Optimal. Leaf size=108 \[ \frac {505}{84} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {475}{36} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{63 \sqrt {7}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 108, 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 {475}{36} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{63 \sqrt {7}}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}+\frac {505}{84} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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}}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {1}{7} \int \frac {\sqrt {3+5 x} \left (168+\frac {505 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {505}{84} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {1}{42} \int \frac {-\frac {5543}{2}-\frac {16625 x}{4}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {505}{84} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {1}{63} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx-\frac {2375}{72} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {505}{84} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {2}{63} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {1}{36} \left (475 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {505}{84} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {475}{36} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{63 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 99, normalized size = 0.92 \begin {gather*} \frac {42 (901-350 x) \sqrt {3+5 x}+23275 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+16 \sqrt {7-14 x} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3528 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 146, normalized size = 1.35
method | result | size |
default | \(-\frac {\left (46550 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +32 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -23275 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-16 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-29400 x \sqrt {-10 x^{2}-x +3}+75684 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{7056 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.55, size = 86, normalized size = 0.80 \begin {gather*} -\frac {125 \, x^{2}}{6 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {475}{144} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{441} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3455 \, x}{84 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {901}{28 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.72, size = 127, normalized size = 1.18 \begin {gather*} \frac {23275 \, \sqrt {5} \sqrt {2} {\left (2 \, x - 1\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 ) + 16 \, \sqrt {7} {\left (2 \, x - 1\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 (350 \, x - 901\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7056 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \cdot \left (3 x + 2\right )}\, dx \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. 180 vs.
\(2 (76) = 152\).
time = 1.12, size = 180, normalized size = 1.67 \begin {gather*} -\frac {1}{4410} \, \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 {475}{144} \, \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 {{\left (70 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1111 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{420 \, {\left (2 \, x - 1\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 )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________