Optimal. Leaf size=156 \[ \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {808 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 \sqrt {2+3 x}}-\frac {5594 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773 \sqrt {33}}-\frac {1196 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 {1196 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773 \sqrt {33}}-\frac {5594 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773 \sqrt {33}}+\frac {5594 \sqrt {1-2 x} \sqrt {5 x+3}}{41503 \sqrt {3 x+2}}+\frac {808 \sqrt {5 x+3}}{17787 \sqrt {1-2 x} \sqrt {3 x+2}}+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} \sqrt {3 x+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)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} \sqrt {2+3 x}}-\frac {2}{231} \int \frac {-\frac {157}{2}-45 x}{(1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {808 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {4 \int \frac {\frac {6837}{4}+1515 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{17787}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {808 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 \sqrt {2+3 x}}+\frac {8 \int \frac {8760+\frac {41955 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{124509}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {808 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 \sqrt {2+3 x}}+\frac {5594 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{41503}+\frac {598 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3773}\\ &=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} \sqrt {2+3 x}}+\frac {808 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} \sqrt {2+3 x}}+\frac {5594 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 \sqrt {2+3 x}}-\frac {5594 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773 \sqrt {33}}-\frac {1196 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.42, size = 98, normalized size = 0.63 \begin {gather*} \frac {2 \left (\frac {\sqrt {3+5 x} \left (12297-39220 x+33564 x^2\right )}{(1-2 x)^{3/2} \sqrt {2+3 x}}+\sqrt {2} \left (2797 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+7070 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{124509} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 224, normalized size = 1.44
method | result | size |
default | \(-\frac {2 \left (19734 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5594 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9867 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2797 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-167820 x^{3}+95408 x^{2}+56175 x -36891\right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {1-2 x}}{124509 \left (15 x^{2}+19 x +6\right ) \left (-1+2 x \right )^{2}}\) | \(224\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {940 \left (-30 x^{2}-38 x -12\right )}{124509 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {23360 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{871563 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {27970 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{871563 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {540}{343} x^{2}-\frac {54}{343} x +\frac {162}{343}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1617 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
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 = 0.17, size = 50, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (33564 \, x^{2} - 39220 \, x + 12297\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{124509 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\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 {5}{2}} \left (3 x + 2\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, 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 )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________