Optimal. Leaf size=288 \[ \frac {103964 \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1918683 \sqrt {253} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {358120 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2319687747 \sqrt {2 x-5}}-\frac {895300 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2319687747 \sqrt {5 x+7}}-\frac {50 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{83421 (5 x+7)^{3/2}}-\frac {179060 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{59479173 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {172, 1599, 1602, 12, 170, 418, 176, 424} \[ \frac {358120 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2319687747 \sqrt {2 x-5}}-\frac {895300 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{2319687747 \sqrt {5 x+7}}-\frac {50 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{83421 (5 x+7)^{3/2}}+\frac {103964 \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{1918683 \sqrt {253} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {179060 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{59479173 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 170
Rule 172
Rule 176
Rule 418
Rule 424
Rule 1599
Rule 1602
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{5/2}} \, dx &=-\frac {50 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{83421 (7+5 x)^{3/2}}+\frac {\int \frac {11928-4270 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx}{83421}\\ &=-\frac {50 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{83421 (7+5 x)^{3/2}}-\frac {895300 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}+\frac {\int \frac {41179978+16294460 x-21487200 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{2319687747}\\ &=-\frac {50 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{83421 (7+5 x)^{3/2}}-\frac {895300 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}+\frac {358120 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2319687747 \sqrt {-5+2 x}}-\frac {\int -\frac {15083097120}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{556725059280}+\frac {1969660 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{59479173}\\ &=-\frac {50 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{83421 (7+5 x)^{3/2}}-\frac {895300 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}+\frac {358120 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2319687747 \sqrt {-5+2 x}}+\frac {51982 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1918683}-\frac {\left (179060 \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 )}{59479173 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}}\\ &=-\frac {50 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{83421 (7+5 x)^{3/2}}-\frac {895300 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}+\frac {358120 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2319687747 \sqrt {-5+2 x}}-\frac {179060 \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 )}{59479173 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (51982 \sqrt {\frac {2}{253}} \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 )}{1918683 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}}\\ &=-\frac {50 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{83421 (7+5 x)^{3/2}}-\frac {895300 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{2319687747 \sqrt {7+5 x}}+\frac {358120 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2319687747 \sqrt {-5+2 x}}-\frac {179060 \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 )}{59479173 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {103964 \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 )}{1918683 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.83, size = 246, normalized size = 0.85 \[ -\frac {2 \sqrt {2 x-5} \sqrt {4 x+1} \left (-28819 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )-984830 \sqrt {682} (3 x-2) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^2 E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )+1705 \sqrt {\frac {5 x+7}{3 x-2}} \left (608600 x^3-294854 x^2-2797991 x-671560\right )\right )}{25516565217 \sqrt {2-3 x} (5 x+7)^{3/2} \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{3000 \, x^{6} + 3850 \, x^{5} - 16485 \, x^{4} - 30943 \, x^{3} - 3325 \, x^{2} + 14553 \, x + 3430}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 7\right )}^{\frac {5}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 786, normalized size = 2.73 \[ \frac {2 \left (78786400 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{3} \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+50128960 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{3} \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+496006500 x^{3}+149694160 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{2} \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+95245024 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{2} \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-665223020 x^{2}+60074630 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+38223332 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-2040625895 x +6893810 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+4386284 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+1509107050\right ) \sqrt {4 x +1}\, \sqrt {2 x -5}\, \sqrt {-3 x +2}}{25516565217 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right ) \sqrt {5 x +7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 7\right )}^{\frac {5}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{5/2}} \,d x \]
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]
________________________________________________________________________________________