3.102 \(\int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\)

Optimal. Leaf size=469 \[ \frac {65 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {895 \sqrt {\frac {11}{62}} \sqrt {2-3 x} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right ),\frac {39}{62}\right )}{48 \sqrt {-\frac {2-3 x}{4 x+1}} \sqrt {4 x+1}}-\frac {5 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{12 \sqrt {2 x-5}}+\frac {5 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {23 \sqrt {\frac {31}{22}} \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {4 x+1}}{\sqrt {5 x+7}}\right )|\frac {39}{62}\right )}{6 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {4117 \sqrt {2-3 x} \Pi \left (\frac {78}{55};\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right )|\frac {39}{62}\right )}{48 \sqrt {682} \sqrt {-\frac {2-3 x}{4 x+1}} \sqrt {4 x+1}} \]

[Out]

-895/2976*(1/(529+506*(7+5*x)/(-5+2*x)))^(1/2)*(529+506*(7+5*x)/(-5+2*x))^(1/2)*EllipticF(506^(1/2)*(7+5*x)^(1
/2)/(-5+2*x)^(1/2)/(529+506*(7+5*x)/(-5+2*x))^(1/2),1/62*2418^(1/2))*682^(1/2)*(2-3*x)^(1/2)/((-2+3*x)/(1+4*x)
)^(1/2)/(1+4*x)^(1/2)-4117/32736*(1/(529+506*(7+5*x)/(-5+2*x)))^(1/2)*(529+506*(7+5*x)/(-5+2*x))^(1/2)*Ellipti
cPi(506^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)/(529+506*(7+5*x)/(-5+2*x))^(1/2),78/55,1/62*2418^(1/2))*(2-3*x)^(1/
2)*682^(1/2)/((-2+3*x)/(1+4*x))^(1/2)/(1+4*x)^(1/2)+23/132*(7+5*x)*EllipticPi(1/11*341^(1/2)*(1+4*x)^(1/2)/(7+
5*x)^(1/2),55/124,1/62*2418^(1/2))*682^(1/2)*((2-3*x)/(7+5*x))^(1/2)*((5-2*x)/(7+5*x))^(1/2)/(2-3*x)^(1/2)/(-5
+2*x)^(1/2)-5/12*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)+65/184*(1/(4+2*(1+4*x)/(2-3*x)))^(1/
2)*(4+2*(1+4*x)/(2-3*x))^(1/2)*EllipticF((1+4*x)^(1/2)*2^(1/2)/(2-3*x)^(1/2)/(4+2*(1+4*x)/(2-3*x))^(1/2),1/23*
I*897^(1/2))*253^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)/((7+5*x)/(5-2*x))^(1/2)+5/24*EllipticE(1/23*897^(1/2)*(1+4
*x)^(1/2)/(-5+2*x)^(1/2),1/39*I*897^(1/2))*429^(1/2)*(2-3*x)^(1/2)*((7+5*x)/(5-2*x))^(1/2)/((2-3*x)/(5-2*x))^(
1/2)/(7+5*x)^(1/2)

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Rubi [A]  time = 0.28, antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {166, 173, 176, 424, 170, 418, 165, 536, 539, 537} \[ -\frac {5 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{12 \sqrt {2 x-5}}+\frac {65 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{8 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {895 \sqrt {\frac {11}{62}} \sqrt {2-3 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right )|\frac {39}{62}\right )}{48 \sqrt {-\frac {2-3 x}{4 x+1}} \sqrt {4 x+1}}+\frac {5 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {23 \sqrt {\frac {31}{22}} \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {4 x+1}}{\sqrt {5 x+7}}\right )|\frac {39}{62}\right )}{6 \sqrt {2-3 x} \sqrt {2 x-5}}-\frac {4117 \sqrt {2-3 x} \Pi \left (\frac {78}{55};\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {5 x+7}}{\sqrt {2 x-5}}\right )|\frac {39}{62}\right )}{48 \sqrt {682} \sqrt {-\frac {2-3 x}{4 x+1}} \sqrt {4 x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(7 + 5*x)^(3/2)/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

