Optimal. Leaf size=116 \[ \frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222}
\begin {gather*} \frac {240911 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}}-\frac {3}{40} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {181}{480} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {1991 (1-2 x)^{3/2} \sqrt {5 x+3}}{1280}+\frac {21901 \sqrt {1-2 x} \sqrt {5 x+3}}{12800} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 56
Rule 81
Rule 222
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2} \, dx &=-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {181}{80} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1991}{320} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {21901 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800 \sqrt {5}}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 78, normalized size = 0.67 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-190161-18795 x+1233700 x^2+1660000 x^3+720000 x^4\right )-722733 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{384000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 104, normalized size = 0.90
method | result | size |
risch | \(-\frac {\left (144000 x^{3}+245600 x^{2}+99380 x -63387\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{38400 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {240911 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (2880000 x^{3} \sqrt {-10 x^{2}-x +3}+4912000 x^{2} \sqrt {-10 x^{2}-x +3}+722733 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1987600 x \sqrt {-10 x^{2}-x +3}-1267740 \sqrt {-10 x^{2}-x +3}\right )}{768000 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.59, size = 70, normalized size = 0.60 \begin {gather*} -\frac {3}{8} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {289}{480} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1991}{640} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {240911}{256000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1991}{12800} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.05, size = 72, normalized size = 0.62 \begin {gather*} \frac {1}{38400} \, {\left (144000 \, x^{3} + 245600 \, x^{2} + 99380 \, x - 63387\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {240911}{256000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 19.69, size = 376, normalized size = 3.24 \begin {gather*} - \frac {77 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} + \frac {17 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{2} - \frac {15 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} \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. 203 vs.
\(2 (83) = 166\).
time = 0.59, size = 203, normalized size = 1.75 \begin {gather*} \frac {1}{128000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {7}{6000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {87}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\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 \sqrt {1-2\,x}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________