Optimal. Leaf size=156 \[ \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {3896 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {164 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164,
114, 120} \begin {gather*} -\frac {164 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {3896 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}+\frac {19480 \sqrt {1-2 x} \sqrt {3 x+2}}{27951 \sqrt {5 x+3}}-\frac {410 \sqrt {1-2 x} \sqrt {3 x+2}}{2541 (5 x+3)^{3/2}}+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx &=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {95}{2}-45 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {4 \int \frac {-\frac {605}{2}+\frac {615 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2541}\\ &=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {18885}{4}-7305 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{27951}\\ &=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}+\frac {82}{847} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {3896 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{9317}\\ &=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {3896 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {164 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.48, size = 98, normalized size = 0.63 \begin {gather*} \frac {2 \left (\frac {\sqrt {2+3 x} \left (27691-5230 x-97400 x^2\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}}+\sqrt {2} \left (1948 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-595 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{27951} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 215, normalized size = 1.38
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (6765 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9740 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+4059 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-5844 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-292200 x^{3}-210490 x^{2}+72613 x +55382\right )}{27951 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {8 \left (-30 x^{2}-38 x -12\right )}{9317 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {12590 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{195657 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {19480 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{195657 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {5600}{1331} x^{2}-\frac {2800}{3993} x +\frac {5600}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.25, size = 50, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (97400 \, x^{2} + 5230 \, x - 27691\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27951 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \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 {1}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________