\(\int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx\) [82]

Optimal result

Integrand size = 37, antiderivative size = 391 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^{3/2}}+\frac {17906 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{417105 \sqrt {7+5 x}}-\frac {35812 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{2085525 \sqrt {-5+2 x}}+\frac {17906 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{53475 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}-\frac {496 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1725 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {496 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{125 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]

[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)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {166, 1618, 1616, 1612, 176, 429, 171, 551, 182, 435} \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\frac {17906 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{53475 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {496 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{125 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}}-\frac {496 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1725 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {35812 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{2085525 \sqrt {2 x-5}}+\frac {17906 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{417105 \sqrt {5 x+7}}-\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^{3/2}} \]

[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 166

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)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*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], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && Lt
Q[m, -1]

Rule 171

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 176

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 182

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 429

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

Rule 435

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 1612

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 1616

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/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*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], 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 1618

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*} \text {integral}& = -\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 ) \text {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 ) \text {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 ) \text {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*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 17.09 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (34864+44765 x)}{417105 (7+5 x)^{3/2}}+\frac {\frac {3571978410 \sqrt {1+4 x} \sqrt {7+5 x} \sqrt {-75+30 x}}{\sqrt {2-3 x}}-\frac {3571978410 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}+\frac {5251113560 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {6160344428 \sqrt {715} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{-2+3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{\sqrt {\frac {5-2 x}{2-3 x}} \sqrt {1+4 x}}-\frac {344407635 i \sqrt {10230} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{-5+2 x}} \operatorname {EllipticPi}\left (-\frac {23}{55},i \text {arcsinh}\left (\frac {\sqrt {\frac {22}{23}} \sqrt {7+5 x}}{\sqrt {-5+2 x}}\right ),\frac {23}{62}\right )}{\sqrt {\frac {2-3 x}{5-2 x}} \sqrt {1+4 x}}-\frac {371344545 \sqrt {10230} \sqrt {2-3 x} \sqrt {\frac {-5+2 x}{1+4 x}} \operatorname {EllipticPi}\left (\frac {78}{55},\arcsin \left (\frac {\sqrt {\frac {22}{39}} \sqrt {7+5 x}}{\sqrt {1+4 x}}\right ),\frac {39}{62}\right )}{\sqrt {-5+2 x} \sqrt {\frac {-2+3 x}{1+4 x}}}}{138676984875 \sqrt {15}} \]

[In]

Integrate[(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]*(34864 + 44765*x))/(417105*(7 + 5*x)^(3/2)) + ((3571978410*Sqrt[
1 + 4*x]*Sqrt[7 + 5*x]*Sqrt[-75 + 30*x])/Sqrt[2 - 3*x] - (3571978410*Sqrt[715]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(
-2 + 3*x)]*EllipticE[ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(Sqrt[(5 - 2*x)/(2 - 3*x)]*Sq
rt[1 + 4*x]) + (5251113560*Sqrt[715]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(-2 + 3*x)]*EllipticF[ArcSin[(Sqrt[11/23]*S
qrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[1 + 4*x]) - (6160344428*Sqrt[715]*Sqrt[
-5 + 2*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])/(Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[1 + 4*x]) - ((344407635*I)*Sqrt[10230]*Sqrt[2 - 3*x]*Sqrt[(1 + 4*x)/(-5
+ 2*x)]*EllipticPi[-23/55, I*ArcSinh[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 23/62])/(Sqrt[(2 - 3*x)/(5 -
2*x)]*Sqrt[1 + 4*x]) - (371344545*Sqrt[10230]*Sqrt[2 - 3*x]*Sqrt[(-5 + 2*x)/(1 + 4*x)]*EllipticPi[78/55, ArcS
in[(Sqrt[22/39]*Sqrt[7 + 5*x])/Sqrt[1 + 4*x]], 39/62])/(Sqrt[-5 + 2*x]*Sqrt[(-2 + 3*x)/(1 + 4*x)]))/(138676984
875*Sqrt[15])

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.19

method	result	size
elliptic	\(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {2 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{375 \left (x +\frac {7}{5}\right )^{2}}+\frac {-\frac {143248}{139035} x^{3}+\frac {250684}{83421} x^{2}-\frac {125342}{139035} x -\frac {35812}{83421}}{\sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}+\frac {81284 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{127582826085 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {22348 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{1962812709 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}+\frac {\frac {71624 \left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}{139035}-\frac {35812 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {181 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{341}-\frac {117 E\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{62}+\frac {91 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{55}\right )}{5612147775}}{\sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\)	\(464\)
default	\(\text {Expression too large to display}\)	\(1077\)

[In]

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

[Out]

(-(7+5*x)*(-2+3*x)*(-5+2*x)*(1+4*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2)/(7+5*x)^(1/2)*(-2/375*(-
120*x^4+182*x^3+385*x^2-197*x-70)^(1/2)/(x+7/5)^2+17906/2085525*(-120*x^3+350*x^2-105*x-50)/((x+7/5)*(-120*x^3
+350*x^2-105*x-50))^(1/2)+81284/127582826085*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/
3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)/(-30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2)*EllipticF(1/69*(-
3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-22348/1962812709*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^
(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)/(-30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2
)*(2/3*EllipticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-31/15*EllipticPi(1/69*(-3795*(x+7/5)/(-
2/3+x))^(1/2),-69/55,1/39*I*897^(1/2)))+71624/139035*((x+7/5)*(x-5/2)*(x+1/4)-1/80730*(-3795*(x+7/5)/(-2/3+x))
^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)*(181/341*EllipticF(1/
69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))-117/62*EllipticE(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*
I*897^(1/2))+91/55*EllipticPi(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2))))/(-30*(x+7/5)*(-2/
3+x)*(x-5/2)*(x+1/4))^(1/2))

Fricas [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}{\left (5 x + 7\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {5}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^{5/2}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^{5/2}} \,d x \]

[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)