Optimal. Leaf size=160 \[ \frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {5057 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 160, 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 = {100, 159, 164,
114, 120} \begin {gather*} \frac {5057 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {7 \sqrt {5 x+3} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}+\frac {312}{275} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {14517 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{2750} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 100
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x)^{3/2} \left (\frac {381}{2}+312 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {1}{275} \int \frac {\left (-13425-\frac {43551 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {\int \frac {\frac {1915857}{4}+\frac {1513107 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4125}\\ &=\frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {15171 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2500}-\frac {504369 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{13750}\\ &=\frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {5057 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.56, size = 110, normalized size = 0.69 \begin {gather*} \frac {-10 \sqrt {2+3 x} \sqrt {3+5 x} \left (-27757+11154 x+2970 x^2\right )-336246 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+169365 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{27500 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 143, normalized size = 0.89
method | result | size |
default | \(\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (166881 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-336246 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+445500 x^{4}+2237400 x^{3}-1866090 x^{2}-4604590 x -1665420\right )}{825000 x^{3}+632500 x^{2}-192500 x -165000}\) | \(143\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {27 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50}+\frac {1149 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{500}-\frac {212873 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{38500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {168123 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{19250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {343 \left (-30 x^{2}-38 x -12\right )}{88 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(240\) |
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 = 0.18, size = 40, normalized size = 0.25 \begin {gather*} \frac {{\left (2970 \, x^{2} + 11154 \, x - 27757\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2750 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{7/2}}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________