Optimal. Leaf size=120 \[ -\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac {39185 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 150, 44,
65, 212} \begin {gather*} \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^4}+\frac {\sqrt {1-2 x} (3789 x+2395)}{1764 (3 x+2)^4}-\frac {39185 \sqrt {1-2 x}}{57624 (3 x+2)}-\frac {39185 \sqrt {1-2 x}}{24696 (3 x+2)^2}-\frac {39185 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 100
Rule 150
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {1}{7} \int \frac {(-173-325 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac {39185 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1764}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac {39185 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{8232}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac {39185 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{57624}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac {39185 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{57624}\\ &=-\frac {39185 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {39185 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {\sqrt {1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac {39185 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 70, normalized size = 0.58 \begin {gather*} \frac {\frac {21 \left (213998+1534434 x+4093057 x^2+4819755 x^3+2115990 x^4\right )}{2 \sqrt {1-2 x} (2+3 x)^4}-39185 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{605052} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 75, normalized size = 0.62
method | result | size |
risch | \(\frac {2115990 x^{4}+4819755 x^{3}+4093057 x^{2}+1534434 x +213998}{57624 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {39185 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}\) | \(56\) |
derivativedivides | \(\frac {5324}{16807 \sqrt {1-2 x}}+\frac {\frac {743679 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {717277 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {691955 \left (1-2 x \right )^{\frac {3}{2}}}{4116}-\frac {74185 \sqrt {1-2 x}}{588}}{\left (-4-6 x \right )^{4}}-\frac {39185 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}\) | \(75\) |
default | \(\frac {5324}{16807 \sqrt {1-2 x}}+\frac {\frac {743679 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {717277 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {691955 \left (1-2 x \right )^{\frac {3}{2}}}{4116}-\frac {74185 \sqrt {1-2 x}}{588}}{\left (-4-6 x \right )^{4}}-\frac {39185 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}\) | \(75\) |
trager | \(-\frac {\left (2115990 x^{4}+4819755 x^{3}+4093057 x^{2}+1534434 x +213998\right ) \sqrt {1-2 x}}{57624 \left (2+3 x \right )^{4} \left (-1+2 x \right )}-\frac {39185 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1210104}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.55, size = 119, normalized size = 0.99 \begin {gather*} \frac {39185}{1210104} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1057995 \, {\left (2 \, x - 1\right )}^{4} + 9051735 \, {\left (2 \, x - 1\right )}^{3} + 28993349 \, {\left (2 \, x - 1\right )}^{2} + 82402418 \, x - 19287625}{28812 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2401 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.62, size = 114, normalized size = 0.95 \begin {gather*} \frac {39185 \, \sqrt {21} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2115990 \, x^{4} + 4819755 \, x^{3} + 4093057 \, x^{2} + 1534434 \, x + 213998\right )} \sqrt {-2 \, x + 1}}{1210104 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\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 = 1.32, size = 109, normalized size = 0.91 \begin {gather*} \frac {39185}{1210104} \, \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 {5324}{16807 \, \sqrt {-2 \, x + 1}} - \frac {2231037 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 15062817 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 33905795 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 25445455 \, \sqrt {-2 \, x + 1}}{3226944 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.25, size = 98, normalized size = 0.82 \begin {gather*} \frac {\frac {840841\,x}{23814}+\frac {591701\,{\left (2\,x-1\right )}^2}{47628}+\frac {431035\,{\left (2\,x-1\right )}^3}{111132}+\frac {39185\,{\left (2\,x-1\right )}^4}{86436}-\frac {393625}{47628}}{\frac {2401\,\sqrt {1-2\,x}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{5/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{7/2}}{3}+{\left (1-2\,x\right )}^{9/2}}-\frac {39185\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{605052} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________