Optimal. Leaf size=122 \[ \frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {25}{9} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{441 \sqrt {7}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 122, 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, 154, 163,
56, 222, 95, 210} \begin {gather*} -\frac {25}{9} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {169 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{441 \sqrt {7}}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}+\frac {32 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 56
Rule 95
Rule 100
Rule 154
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {1}{7} \int \frac {\sqrt {3+5 x} \left (69+\frac {175 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {1}{147} \int \frac {\frac {4027}{2}+\frac {6125 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {169}{882} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx-\frac {125}{18} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {169}{441} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {1}{9} \left (25 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {25}{9} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{441 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 116, normalized size = 0.95 \begin {gather*} \frac {42 \sqrt {3+5 x} (725+1091 x)+8575 \sqrt {10-20 x} (2+3 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-338 \sqrt {7-14 x} (2+3 x) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6174 \sqrt {1-2 x} (2+3 x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs.
\(2(90)=180\).
time = 0.08, size = 198, normalized size = 1.62
method | result | size |
default | \(-\frac {\left (51450 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-2028 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+8575 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -338 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -17150 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+676 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+91644 x \sqrt {-10 x^{2}-x +3}+60900 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{12348 \left (2+3 x \right ) \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 103, normalized size = 0.84 \begin {gather*} -\frac {25}{36} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {169}{6174} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {5455 \, x}{441 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {9784}{1323 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{189 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.01, size = 136, normalized size = 1.11 \begin {gather*} \frac {8575 \, \sqrt {5} \sqrt {2} {\left (6 \, x^{2} + 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 ) - 338 \, \sqrt {7} {\left (6 \, x^{2} + 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 (1091 \, x + 725\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12348 \, {\left (6 \, x^{2} + x - 2\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. 286 vs.
\(2 (90) = 180\).
time = 1.17, size = 286, normalized size = 2.34 \begin {gather*} \frac {169}{61740} \, \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 {25}{36} \, \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 {121 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{245 \, {\left (2 \, x - 1\right )}} - \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 )}}{147 \, {\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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________