Optimal. Leaf size=158 \[ \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {8 \sqrt {1-2 x} \sqrt {3+5 x}}{49 (2+3 x)^{3/2}}+\frac {38 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}-\frac {38}{343} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {212 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 158, 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 = {101, 157, 164,
114, 120} \begin {gather*} -\frac {212 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}}-\frac {38}{343} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{343 \sqrt {3 x+2}}-\frac {8 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{3/2}}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 101
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {2}{7} \int \frac {-13-\frac {45 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {8 \sqrt {1-2 x} \sqrt {3+5 x}}{49 (2+3 x)^{3/2}}-\frac {4}{147} \int \frac {-\frac {99}{4}-30 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {8 \sqrt {1-2 x} \sqrt {3+5 x}}{49 (2+3 x)^{3/2}}+\frac {38 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}-\frac {8 \int \frac {-\frac {165}{2}-\frac {285 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {8 \sqrt {1-2 x} \sqrt {3+5 x}}{49 (2+3 x)^{3/2}}+\frac {38 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}+\frac {38}{343} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {106}{343} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {8 \sqrt {1-2 x} \sqrt {3+5 x}}{49 (2+3 x)^{3/2}}+\frac {38 \sqrt {1-2 x} \sqrt {3+5 x}}{343 \sqrt {2+3 x}}-\frac {38}{343} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {212 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{343 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.19, size = 99, normalized size = 0.63 \begin {gather*} \frac {2 \left (-\frac {3 \sqrt {3+5 x} \left (-59-37 x+114 x^2\right )}{\sqrt {1-2 x} (2+3 x)^{3/2}}+\sqrt {2} \left (19 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+140 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{1029} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 215, normalized size = 1.36
method | result | size |
default | \(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (477 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-57 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+318 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-38 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1710 x^{3}-471 x^{2}+1218 x +531\right )}{1029 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 \left (-30 x^{2}-38 x -12\right )}{343 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {220 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {190 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7203 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {500}{343} x^{2}-\frac {50}{343} x +\frac {150}{343}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{441 \left (\frac {2}{3}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.19, size = 50, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (114 \, x^{2} - 37 \, x - 59\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{343 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________