(-5*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(12*Sqrt[-5 + 2*x]) + (5*Sqrt[143/3]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*
x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(8*Sqrt[(2 - 3*x)/(5 - 2*
x)]*Sqrt[7 + 5*x]) + (65*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -3
9/23])/(8*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) - (895*Sqrt[11/62]*Sqrt[2 - 3*x]*EllipticF[ArcTan[(Sqrt[22
/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(48*Sqrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[1 + 4*x]) + (23*Sqrt[31/22]
*Sqrt[(2 - 3*x)/(7 + 5*x)]*Sqrt[(5 - 2*x)/(7 + 5*x)]*(7 + 5*x)*EllipticPi[55/124, ArcSin[(Sqrt[31/11]*Sqrt[1 +
4*x])/Sqrt[7 + 5*x]], 39/62])/(6*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]) - (4117*Sqrt[2 - 3*x]*EllipticPi[78/55, ArcTan
[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(48*Sqrt[682]*Sqrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[1 + 4*x]
)

Rule 165

Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S
ymbol] :> Dist[(2*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Sqrt[((b*g - a*h)*(e + f*x))
/((f*g - e*h)*(a + b*x))])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Subst[Int[1/((h - b*x^2)*Sqrt[1 + ((b*c - a*d)*x^2)/
(d*g - c*h)]*Sqrt[1 + ((b*e - a*f)*x^2)/(f*g - e*h)]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b,
c, d, e, f, g, h}, x]

Rule 166

Int[((a_.) + (b_.)*(x_))^(3/2)/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[b/d, Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/(Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] - Dist[(b*c - a*
d)/d, Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x]

Rule 170

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(2*Sqrt[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((f*g - e*h)*Sqrt[c +
d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]), Subst[Int[1/(Sqrt[1 + ((b*c - a*d)*x^2)/(d*e
- c*f)]*Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d
, e, f, g, h}, x]

Rule 173

Int[(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)])/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Simp[(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x])/(h*Sqrt[e + f*x]), x] + (-Dist[((d*e - c*f)*(f*g
- e*h))/(2*f*h), Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*(e + f*x)^(3/2)*Sqrt[g + h*x]), x], x] + Dist[((d*e - c*f)*(
b*f*g + b*e*h - 2*a*f*h))/(2*f^2*h), Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] +
Dist[(a*d*f*h - b*(d*f*g + d*e*h - c*f*h))/(2*f^2*h), Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g +
h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 176

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(-2*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])/((b*e - a*f)*Sqrt
[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]), Subst[Int[Sqrt[1 + ((b*c - a*d)*x^2)/(d*e -
c*f)]/Sqrt[1 - ((b*g - a*h)*x^2)/(f*g - e*h)], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e
, f, g, h}, x]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 536

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> -Dist[f/(b*e -
a*f), Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[b/(b*e - a*f), Int[Sqrt[e + f*x^2]/((a + b*x^2)*
Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/c, 0] && GtQ[f/e, 0] &&  !SimplerSqrtQ[d/c,
f/e]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rubi steps

