Optimal. Leaf size=370 \[ \frac {258506776 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1618368818157 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{1956607901151813 \sqrt {2 x-5}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1956607901151813 \sqrt {5 x+7}}-\frac {3217468 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{50259901185 (5 x+7)^{3/2}}+\frac {98 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1807455 (5 x+7)^{5/2}}+\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{273 (5 x+7)^{7/2}}-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.52, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {164, 1604, 1599, 1602, 12, 170, 418, 176, 424} \[ \frac {16377776536 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{1956607901151813 \sqrt {2 x-5}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1956607901151813 \sqrt {5 x+7}}-\frac {3217468 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{50259901185 (5 x+7)^{3/2}}+\frac {98 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1807455 (5 x+7)^{5/2}}+\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{273 (5 x+7)^{7/2}}+\frac {258506776 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{1618368818157 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 164
Rule 170
Rule 176
Rule 418
Rule 424
Rule 1599
Rule 1602
Rule 1604
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}-\frac {1}{273} \int \frac {-49+70 x+96 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{7/2}} \, dx\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {\int \frac {-958104+2280510 x+49392 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{5/2}} \, dx}{37956555}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {\int \frac {-11461434930+18134687340 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx}{3166373774655}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}-\frac {\int \frac {-32763839696280-33533497457460 x+44219996647200 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{88047355551831585}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}+\frac {\int \frac {18564560715729600}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{21131365332439580400}+\frac {90077770948 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{50169433362867}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}+\frac {1421787268 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1618368818157}-\frac {\left (8188888268 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {-\frac {7+5 x}{-5+2 x}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-\frac {39 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )}{50169433362867 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (129253388 \sqrt {\frac {22}{23}} \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {7+5 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2}} \sqrt {1+\frac {31 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {2-3 x}}\right )}{1618368818157 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}}\\ &=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {258506776 \sqrt {\frac {11}{23}} \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{1618368818157 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.45, size = 258, normalized size = 0.70 \[ \frac {2 \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} \left (\frac {(3 x-2) \left (2559027583750 x^3+12313608173580 x^2+19165803061167 x+2552362046246\right )}{(5 x+7)^4}-\frac {22 \left (71545594 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )+558333291 \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )-186111097 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )\right )}{\sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )}\right )}{1956607901151813 \sqrt {2-3 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{6250 \, x^{6} + 28125 \, x^{5} + 13125 \, x^{4} - 134750 \, x^{3} - 308700 \, x^{2} - 266511 \, x - 84035}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 1160, normalized size = 3.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}} \sqrt {2 \, x - 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{9/2}} \,d x \]
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]
________________________________________________________________________________________