Optimal. Leaf size=134 \[ -\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 134, 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 = {99, 154, 150,
65, 212} \begin {gather*} \frac {3 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^4}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{315 (3 x+2)^3}-\frac {2 \sqrt {1-2 x} (15074 x+9529)}{9261 (3 x+2)^2}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 99
Rule 150
Rule 154
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {1}{180} \int \frac {(3+5 x)^2 (-684+180 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {\int \frac {(3+5 x) (-48960+6840 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{11340}\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}+\frac {6946 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{9261}\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {6946 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{9261}\\ &=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.31, size = 70, normalized size = 0.52 \begin {gather*} \frac {4 \left (\frac {21 \sqrt {1-2 x} \left (300049+2619854 x+7992771 x^2+10375830 x^3+4904370 x^4\right )}{4 (2+3 x)^5}-17365 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{972405} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 75, normalized size = 0.56
method | result | size |
risch | \(-\frac {9808740 x^{5}+15847290 x^{4}+5609712 x^{3}-2753063 x^{2}-2019756 x -300049}{46305 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {13892 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(61\) |
derivativedivides | \(\frac {-\frac {217972 \left (1-2 x \right )^{\frac {9}{2}}}{1029}+\frac {36616 \left (1-2 x \right )^{\frac {7}{2}}}{21}-\frac {1682344 \left (1-2 x \right )^{\frac {5}{2}}}{315}+\frac {194488 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {97244 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{5}}-\frac {13892 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(75\) |
default | \(\frac {-\frac {217972 \left (1-2 x \right )^{\frac {9}{2}}}{1029}+\frac {36616 \left (1-2 x \right )^{\frac {7}{2}}}{21}-\frac {1682344 \left (1-2 x \right )^{\frac {5}{2}}}{315}+\frac {194488 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {97244 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{5}}-\frac {13892 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(75\) |
trager | \(\frac {\left (4904370 x^{4}+10375830 x^{3}+7992771 x^{2}+2619854 x +300049\right ) \sqrt {1-2 x}}{46305 \left (2+3 x \right )^{5}}-\frac {6946 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{194481}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 128, normalized size = 0.96 \begin {gather*} \frac {6946}{194481} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (2452185 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 20184570 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 61826142 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}\right )}}{46305 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.27, size = 114, normalized size = 0.85 \begin {gather*} \frac {34730 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (4904370 \, x^{4} + 10375830 \, x^{3} + 7992771 \, x^{2} + 2619854 \, x + 300049\right )} \sqrt {-2 \, x + 1}}{972405 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 1.14, size = 116, normalized size = 0.87 \begin {gather*} \frac {6946}{194481} \, \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 {2452185 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 20184570 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 61826142 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}}{370440 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.08, size = 107, normalized size = 0.80 \begin {gather*} \frac {\frac {97244\,\sqrt {1-2\,x}}{6561}-\frac {194488\,{\left (1-2\,x\right )}^{3/2}}{6561}+\frac {1682344\,{\left (1-2\,x\right )}^{5/2}}{76545}-\frac {36616\,{\left (1-2\,x\right )}^{7/2}}{5103}+\frac {217972\,{\left (1-2\,x\right )}^{9/2}}{250047}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}}-\frac {13892\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{194481} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________