\begin {align*} \int \frac {(7+5 x)^{3/2}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx &=-\left (\frac {5}{3} \int \frac {\sqrt {2-3 x} \sqrt {7+5 x}}{\sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\right )+\frac {31}{3} \int \frac {\sqrt {7+5 x}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{12 \sqrt {-5+2 x}}+\frac {895}{48} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx+\frac {715}{16} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx-\frac {715}{8} \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx+\frac {\left (713 \sqrt {2} \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {-\frac {-5+2 x}{7+5 x}} (7+5 x)\right ) \operatorname {Subst}\left (\int \frac {1}{\left (4-5 x^2\right ) \sqrt {1-\frac {31 x^2}{11}} \sqrt {1-\frac {39 x^2}{22}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {7+5 x}}\right )}{33 \sqrt {2-3 x} \sqrt {-5+2 x}}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{12 \sqrt {-5+2 x}}+\frac {23 \sqrt {\frac {31}{22}} \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {\frac {5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{6 \sqrt {2-3 x} \sqrt {-5+2 x}}+\frac {\left (11635 \sqrt {-\frac {2-3 x}{-5+2 x}} (-5+2 x) \sqrt {\frac {1+4 x}{-5+2 x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (5-2 x^2\right ) \sqrt {1+\frac {11 x^2}{31}} \sqrt {1+\frac {22 x^2}{23}}} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )}{8 \sqrt {713} \sqrt {2-3 x} \sqrt {1+4 x}}+\frac {\left (65 \sqrt {\frac {11}{46}} \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 )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}}+\frac {\left (65 \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 )}{8 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{12 \sqrt {-5+2 x}}+\frac {5 \sqrt {\frac {143}{3}} \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 )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {65 \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 )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {23 \sqrt {\frac {31}{22}} \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {\frac {5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{6 \sqrt {2-3 x} \sqrt {-5+2 x}}+\frac {\left (895 \sqrt {\frac {23}{31}} \sqrt {-\frac {2-3 x}{-5+2 x}} (-5+2 x) \sqrt {\frac {1+4 x}{-5+2 x}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {22 x^2}{23}}}{\left (5-2 x^2\right ) \sqrt {1+\frac {11 x^2}{31}}} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )}{48 \sqrt {2-3 x} \sqrt {1+4 x}}+\frac {\left (9845 \sqrt {-\frac {2-3 x}{-5+2 x}} (-5+2 x) \sqrt {\frac {1+4 x}{-5+2 x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {11 x^2}{31}} \sqrt {1+\frac {22 x^2}{23}}} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )}{48 \sqrt {713} \sqrt {2-3 x} \sqrt {1+4 x}}\\ &=-\frac {5 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{12 \sqrt {-5+2 x}}+\frac {5 \sqrt {\frac {143}{3}} \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 )}{8 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {65 \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 )}{8 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {895 \sqrt {\frac {11}{62}} \sqrt {2-3 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )|\frac {39}{62}\right )}{48 \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}}+\frac {23 \sqrt {\frac {31}{22}} \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {\frac {5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{6 \sqrt {2-3 x} \sqrt {-5+2 x}}-\frac {4117 \sqrt {2-3 x} \Pi \left (\frac {78}{55};\tan ^{-1}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right )|\frac {39}{62}\right )}{48 \sqrt {682} \sqrt {-\frac {2-3 x}{1+4 x}} \sqrt {1+4 x}}\\ \end {align*}

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Mathematica [A]  time = 1.32, size = 347, normalized size = 0.74 \[ \frac {\sqrt {2 x-5} \left (-6969 \sqrt {341} \sqrt {\frac {3 x-2}{4 x+1}} \sqrt {\frac {5 x+7}{4 x+1}} \left (8 x^2-18 x-5\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {22}{39}} \sqrt {\frac {5 x+7}{4 x+1}}\right ),\frac {39}{62}\right )+6820 \sqrt {341} \sqrt {\frac {3 x-2}{4 x+1}} \sqrt {\frac {5 x+7}{4 x+1}} \left (8 x^2-18 x-5\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {22}{39}} \sqrt {\frac {5 x+7}{4 x+1}}\right )|\frac {39}{62}\right )+\sqrt {\frac {2 x-5}{4 x+1}} \left (9821 \sqrt {341} \sqrt {\frac {3 x-2}{4 x+1}} \sqrt {\frac {10 x^2-11 x-35}{(4 x+1)^2}} (4 x+1)^2 \Pi \left (\frac {78}{55};\sin ^{-1}\left (\sqrt {\frac {22}{39}} \sqrt {\frac {5 x+7}{4 x+1}}\right )|\frac {39}{62}\right )+13640 \sqrt {2} \left (30 x^3-53 x^2-83 x+70\right )\right )\right )}{16368 \sqrt {4-6 x} \left (\frac {2 x-5}{4 x+1}\right )^{3/2} (4 x+1)^{3/2} \sqrt {5 x+7}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(7 + 5*x)^(3/2)/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]),x]

[Out]

