\begin {gather*} -\frac {73205 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}}+\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{4 (3 x+2)^4}+\frac {55 (5 x+3)^{3/2} (1-2 x)^{3/2}}{24 (3 x+2)^3}+\frac {605 (5 x+3)^{3/2} \sqrt {1-2 x}}{32 (3 x+2)^2}-\frac {6655 \sqrt {5 x+3} \sqrt {1-2 x}}{448 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 95
Rule 96
Rule 210
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx &=\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55}{8} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605}{16} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}+\frac {6655}{64} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}+\frac {73205}{896} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}+\frac {73205}{448} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {6655 \sqrt {1-2 x} \sqrt {3+5 x}}{448 (2+3 x)}+\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{4 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^{3/2}}{24 (2+3 x)^3}+\frac {605 \sqrt {1-2 x} (3+5 x)^{3/2}}{32 (2+3 x)^2}-\frac {73205 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.27, size = 79, normalized size = 0.52 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (164688+723428 x+1059032 x^2+518715 x^3\right )}{(2+3 x)^4}-219615 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9408} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs.
\(2(118)=236\).
time = 0.11, size = 250, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (518715 x^{3}+1059032 x^{2}+723428 x +164688\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1344 \left (2+3 x \right )^{4} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {73205 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6272 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (17788815 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+47436840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+47436840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+7262010 x^{3} \sqrt {-10 x^{2}-x +3}+21083040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +14826448 x^{2} \sqrt {-10 x^{2}-x +3}+3513840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+10127992 x \sqrt {-10 x^{2}-x +3}+2305632 \sqrt {-10 x^{2}-x +3}\right )}{18816 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 157, normalized size = 1.04 \begin {gather*} \frac {73205}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3025}{336} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {17 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1815 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{224 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {22385 \, \sqrt {-10 \, x^{2} - x + 3}}{1344 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.52, size = 116, normalized size = 0.77 \begin {gather*} -\frac {219615 \, \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 (518715 \, x^{3} + 1059032 \, x^{2} + 723428 \, x + 164688\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{18816 \, {\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 368 vs.
\(2 (118) = 236\).
time = 0.82, size = 368, normalized size = 2.44 \begin {gather*} \frac {14641}{12544} \, \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 {73205 \, \sqrt {10} {\left (3 \, {\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} - 4088 \, {\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} - 862400 \, {\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 {65856000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {263424000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{672 \, {\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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________