Optimal. Leaf size=153 \[ \frac {2788127 \sqrt {1-2 x}}{2058 (3 x+2)}+\frac {120077 \sqrt {1-2 x}}{882 (3 x+2)^2}+\frac {5732 \sqrt {1-2 x}}{315 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{15 (3 x+2)^4}+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}+\frac {96169877 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}-2750 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 151, 156, 63, 206} \begin {gather*} \frac {2788127 \sqrt {1-2 x}}{2058 (3 x+2)}+\frac {120077 \sqrt {1-2 x}}{882 (3 x+2)^2}+\frac {5732 \sqrt {1-2 x}}{315 (3 x+2)^3}+\frac {41 \sqrt {1-2 x}}{15 (3 x+2)^4}+\frac {7 \sqrt {1-2 x}}{15 (3 x+2)^5}+\frac {96169877 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}-2750 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 98
Rule 151
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {186-295 x}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {1}{420} \int \frac {26712-40180 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {5732 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {\int \frac {2928660-4012400 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{8820}\\ &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {5732 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {120077 \sqrt {1-2 x}}{882 (2+3 x)^2}+\frac {\int \frac {222229980-252161700 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{123480}\\ &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {5732 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {120077 \sqrt {1-2 x}}{882 (2+3 x)^2}+\frac {2788127 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {\int \frac {9560404980-5855066700 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{864360}\\ &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {5732 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {120077 \sqrt {1-2 x}}{882 (2+3 x)^2}+\frac {2788127 \sqrt {1-2 x}}{2058 (2+3 x)}-\frac {96169877 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2058}+75625 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {5732 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {120077 \sqrt {1-2 x}}{882 (2+3 x)^2}+\frac {2788127 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {96169877 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2058}-75625 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7 \sqrt {1-2 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {5732 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {120077 \sqrt {1-2 x}}{882 (2+3 x)^2}+\frac {2788127 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {96169877 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}-2750 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 93, normalized size = 0.61 \begin {gather*} \frac {\sqrt {1-2 x} \left (1129191435 x^4+3049001415 x^3+3088510878 x^2+1391064622 x+235067382\right )}{10290 (3 x+2)^5}+\frac {96169877 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}-2750 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.74, size = 113, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {1-2 x} \left (1129191435 (1-2 x)^4-10614768570 (1-2 x)^3+37423200612 (1-2 x)^2-58647378230 (1-2 x)+34470832865\right )}{5145 (3 (1-2 x)-7)^5}+\frac {96169877 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}-2750 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.06, size = 170, normalized size = 1.11 \begin {gather*} \frac {297123750 \, \sqrt {55} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 480849385 \, \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 (1129191435 \, x^{4} + 3049001415 \, x^{3} + 3088510878 \, x^{2} + 1391064622 \, x + 235067382\right )} \sqrt {-2 \, x + 1}}{216090 \, {\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]
________________________________________________________________________________________
giac [A] time = 1.02, size = 155, normalized size = 1.01 \begin {gather*} 1375 \, \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 {96169877}{43218} \, \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 {1129191435 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 10614768570 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 37423200612 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 58647378230 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 34470832865 \, \sqrt {-2 \, x + 1}}{164640 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 93, normalized size = 0.61 \begin {gather*} \frac {96169877 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21609}-2750 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )-\frac {486 \left (\frac {2788127 \left (-2 x +1\right )^{\frac {9}{2}}}{6174}-\frac {2406977 \left (-2 x +1\right )^{\frac {7}{2}}}{567}+\frac {127289798 \left (-2 x +1\right )^{\frac {5}{2}}}{8505}-\frac {17098361 \left (-2 x +1\right )^{\frac {3}{2}}}{729}+\frac {20099611 \sqrt {-2 x +1}}{1458}\right )}{\left (-6 x -4\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.18, size = 164, normalized size = 1.07 \begin {gather*} 1375 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {96169877}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1129191435 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 10614768570 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 37423200612 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 58647378230 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 34470832865 \, \sqrt {-2 \, x + 1}}{5145 \, {\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]
________________________________________________________________________________________
mupad [B] time = 1.20, size = 125, normalized size = 0.82 \begin {gather*} \frac {96169877\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}-2750\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {20099611\,\sqrt {1-2\,x}}{729}-\frac {34196722\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {254579596\,{\left (1-2\,x\right )}^{5/2}}{8505}-\frac {4813954\,{\left (1-2\,x\right )}^{7/2}}{567}+\frac {2788127\,{\left (1-2\,x\right )}^{9/2}}{3087}}{\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] 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]
________________________________________________________________________________________