Optimal. Leaf size=222 \[ -\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}-\frac {3545996 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}-\frac {95264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157,
164, 114, 120} \begin {gather*} -\frac {95264 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}-\frac {3545996 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{567 (3 x+2)^{7/2}}+\frac {1532 (5 x+3)^{3/2} \sqrt {1-2 x}}{567 (3 x+2)^{5/2}}+\frac {3545996 \sqrt {5 x+3} \sqrt {1-2 x}}{250047 \sqrt {3 x+2}}-\frac {104036 \sqrt {5 x+3} \sqrt {1-2 x}}{35721 (3 x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-\frac {1905}{2}+\frac {15 x}{2}\right )}{(2+3 x)^{7/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {8 \int \frac {\left (\frac {91845}{4}-14325 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{8505}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {16 \int \frac {\frac {3022395}{8}-\frac {1057575 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{535815}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {32 \int \frac {\frac {41857275}{8}+\frac {66487425 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3750705}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}+\frac {523952 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{250047}+\frac {3545996 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{250047}\\ &=-\frac {104036 \sqrt {1-2 x} \sqrt {3+5 x}}{35721 (2+3 x)^{3/2}}+\frac {3545996 \sqrt {1-2 x} \sqrt {3+5 x}}{250047 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{567 (2+3 x)^{7/2}}+\frac {1532 \sqrt {1-2 x} (3+5 x)^{3/2}}{567 (2+3 x)^{5/2}}-\frac {3545996 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}-\frac {95264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{250047}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 9.18, size = 111, normalized size = 0.50 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (29785139+176436240 x+391601529 x^2+386630766 x^3+143612838 x^4\right )}{2 (2+3 x)^{9/2}}+\sqrt {2} \left (886499 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-493535 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{750141} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs.
\(2(162)=324\).
time = 0.10, size = 494, normalized size = 2.23
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {5930 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{137781 \left (\frac {2}{3}+x \right )^{3}}+\frac {38764 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321489 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {35459960}{250047} x^{2}-\frac {3545996}{250047} x +\frac {3545996}{83349}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {11161940 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5250987 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {17729980 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{5250987 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{177147 \left (\frac {2}{3}+x \right )^{5}}-\frac {776 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59049 \left (\frac {2}{3}+x \right )^{4}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(308\) |
default | \(-\frac {2 \left (63660168 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-143612838 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+169760448 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-382967568 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+169760448 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-382967568 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+75449088 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-170207808 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4308385140 x^{6}+12574848 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-28367968 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-12029761494 x^{5}-11615422626 x^{4}-2988214893 x^{3}+2101550871 x^{2}+1498570743 x +268066251\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{750141 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(494\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.23, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (143612838 \, x^{4} + 386630766 \, x^{3} + 391601529 \, x^{2} + 176436240 \, x + 29785139\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{250047 \, {\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________