Optimal. Leaf size=96 \[ \frac {5}{49 (1-2 x)^{3/2}}+\frac {45}{343 \sqrt {1-2 x}}+\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {45}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
[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, 44, 53, 65,
212} \begin {gather*} \frac {45}{343 \sqrt {1-2 x}}-\frac {3}{14 (1-2 x)^{3/2} (3 x+2)}+\frac {5}{49 (1-2 x)^{3/2}}+\frac {1}{42 (1-2 x)^{3/2} (3 x+2)^2}-\frac {45}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}+\frac {3}{2} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\\ &=\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {15}{14} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)} \, dx\\ &=\frac {5}{49 (1-2 x)^{3/2}}+\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {45}{98} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac {5}{49 (1-2 x)^{3/2}}+\frac {45}{343 \sqrt {1-2 x}}+\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {135}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {5}{49 (1-2 x)^{3/2}}+\frac {45}{343 \sqrt {1-2 x}}+\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {135}{686} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {5}{49 (1-2 x)^{3/2}}+\frac {45}{343 \sqrt {1-2 x}}+\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {45}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 65, normalized size = 0.68 \begin {gather*} \frac {-\frac {7 \left (-1087-2277 x+2160 x^2+4860 x^3\right )}{2 (1-2 x)^{3/2} (2+3 x)^2}-135 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7203} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 66, normalized size = 0.69
method | result | size |
risch | \(\frac {4860 x^{3}+2160 x^{2}-2277 x -1087}{2058 \left (2+3 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {45 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(58\) |
derivativedivides | \(\frac {44}{1029 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {256}{2401 \sqrt {1-2 x}}+\frac {\frac {531 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {171 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {45 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(66\) |
default | \(\frac {44}{1029 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {256}{2401 \sqrt {1-2 x}}+\frac {\frac {531 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {171 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {45 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(66\) |
trager | \(-\frac {\left (4860 x^{3}+2160 x^{2}-2277 x -1087\right ) \sqrt {1-2 x}}{2058 \left (6 x^{2}+x -2\right )^{2}}-\frac {45 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{4802}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.87, size = 92, normalized size = 0.96 \begin {gather*} \frac {45}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1215 \, {\left (2 \, x - 1\right )}^{3} + 4725 \, {\left (2 \, x - 1\right )}^{2} + 7056 \, x - 5684}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.14, size = 105, normalized size = 1.09 \begin {gather*} \frac {135 \, \sqrt {7} \sqrt {3} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (4860 \, x^{3} + 2160 \, x^{2} - 2277 \, x - 1087\right )} \sqrt {-2 \, x + 1}}{14406 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 = 1.65, size = 89, normalized size = 0.93 \begin {gather*} \frac {45}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (384 \, x - 269\right )}}{7203 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {9 \, {\left (59 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 133 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 72, normalized size = 0.75 \begin {gather*} -\frac {45\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {16\,x}{21}+\frac {25\,{\left (2\,x-1\right )}^2}{49}+\frac {45\,{\left (2\,x-1\right )}^3}{343}-\frac {116}{189}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________