Optimal. Leaf size=128 \[ \frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {9145 \sqrt {1-2 x}}{57624 (2+3 x)}-\frac {9145 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 44, 65,
212} \begin {gather*} -\frac {9145 \sqrt {1-2 x}}{57624 (3 x+2)}-\frac {9145 \sqrt {1-2 x}}{24696 (3 x+2)^2}-\frac {1829 \sqrt {1-2 x}}{1764 (3 x+2)^3}-\frac {2179 \sqrt {1-2 x}}{588 (3 x+2)^4}+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^4}-\frac {9145 \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 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {1}{14} \int \frac {-1336+175 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}+\frac {1829}{84} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}+\frac {9145 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1764}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}+\frac {9145 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{8232}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {9145 \sqrt {1-2 x}}{57624 (2+3 x)}+\frac {9145 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{57624}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {9145 \sqrt {1-2 x}}{57624 (2+3 x)}-\frac {9145 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{57624}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^4}-\frac {2179 \sqrt {1-2 x}}{588 (2+3 x)^4}-\frac {1829 \sqrt {1-2 x}}{1764 (2+3 x)^3}-\frac {9145 \sqrt {1-2 x}}{24696 (2+3 x)^2}-\frac {9145 \sqrt {1-2 x}}{57624 (2+3 x)}-\frac {9145 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812 \sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 70, normalized size = 0.55 \begin {gather*} \frac {\frac {21 \left (6486+218578 x+843169 x^2+1124835 x^3+493830 x^4\right )}{2 \sqrt {1-2 x} (2+3 x)^4}-9145 \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 = 0.14, size = 75, normalized size = 0.59
method | result | size |
risch | \(\frac {493830 x^{4}+1124835 x^{3}+843169 x^{2}+218578 x +6486}{57624 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {9145 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}\) | \(56\) |
derivativedivides | \(\frac {968}{16807 \sqrt {1-2 x}}+\frac {\frac {262503 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {285989 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {307355 \left (1-2 x \right )^{\frac {3}{2}}}{4116}-\frac {36305 \sqrt {1-2 x}}{588}}{\left (-4-6 x \right )^{4}}-\frac {9145 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}\) | \(75\) |
default | \(\frac {968}{16807 \sqrt {1-2 x}}+\frac {\frac {262503 \left (1-2 x \right )^{\frac {7}{2}}}{67228}-\frac {285989 \left (1-2 x \right )^{\frac {5}{2}}}{9604}+\frac {307355 \left (1-2 x \right )^{\frac {3}{2}}}{4116}-\frac {36305 \sqrt {1-2 x}}{588}}{\left (-4-6 x \right )^{4}}-\frac {9145 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{605052}\) | \(75\) |
trager | \(-\frac {\left (493830 x^{4}+1124835 x^{3}+843169 x^{2}+218578 x +6486\right ) \sqrt {1-2 x}}{57624 \left (2+3 x \right )^{4} \left (-1+2 x \right )}+\frac {9145 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{1210104}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.53, size = 119, normalized size = 0.93 \begin {gather*} \frac {9145}{1210104} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {246915 \, {\left (2 \, x - 1\right )}^{4} + 2112495 \, {\left (2 \, x - 1\right )}^{3} + 6542333 \, {\left (2 \, x - 1\right )}^{2} + 17218306 \, x - 4624865}{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 = 0.95, size = 114, normalized size = 0.89 \begin {gather*} \frac {9145 \, \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 (493830 \, x^{4} + 1124835 \, x^{3} + 843169 \, x^{2} + 218578 \, x + 6486\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 = 1.71, size = 109, normalized size = 0.85 \begin {gather*} \frac {9145}{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 {968}{16807 \, \sqrt {-2 \, x + 1}} - \frac {787509 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 6005769 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 15060395 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 12452615 \, \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 = 0.08, size = 98, normalized size = 0.77 \begin {gather*} \frac {\frac {175697\,x}{23814}+\frac {133517\,{\left (2\,x-1\right )}^2}{47628}+\frac {100595\,{\left (2\,x-1\right )}^3}{111132}+\frac {9145\,{\left (2\,x-1\right )}^4}{86436}-\frac {94385}{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 {9145\,\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]
________________________________________________________________________________________