Optimal. Leaf size=125 \[ \frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{231 (1-2 x)^{3/2}}+\frac {544 \sqrt {2+3 x} \sqrt {3+5 x}}{17787 \sqrt {1-2 x}}+\frac {272 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {202 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {202 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}+\frac {272 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}+\frac {544 \sqrt {3 x+2} \sqrt {5 x+3}}{17787 \sqrt {1-2 x}}+\frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{231 (1-2 x)^{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)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {121}{2}-15 x}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{231 (1-2 x)^{3/2}}+\frac {544 \sqrt {2+3 x} \sqrt {3+5 x}}{17787 \sqrt {1-2 x}}+\frac {4 \int \frac {\frac {885}{4}-1020 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{17787}\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{231 (1-2 x)^{3/2}}+\frac {544 \sqrt {2+3 x} \sqrt {3+5 x}}{17787 \sqrt {1-2 x}}-\frac {272 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5929}+\frac {101}{539} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{231 (1-2 x)^{3/2}}+\frac {544 \sqrt {2+3 x} \sqrt {3+5 x}}{17787 \sqrt {1-2 x}}+\frac {272 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {202 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.67, size = 115, normalized size = 0.92 \begin {gather*} -\frac {4 \sqrt {2+3 x} \sqrt {3+5 x} (-213+272 x)-272 \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 )+3605 \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 )}{17787 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(223\) vs.
\(2(93)=186\).
time = 0.10, size = 224, normalized size = 1.79
method | result | size |
default | \(-\frac {\left (6666 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+544 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3333 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-272 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+16320 x^{3}+7892 x^{2}-9660 x -5112\right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {1-2 x}}{17787 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(224\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {272 \left (-30 x^{2}-38 x -12\right )}{17787 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {295 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1360 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{231 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(225\) |
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 = 40, normalized size = 0.32 \begin {gather*} -\frac {4 \, {\left (272 \, x - 213\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{17787 \, {\left (4 \, x^{2} - 4 \, x + 1\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}} \sqrt {3 x + 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}\,\sqrt {3\,x+2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________