Optimal. Leaf size=189 \[ -91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {91}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159,
164, 114, 120} \begin {gather*} -\frac {91}{5} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {137 \sqrt {3 x+2} (5 x+3)^{5/2}}{33 \sqrt {1-2 x}}-\frac {817}{66} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-91 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {77}{2}+60 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {8033}{2}-\frac {12255 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1}{495} \int \frac {\sqrt {3+5 x} \left (\frac {1053855}{4}+405405 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {-\frac {34129755}{4}-\frac {53909955 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4455}\\ &=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1001}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {12101}{20} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {91}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 7.76, size = 125, normalized size = 0.66 \begin {gather*} -\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (957-2579 x+438 x^2+90 x^3\right )+12101 \sqrt {2-4 x} (-1+2 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-6095 \sqrt {2-4 x} (-1+2 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{60 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 234, normalized size = 1.24
method | result | size |
default | \(-\frac {\left (12012 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-24202 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-6006 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+12101 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+13500 x^{5}+82800 x^{4}-298230 x^{3}-320180 x^{2}+27090 x +57420\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}}{60 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(234\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {15 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4}-22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}+\frac {7661 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{84 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {12101 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{84 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {20735}{8} x^{2}-\frac {78793}{24} x -\frac {4147}{4}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {847 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(276\) |
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 = 0.17, size = 50, normalized size = 0.26 \begin {gather*} -\frac {{\left (90 \, x^{3} + 438 \, x^{2} - 2579 \, x + 957\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{6 \, {\left (4 \, x^{2} - 4 \, x + 1\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 [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 = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________