Optimal. Leaf size=96 \[ -\frac {845}{88} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {169}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 49, 52, 56,
222} \begin {gather*} \frac {169}{8} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {7 (5 x+3)^{5/2}}{33 (1-2 x)^{3/2}}-\frac {169 (5 x+3)^{3/2}}{66 \sqrt {1-2 x}}-\frac {845}{88} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 49
Rule 52
Rule 56
Rule 79
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}-\frac {169}{66} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {845}{44} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {845}{88} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {845}{16} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {845}{88} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {1}{8} \left (169 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {845}{88} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {169 (3+5 x)^{3/2}}{66 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{5/2}}{33 (1-2 x)^{3/2}}+\frac {169}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 73, normalized size = 0.76 \begin {gather*} \frac {-2 \sqrt {3+5 x} \left (369-1136 x+180 x^2\right )+507 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{48 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 120, normalized size = 1.25
method | result | size |
default | \(\frac {\left (2028 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-2028 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -720 x^{2} \sqrt {-10 x^{2}-x +3}+507 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+4544 x \sqrt {-10 x^{2}-x +3}-1476 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{96 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 119, normalized size = 1.24 \begin {gather*} \frac {169}{32} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {77 \, \sqrt {-10 \, x^{2} - x + 3}}{24 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {271 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.47, size = 97, normalized size = 1.01 \begin {gather*} -\frac {507 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, 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 ) + 4 \, {\left (180 \, x^{2} - 1136 \, x + 369\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{96 \, {\left (4 \, x^{2} - 4 \, 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 (3 x + 2\right ) \left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.53, size = 71, normalized size = 0.74 \begin {gather*} \frac {169}{16} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9 \, \sqrt {5} {\left (5 \, x + 3\right )} - 338 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 5577 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{600 \, {\left (2 \, x - 1\right )}^{2}} \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 (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________