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

Optimal. Leaf size=351 \[ \frac {8959 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{720 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}+\frac {509 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{240 \sqrt {2 x-5}}-\frac {509 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{160 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {2198489 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{3600 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]

[Out]

2198489/1544400*(2-3*x)*EllipticPi(1/23*253^(1/2)*(7+5*x)^(1/2)/(2-3*x)^(1/2),-69/55,1/39*I*897^(1/2))*((5-2*x
)/(2-3*x))^(1/2)*((-1-4*x)/(2-3*x))^(1/2)*429^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2)+509/240*(2-3*x)^(1/2)*(1+4*x)
^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)+1/4*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)+8959/16560*(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)-509/480
*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 = 0.32, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {161, 1602, 1598, 170, 418, 165, 537, 176, 424} \[ \frac {1}{4} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}+\frac {509 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{240 \sqrt {2 x-5}}+\frac {8959 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{720 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {509 \sqrt {\frac {143}{3}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{160 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {2198489 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{3600 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(509*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(240*Sqrt[-5 + 2*x]) + (Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 +
4*x]*Sqrt[7 + 5*x])/4 - (509*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])/(160*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (8959*Sqrt[11/23]*Sq
rt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(720*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*
x)/(5 - 2*x)]) + (2198489*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55,
ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(3600*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

Rule 161

Int[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_Sy
mbol] :> Simp[(2*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(2*m + 5)), x] + Dist[1/(b*(2
*m + 5)), Int[((a + b*x)^m*Simp[3*b*c*e*g - a*(d*e*g + c*f*g + c*e*h) + 2*(b*(d*e*g + c*f*g + c*e*h) - a*(d*f*
g + d*e*h + c*f*h))*x - (3*a*d*f*h - b*(d*f*g + d*e*h + c*f*h))*x^2, x])/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g +
h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] &&  !LtQ[m, -1]

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 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 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 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 1598

Int[((A_.) + (B_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.
) + (h_.)*(x_)]), x_Symbol] :> Dist[(A*b - a*B)/b, Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h
*x]), x], x] + Dist[B/b, 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, A, B}, x]

Rule 1602

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

Rubi steps

\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{\sqrt {-5+2 x}} \, dx &=\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {1}{8} \int \frac {309-410 x-1018 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx\\ &=\frac {509 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{240 \sqrt {-5+2 x}}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {\int \frac {-320516+283676 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1920}+\frac {72787}{160} \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx\\ &=\frac {509 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{240 \sqrt {-5+2 x}}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}+\frac {70919 \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1440}+\frac {98549 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1440}-\frac {\left (6617 \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 )}{160 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}}\\ &=\frac {509 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{240 \sqrt {-5+2 x}}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {509 \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 )}{160 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (2198489 (2-3 x) \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {11 x^2}{23}} \sqrt {1+\frac {11 x^2}{39}} \left (5+3 x^2\right )} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {2-3 x}}\right )}{720 \sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}}+\frac {\left (8959 \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 )}{720 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}}\\ &=\frac {509 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{240 \sqrt {-5+2 x}}+\frac {1}{4} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}-\frac {509 \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 )}{160 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {8959 \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 )}{720 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {2198489 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{3600 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}}\\ \end {align*}

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Mathematica [A]  time = 4.21, size = 347, normalized size = 0.99 \[ \frac {\sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} \left (66960 (2-3 x)-\frac {3 \left (76756 \sqrt {682} \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )+94674 \sqrt {682} (2-3 x) (5 x+7) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )+\sqrt {\frac {5 x+7}{3 x-2}} \left (70919 \sqrt {682} \sqrt {\frac {4 x+1}{3 x-2}} \sqrt {\frac {10 x^2-11 x-35}{(2-3 x)^2}} (2-3 x)^2 \Pi \left (\frac {117}{62};\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )+284022 \left (40 x^3-34 x^2-151 x-35\right )\right )\right )}{(2-3 x) \left (\frac {5 x+7}{3 x-2}\right )^{3/2} \left (-8 x^2+18 x+5\right )}\right )}{267840 \sqrt {2-3 x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*(66960*(2 - 3*x) - (3*(94674*Sqrt[682]*(2 - 3*x)*(7 + 5*x)*Sqrt[(-
5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + 76756*Sqrt[
682]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticF[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/
(-2 + 3*x)]], 39/62] + Sqrt[(7 + 5*x)/(-2 + 3*x)]*(284022*(-35 - 151*x - 34*x^2 + 40*x^3) + 70919*Sqrt[682]*(2
- 3*x)^2*Sqrt[(1 + 4*x)/(-2 + 3*x)]*Sqrt[(-35 - 11*x + 10*x^2)/(2 - 3*x)^2]*EllipticPi[117/62, ArcSin[Sqrt[31
/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62])))/((2 - 3*x)*((7 + 5*x)/(-2 + 3*x))^(3/2)*(5 + 18*x - 8*x^2))))/(26
7840*Sqrt[2 - 3*x])

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 880, normalized size = 2.51 \[ -\frac {\sqrt {-3 x +2}\, \sqrt {4 x +1}\, \sqrt {5 x +7}\, \sqrt {2 x -5}\, \left (-61776000 x^{4}-168339600 x^{3}-34937760 \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 )+3622960 \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 )-26098192 \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 )[phone] x^{2}-17468880 \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 )+1811480 \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 )-13049096 \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 )[phone] x -2183610 \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 )+226435 \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 )-1631137 \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 )-647446800\right )}{2059200 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2059200*(-3*x+2)^(1/2)*(4*x+1)^(1/2)*(5*x+7)^(1/2)*(2*x-5)^(1/2)*(3622960*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))-26098192*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))-34937760*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))+1811480*
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))-13049096*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))-17468880*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))+226435*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)*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))-1631137*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))-2183610*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))-168339600*x^3-61776000*x^4+661123320*x^2+623542920*x-647446800)/(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 \[ \int \frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{\sqrt {2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1} \sqrt {5 x + 7}}{\sqrt {2 x - 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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