(Sqrt[-5 + 2*x]*(6820*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*Sqrt[(7 + 5*x)/(1 + 4*x)]*(-5 - 18*x + 8*x^2)*Ellip
ticE[ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 39/62] - 6969*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*Sqrt[(7
+ 5*x)/(1 + 4*x)]*(-5 - 18*x + 8*x^2)*EllipticF[ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 39/62] + Sqrt[
(-5 + 2*x)/(1 + 4*x)]*(13640*Sqrt[2]*(70 - 83*x - 53*x^2 + 30*x^3) + 9821*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]
*(1 + 4*x)^2*Sqrt[(-35 - 11*x + 10*x^2)/(1 + 4*x)^2]*EllipticPi[78/55, ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 +
4*x)]], 39/62])))/(16368*Sqrt[4 - 6*x]*((-5 + 2*x)/(1 + 4*x))^(3/2)*(1 + 4*x)^(3/2)*Sqrt[7 + 5*x])

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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{24 \, x^{3} - 70 \, x^{2} + 21 \, x + 10}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7+5*x)^(3/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-(5*x + 7)^(3/2)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(24*x^3 - 70*x^2 + 21*x + 10), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7+5*x)^(3/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="giac")

[Out]

integrate((5*x + 7)^(3/2)/(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), x)

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maple [A]  time = 0.03, size = 875, normalized size = 1.87 \[ \frac {\sqrt {5 x +7}\, \sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (-514800 x^{3}-68640 \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 )+71024 \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 )-157136 \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} \EllipticPi \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {124}{55}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+909480 x^{2}-34320 \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 )+35512 \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 )-78568 \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 \EllipticPi \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {124}{55}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+1424280 x -4290 \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 )+4439 \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 )-9821 \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}}\, \EllipticPi \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {124}{55}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-1201200\right )}{2471040 x^{4}-3747744 x^{3}-7927920 x^{2}+4056624 x +1441440} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+7)^(3/2)/(-3*x+2)^(1/2)/(2*x-5)^(1/2)/(4*x+1)^(1/2),x)

[Out]

1/20592*(5*x+7)^(1/2)*(-3*x+2)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(71024*11^(1/2)*((5*x+7)/(4*x+1))^(1/2)*3^(1/
2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((3*x-2)/(4*x+1))^(1/2)*x^2*EllipticF(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x
+1))^(1/2),1/39*31^(1/2)*78^(1/2))-157136*11^(1/2)*((5*x+7)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^
(1/2)*((3*x-2)/(4*x+1))^(1/2)*x^2*EllipticPi(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),124/55,1/39*31^(1/
2)*78^(1/2))-68640*11^(1/2)*((5*x+7)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((3*x-2)/(4*x+1))
^(1/2)*x^2*EllipticE(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))+35512*11^(1/2)*((5
*x+7)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((3*x-2)/(4*x+1))^(1/2)*x*EllipticF(1/31*31^(1/2
)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-78568*11^(1/2)*((5*x+7)/(4*x+1))^(1/2)*3^(1/2)*13^(
1/2)*((2*x-5)/(4*x+1))^(1/2)*((3*x-2)/(4*x+1))^(1/2)*x*EllipticPi(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/
2),124/55,1/39*31^(1/2)*78^(1/2))-34320*11^(1/2)*((5*x+7)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1
/2)*((3*x-2)/(4*x+1))^(1/2)*x*EllipticE(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))
+4439*11^(1/2)*((5*x+7)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((3*x-2)/(4*x+1))^(1/2)*Ellipt
icF(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-9821*11^(1/2)*((5*x+7)/(4*x+1))^(1/
2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((3*x-2)/(4*x+1))^(1/2)*EllipticPi(1/31*31^(1/2)*11^(1/2)*((5*x+7)
/(4*x+1))^(1/2),124/55,1/39*31^(1/2)*78^(1/2))-4290*11^(1/2)*((5*x+7)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)
/(4*x+1))^(1/2)*((3*x-2)/(4*x+1))^(1/2)*EllipticE(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),1/39*31^(1/2)
*78^(1/2))-514800*x^3+909480*x^2+1424280*x-1201200)/(120*x^4-182*x^3-385*x^2+197*x+70)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 7\right )}^{\frac {3}{2}}}{\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((7+5*x)^(3/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((5*x + 7)^(3/2)/(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (5\,x+7\right )}^{3/2}}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 7)^(3/2)/((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2)),x)

[Out]

int((5*x + 7)^(3/2)/((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2)), x)

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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]

integrate((7+5*x)**(3/2)/(2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

Timed out

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