3.82 \(\int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx\)
Optimal. Leaf size=391 \[ -\frac {496 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1725 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {35812 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2085525 \sqrt {2 x-5}}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{417105 \sqrt {5 x+7}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}+\frac {17906 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{53475 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {496 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{125 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]
[Out]
-2/15*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(3/2)+496/53625*(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)+17906/417105*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(1/2)-35812/208552
5*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)-496/39675*(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)+17906/2085525*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 = 0.43, antiderivative size = 391, normalized size of antiderivative = 1.00,
number of steps used = 10, number of rules used = 10, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used
= {160, 1604, 1602, 1598, 170, 418, 165, 537, 176, 424} \[ -\frac {35812 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2085525 \sqrt {2 x-5}}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{417105 \sqrt {5 x+7}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}}-\frac {496 \sqrt {\frac {11}{23}} \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{1725 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {17906 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{53475 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {496 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{125 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x)^(5/2),x]
[Out]
(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(15*(7 + 5*x)^(3/2)) + (17906*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqr
t[1 + 4*x])/(417105*Sqrt[7 + 5*x]) - (35812*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2085525*Sqrt[-5 + 2*x]
) + (17906*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sq
rt[-5 + 2*x]], -23/39])/(53475*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (496*Sqrt[11/23]*Sqrt[7 + 5*x]*Ellip
ticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(1725*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) +
(496*(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])/(125*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])
Rule 160
Int[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_Sy
mbol] :> Simp[((a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(b*(m + 1)), x] - Dist[1/(2*b*(m +
1)), Int[((a + b*x)^(m + 1)*Simp[d*e*g + c*f*g + c*e*h + 2*(d*f*g + d*e*h + c*f*h)*x + 3*d*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] && Lt
Q[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]
Rule 1604
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_
.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]
*Sqrt[e + f*x]*Sqrt[g + h*x])/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)), x] - Dist[1/(2*(m + 1)*(b*c - a*d
)*(b*e - a*f)*(b*g - a*h)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*
d*f*h*(m + 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - (b*B - a*C)*(
a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h +
c*f*h)) - C*(a^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g + c*e*h)))*x + d*f*h
*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[
2*m] && LtQ[m, -1]
Rubi steps
\begin {align*} \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^{3/2}}+\frac {1}{15} \int \frac {-21+140 x-72 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^{3/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{417105 \sqrt {7+5 x}}+\frac {\int \frac {40642-726310 x+429744 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{417105}\\ &=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^{3/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{417105 \sqrt {7+5 x}}-\frac {35812 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2085525 \sqrt {-5+2 x}}-\frac {\int \frac {94243968+96100992 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{100105200}-\frac {196966 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{53475}\\ &=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^{3/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{417105 \sqrt {7+5 x}}-\frac {35812 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2085525 \sqrt {-5+2 x}}+\frac {8}{25} \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx-\frac {2728 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1725}+\frac {\left (17906 \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 )}{53475 \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}}{15 (7+5 x)^{3/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{417105 \sqrt {7+5 x}}-\frac {35812 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2085525 \sqrt {-5+2 x}}+\frac {17906 \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 )}{53475 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (496 (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 )}{25 \sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}}-\frac {\left (248 \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 )}{1725 \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}}{15 (7+5 x)^{3/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{417105 \sqrt {7+5 x}}-\frac {35812 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2085525 \sqrt {-5+2 x}}+\frac {17906 \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 )}{53475 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {496 \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 )}{1725 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {496 (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 )}{125 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}}\\ \end {align*}
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Mathematica [A] time = 3.23, size = 366, normalized size = 0.94 \[ \frac {\sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7} \left (\frac {30 \sqrt {6-9 x} (44765 x+34864)}{(5 x+7)^2}-\frac {2 \left (-37053 \sqrt {682} (3 x-2) (5 x+7) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )+80577 \sqrt {682} (3 x-2) (5 x+7) \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )+\sqrt {\frac {5 x+7}{3 x-2}} \left (48438 \sqrt {682} (2-3 x)^2 \sqrt {\frac {4 x+1}{3 x-2}} \sqrt {\frac {10 x^2-11 x-35}{(2-3 x)^2}} \Pi \left (\frac {117}{62};\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )-241731 \left (40 x^3-34 x^2-151 x-35\right )\right )\right )}{\sqrt {3} (2-3 x)^{3/2} \left (\frac {5 x+7}{3 x-2}\right )^{3/2} \left (-8 x^2+18 x+5\right )}\right )}{6256575 \sqrt {3}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x)^(5/2),x]
[Out]
(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*((30*Sqrt[6 - 9*x]*(34864 + 44765*x))/(7 + 5*x)^2 - (2*(80577*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] - 37053*Sqrt[682]*(-2 + 3*x)*(7 + 5*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticF[Ar
cSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-241731*(-35 - 151*x - 34*
x^2 + 40*x^3) + 48438*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])))/(Sqrt[3]*(2 - 3*x)^(3/2)*((7 +
5*x)/(-2 + 3*x))^(3/2)*(5 + 18*x - 8*x^2))))/(6256575*Sqrt[3])
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(5/2),x, algorithm="fricas")
[Out]
integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(125*x^3 + 525*x^2 + 735*x + 343), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(5/2),x, algorithm="giac")
[Out]
integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(5/2), x)
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maple [B] time = 0.05, size = 1149, normalized size = 2.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-3*x+2)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)/(5*x+7)^(5/2),x)
[Out]
2/114703875*(24389200*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))*x^3+23614560*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))*x^3-39393200*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))*x^3+46339480*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))+44867664*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))-7
4847080*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))+18596765*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))+18006102*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),12
4/55,1/39*31^(1/2)*78^(1/2))-30037315*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))+2
134055*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)*Ellip
ticF(1/31*31^(1/2)*11^(1/2)*((5*x+7)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))+2066274*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))-3446905*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*3
1^(1/2)*78^(1/2))-523292550*x^3+1135538635*x^2+779434810*x-869257400)*(4*x+1)^(1/2)*(2*x-5)^(1/2)*(-3*x+2)^(1/
2)/(120*x^4-182*x^3-385*x^2+197*x+70)/(5*x+7)^(1/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(5/2),x, algorithm="maxima")
[Out]
integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(5/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2))/(5*x + 7)^(5/2),x)
[Out]
int(((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2))/(5*x + 7)^(5/2), 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((2-3*x)**(1/2)*(-5+2*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**(5/2),x)
[Out]
Timed out
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