Optimal. Leaf size=172 \[ -\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 156,
162, 65, 212} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}+\frac {113875 \sqrt {1-2 x}}{6 (5 x+3)}+\frac {1256 \sqrt {1-2 x}}{3 (3 x+2) (5 x+3)^2}+\frac {581 \sqrt {1-2 x}}{27 (3 x+2)^2 (5 x+3)^2}-\frac {169975 \sqrt {1-2 x}}{54 (5 x+3)^2}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 100
Rule 154
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {1}{9} \int \frac {(232-233 x) \sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}-\frac {1}{54} \int \frac {-26260+39738 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}-\frac {1}{378} \int \frac {-2861390+3956400 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {\int \frac {-205876440+235585350 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{8316}\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}-\frac {\int \frac {-8504523720+5208414750 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{91476}\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}-392785 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {1271325}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}+392785 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {1271325}{2} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.36, size = 98, normalized size = 0.57 \begin {gather*} \frac {\sqrt {1-2 x} \left (864074+5401374 x+12649336 x^2+13153400 x^3+5124375 x^4\right )}{2 (2+3 x)^3 (3+5 x)^2}+\frac {785570 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 103, normalized size = 0.60
method | result | size |
risch | \(-\frac {10248750 x^{5}+21182425 x^{4}+12145272 x^{3}-1846588 x^{2}-3673226 x -864074}{2 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}-23115 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {785570 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(86\) |
derivativedivides | \(\frac {-75075 \left (1-2 x \right )^{\frac {3}{2}}+163955 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-23115 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {108 \left (\frac {6883 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {145600 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {342265 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{3}}+\frac {785570 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(103\) |
default | \(\frac {-75075 \left (1-2 x \right )^{\frac {3}{2}}+163955 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-23115 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {108 \left (\frac {6883 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {145600 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {342265 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{3}}+\frac {785570 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(103\) |
trager | \(\frac {\left (5124375 x^{4}+13153400 x^{3}+12649336 x^{2}+5401374 x +864074\right ) \sqrt {1-2 x}}{2 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-\frac {23115 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{2}+\frac {392785 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{21}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 163, normalized size = 0.95 \begin {gather*} \frac {23115}{2} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {392785}{21} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {5124375 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 46804300 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 160263994 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 243823580 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 139064695 \, \sqrt {-2 \, x + 1}}{675 \, {\left (2 \, x - 1\right )}^{5} + 7695 \, {\left (2 \, x - 1\right )}^{4} + 35082 \, {\left (2 \, x - 1\right )}^{3} + 79954 \, {\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.28, size = 170, normalized size = 0.99 \begin {gather*} \frac {485415 \, \sqrt {55} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 785570 \, \sqrt {21} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (5124375 \, x^{4} + 13153400 \, x^{3} + 12649336 \, x^{2} + 5401374 \, x + 864074\right )} \sqrt {-2 \, x + 1}}{42 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 = 0.58, size = 151, normalized size = 0.88 \begin {gather*} \frac {23115}{2} \, \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 {392785}{21} \, \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 {55 \, {\left (1365 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2981 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {61947 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 291200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 342265 \, \sqrt {-2 \, x + 1}}{4 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.10, size = 125, normalized size = 0.73 \begin {gather*} \frac {785570\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21}-23115\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {27812939\,\sqrt {1-2\,x}}{135}-\frac {48764716\,{\left (1-2\,x\right )}^{3/2}}{135}+\frac {160263994\,{\left (1-2\,x\right )}^{5/2}}{675}-\frac {1872172\,{\left (1-2\,x\right )}^{7/2}}{27}+\frac {22775\,{\left (1-2\,x\right )}^{9/2}}{3}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________