Optimal. Leaf size=225 \[ -\frac {6101 \sqrt {5-2 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{231725 \sqrt {66} \sqrt {2 x-5}}-\frac {361 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{481988 (5 x+7)}+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}+\frac {361 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1204970 \sqrt {5-2 x}}-\frac {6655867 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{747081400 \sqrt {11} \sqrt {2 x-5}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {164, 1604, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ -\frac {361 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{481988 (5 x+7)}+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}-\frac {6101 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{231725 \sqrt {66} \sqrt {2 x-5}}+\frac {361 \sqrt {11} \sqrt {2 x-5} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1204970 \sqrt {5-2 x}}-\frac {6655867 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{747081400 \sqrt {11} \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 113
Rule 114
Rule 119
Rule 121
Rule 158
Rule 164
Rule 168
Rule 537
Rule 538
Rule 1604
Rule 1607
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx &=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {1}{156} \int \frac {-37+100 x+24 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx\\ &=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {\int \frac {-272145+485280 x+77976 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{8675784}\\ &=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {\int \frac {\frac {1880568}{25}+\frac {77976 x}{5}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{8675784}+\frac {6655867 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{72298200}\\ &=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {1083 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{1204970}-\frac {6101 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{463450}-\frac {6655867 \operatorname {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {-\frac {11}{3}-\frac {2 x^2}{3}}} \, dx,x,\sqrt {2-3 x}\right )}{36149100}\\ &=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}-\frac {\left (6101 \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{231725 \sqrt {22} \sqrt {-5+2 x}}-\frac {\left (6655867 \sqrt {5-2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {1+\frac {2 x^2}{11}}} \, dx,x,\sqrt {2-3 x}\right )}{12049700 \sqrt {33} \sqrt {-5+2 x}}-\frac {\left (1083 \sqrt {-5+2 x}\right ) \int \frac {\sqrt {\frac {15}{11}-\frac {6 x}{11}}}{\sqrt {2-3 x} \sqrt {\frac {3}{11}+\frac {12 x}{11}}} \, dx}{1204970 \sqrt {5-2 x}}\\ &=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}+\frac {361 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1204970 \sqrt {5-2 x}}-\frac {6101 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{231725 \sqrt {66} \sqrt {-5+2 x}}-\frac {6655867 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{747081400 \sqrt {11} \sqrt {-5+2 x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.58, size = 135, normalized size = 0.60 \[ \frac {-3 \sqrt {55-22 x} \left (-9834812 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+2462020 E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )+6655867 \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )\right )-\frac {17050 \sqrt {2-3 x} (2 x-5) \sqrt {4 x+1} (5415 x-10957)}{(5 x+7)^2}}{24653686200 \sqrt {2 x-5}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{250 \, x^{4} + 425 \, x^{3} - 1155 \, x^{2} - 2989 \, x - 1715}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 461, normalized size = 2.05 \[ -\frac {\sqrt {-3 x +2}\, \sqrt {4 x +1}\, \sqrt {2 x -5}\, \left (2215818000 x^{4}-10946406900 x^{3}-184651500 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+737610900 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-499190025 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x^{2} \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )+15016020250 x^{2}-517024200 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+2065310520 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-1397732070 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, x \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )-2999896350 x -361916940 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticE \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )+1445717364 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticF \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {i \sqrt {2}}{2}\right )-978412449 \sqrt {11}\, \sqrt {-3 x +2}\, \sqrt {-2 x +5}\, \sqrt {4 x +1}\, \EllipticPi \left (\frac {2 \sqrt {-33 x +22}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )-1868168500\right )}{24653686200 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (5 x +7\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________