Optimal. Leaf size=96 \[ -\frac {63}{125} \sqrt {1-2 x}-\frac {21}{275} (1-2 x)^{3/2}-\frac {(1-2 x)^{7/2}}{110 (3+5 x)^2}-\frac {63 (1-2 x)^{5/2}}{550 (3+5 x)}+\frac {63}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 52, 65,
212} \begin {gather*} -\frac {(1-2 x)^{7/2}}{110 (5 x+3)^2}-\frac {63 (1-2 x)^{5/2}}{550 (5 x+3)}-\frac {21}{275} (1-2 x)^{3/2}-\frac {63}{125} \sqrt {1-2 x}+\frac {63}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 52
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{7/2}}{110 (3+5 x)^2}+\frac {63}{110} \int \frac {(1-2 x)^{5/2}}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{110 (3+5 x)^2}-\frac {63 (1-2 x)^{5/2}}{550 (3+5 x)}-\frac {63}{110} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=-\frac {21}{275} (1-2 x)^{3/2}-\frac {(1-2 x)^{7/2}}{110 (3+5 x)^2}-\frac {63 (1-2 x)^{5/2}}{550 (3+5 x)}-\frac {63}{50} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=-\frac {63}{125} \sqrt {1-2 x}-\frac {21}{275} (1-2 x)^{3/2}-\frac {(1-2 x)^{7/2}}{110 (3+5 x)^2}-\frac {63 (1-2 x)^{5/2}}{550 (3+5 x)}-\frac {693}{250} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {63}{125} \sqrt {1-2 x}-\frac {21}{275} (1-2 x)^{3/2}-\frac {(1-2 x)^{7/2}}{110 (3+5 x)^2}-\frac {63 (1-2 x)^{5/2}}{550 (3+5 x)}+\frac {693}{250} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {63}{125} \sqrt {1-2 x}-\frac {21}{275} (1-2 x)^{3/2}-\frac {(1-2 x)^{7/2}}{110 (3+5 x)^2}-\frac {63 (1-2 x)^{5/2}}{550 (3+5 x)}+\frac {63}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.17, size = 63, normalized size = 0.66 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (-1394-3795 x-2280 x^2+400 x^3\right )}{(3+5 x)^2}+126 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1250} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 66, normalized size = 0.69
method | result | size |
risch | \(-\frac {800 x^{4}-4960 x^{3}-5310 x^{2}+1007 x +1394}{250 \left (3+5 x \right )^{2} \sqrt {1-2 x}}+\frac {63 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}\) | \(56\) |
derivativedivides | \(-\frac {4 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {256 \sqrt {1-2 x}}{625}-\frac {44 \left (-\frac {57 \left (1-2 x \right )^{\frac {3}{2}}}{20}+\frac {649 \sqrt {1-2 x}}{100}\right )}{25 \left (-6-10 x \right )^{2}}+\frac {63 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}\) | \(66\) |
default | \(-\frac {4 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {256 \sqrt {1-2 x}}{625}-\frac {44 \left (-\frac {57 \left (1-2 x \right )^{\frac {3}{2}}}{20}+\frac {649 \sqrt {1-2 x}}{100}\right )}{25 \left (-6-10 x \right )^{2}}+\frac {63 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{625}\) | \(66\) |
trager | \(\frac {\left (400 x^{3}-2280 x^{2}-3795 x -1394\right ) \sqrt {1-2 x}}{250 \left (3+5 x \right )^{2}}-\frac {63 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1250}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 92, normalized size = 0.96 \begin {gather*} -\frac {4}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {63}{1250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {256}{625} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (285 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 649 \, \sqrt {-2 \, x + 1}\right )}}{625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.82, size = 86, normalized size = 0.90 \begin {gather*} \frac {63 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 5 \, {\left (400 \, x^{3} - 2280 \, x^{2} - 3795 \, x - 1394\right )} \sqrt {-2 \, x + 1}}{1250 \, {\left (25 \, x^{2} + 30 \, x + 9\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 [A]
time = 0.57, size = 86, normalized size = 0.90 \begin {gather*} -\frac {4}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {63}{1250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {256}{625} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (285 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 649 \, \sqrt {-2 \, x + 1}\right )}}{2500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.08, size = 74, normalized size = 0.77 \begin {gather*} -\frac {256\,\sqrt {1-2\,x}}{625}-\frac {4\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {\frac {7139\,\sqrt {1-2\,x}}{15625}-\frac {627\,{\left (1-2\,x\right )}^{3/2}}{3125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}-\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,63{}\mathrm {i}}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________