Optimal. Leaf size=168 \[ \frac {23 (1-2 x)^{7/2}}{882 (3 x+2)^6}-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}-\frac {467 (1-2 x)^{5/2}}{2646 (3 x+2)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (3 x+2)^4}+\frac {2335 \sqrt {1-2 x}}{3111696 (3 x+2)}+\frac {2335 \sqrt {1-2 x}}{1333584 (3 x+2)^2}-\frac {2335 \sqrt {1-2 x}}{95256 (3 x+2)^3}+\frac {2335 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac {23 (1-2 x)^{7/2}}{882 (3 x+2)^6}-\frac {(1-2 x)^{7/2}}{441 (3 x+2)^7}-\frac {467 (1-2 x)^{5/2}}{2646 (3 x+2)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (3 x+2)^4}+\frac {2335 \sqrt {1-2 x}}{3111696 (3 x+2)}+\frac {2335 \sqrt {1-2 x}}{1333584 (3 x+2)^2}-\frac {2335 \sqrt {1-2 x}}{95256 (3 x+2)^3}+\frac {2335 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 51
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^8} \, dx &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {1}{441} \int \frac {(1-2 x)^{5/2} (1967+3675 x)}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}+\frac {2335}{882} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}-\frac {2335 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5} \, dx}{2646}\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}+\frac {2335 \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx}{10584}\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac {2335 \sqrt {1-2 x}}{95256 (2+3 x)^3}-\frac {2335 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{95256}\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac {2335 \sqrt {1-2 x}}{95256 (2+3 x)^3}+\frac {2335 \sqrt {1-2 x}}{1333584 (2+3 x)^2}-\frac {2335 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{444528}\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac {2335 \sqrt {1-2 x}}{95256 (2+3 x)^3}+\frac {2335 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {2335 \sqrt {1-2 x}}{3111696 (2+3 x)}-\frac {2335 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{3111696}\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac {2335 \sqrt {1-2 x}}{95256 (2+3 x)^3}+\frac {2335 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {2335 \sqrt {1-2 x}}{3111696 (2+3 x)}+\frac {2335 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3111696}\\ &=-\frac {(1-2 x)^{7/2}}{441 (2+3 x)^7}+\frac {23 (1-2 x)^{7/2}}{882 (2+3 x)^6}-\frac {467 (1-2 x)^{5/2}}{2646 (2+3 x)^5}+\frac {2335 (1-2 x)^{3/2}}{31752 (2+3 x)^4}-\frac {2335 \sqrt {1-2 x}}{95256 (2+3 x)^3}+\frac {2335 \sqrt {1-2 x}}{1333584 (2+3 x)^2}+\frac {2335 \sqrt {1-2 x}}{3111696 (2+3 x)}+\frac {2335 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848 \sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 47, normalized size = 0.28 \[ \frac {(1-2 x)^{7/2} \left (\frac {823543 (69 x+44)}{(3 x+2)^7}-149440 \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{726364926} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 145, normalized size = 0.86 \[ \frac {2335 \, \sqrt {21} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1702215 \, x^{6} + 8132805 \, x^{5} - 24492348 \, x^{4} - 23950566 \, x^{3} + 1405308 \, x^{2} + 1415408 \, x - 1107536\right )} \sqrt {-2 \, x + 1}}{65345616 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.15, size = 148, normalized size = 0.88 \[ -\frac {2335}{65345616} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1702215 \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + 26478900 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 8891883 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 386781696 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 951955683 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 784886900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 274710415 \, \sqrt {-2 \, x + 1}}{199148544 \, {\left (3 \, x + 2\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 93, normalized size = 0.55 \[ \frac {2335 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{32672808}-\frac {139968 \left (\frac {2335 \left (-2 x +1\right )^{\frac {13}{2}}}{298722816}-\frac {11675 \left (-2 x +1\right )^{\frac {11}{2}}}{96018048}+\frac {6721 \left (-2 x +1\right )^{\frac {9}{2}}}{164602368}+\frac {571 \left (-2 x +1\right )^{\frac {7}{2}}}{321489}-\frac {132161 \left (-2 x +1\right )^{\frac {5}{2}}}{30233088}+\frac {81725 \left (-2 x +1\right )^{\frac {3}{2}}}{22674816}-\frac {114415 \sqrt {-2 x +1}}{90699264}\right )}{\left (-6 x -4\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.41, size = 164, normalized size = 0.98 \[ -\frac {2335}{65345616} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1702215 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 26478900 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 8891883 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 386781696 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 951955683 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 784886900 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 274710415 \, \sqrt {-2 \, x + 1}}{1555848 \, {\left (2187 \, {\left (2 \, x - 1\right )}^{7} + 35721 \, {\left (2 \, x - 1\right )}^{6} + 250047 \, {\left (2 \, x - 1\right )}^{5} + 972405 \, {\left (2 \, x - 1\right )}^{4} + 2268945 \, {\left (2 \, x - 1\right )}^{3} + 3176523 \, {\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.19, size = 143, normalized size = 0.85 \[ \frac {\frac {81725\,{\left (1-2\,x\right )}^{3/2}}{354294}-\frac {114415\,\sqrt {1-2\,x}}{1417176}-\frac {132161\,{\left (1-2\,x\right )}^{5/2}}{472392}+\frac {36544\,{\left (1-2\,x\right )}^{7/2}}{321489}+\frac {6721\,{\left (1-2\,x\right )}^{9/2}}{2571912}-\frac {11675\,{\left (1-2\,x\right )}^{11/2}}{1500282}+\frac {2335\,{\left (1-2\,x\right )}^{13/2}}{4667544}}{\frac {1647086\,x}{729}+\frac {117649\,{\left (2\,x-1\right )}^2}{81}+\frac {84035\,{\left (2\,x-1\right )}^3}{81}+\frac {12005\,{\left (2\,x-1\right )}^4}{27}+\frac {343\,{\left (2\,x-1\right )}^5}{3}+\frac {49\,{\left (2\,x-1\right )}^6}{3}+{\left (2\,x-1\right )}^7-\frac {1647086}{2187}}+\frac {2335\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{32672808} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________