3.39 \(\int \frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{7+5 x} \, dx\)
Optimal. Leaf size=182 \[ -\frac{1253 \sqrt{\frac{2}{33}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{375 \sqrt{2 x-5}}+\frac{2}{15} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{427 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{225 \sqrt{5-2 x}}-\frac{2691 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{125 \sqrt{11} \sqrt{2 x-5}} \]
[Out]
(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/15 - (427*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 -
3*x])/Sqrt[11]], -1/2])/(225*Sqrt[5 - 2*x]) - (1253*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[
1 + 4*x]], 1/3])/(375*Sqrt[-5 + 2*x]) - (2691*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[1
1]], -1/2])/(125*Sqrt[11]*Sqrt[-5 + 2*x])
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Rubi [A] time = 0.212519, antiderivative size = 182, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used
= {161, 1607, 168, 538, 537, 158, 114, 113, 121, 119} \[ \frac{2}{15} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{1253 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{375 \sqrt{2 x-5}}-\frac{427 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{225 \sqrt{5-2 x}}-\frac{2691 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{125 \sqrt{11} \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x),x]
[Out]
(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/15 - (427*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 -
3*x])/Sqrt[11]], -1/2])/(225*Sqrt[5 - 2*x]) - (1253*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[
1 + 4*x]], 1/3])/(375*Sqrt[-5 + 2*x]) - (2691*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[1
1]], -1/2])/(125*Sqrt[11]*Sqrt[-5 + 2*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 1607
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h
*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q,
x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]
Rule 168
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]
Rule 538
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] && !GtQ[c, 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 158
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]
Rule 114
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) && !LtQ[-((b*c - a*d)/d), 0]
Rule 113
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !LtQ[-((b*c - a*d)/d),
0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)
/b, 0])
Rule 121
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]
Rule 119
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) && !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))
Rubi steps
\begin{align*} \int \frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{7+5 x} \, dx &=\frac{2}{15} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{1}{15} \int \frac{-3-1190 x+854 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx\\ &=\frac{2}{15} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{1}{15} \int \frac{-\frac{11928}{25}+\frac{854 x}{5}}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx+\frac{27807}{125} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx\\ &=\frac{2}{15} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}-\frac{1253}{375} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx+\frac{427}{75} \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx-\frac{55614}{125} \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{-\frac{11}{3}-\frac{2 x^2}{3}}} \, dx,x,\sqrt{2-3 x}\right )\\ &=\frac{2}{15} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}-\frac{\left (1253 \sqrt{\frac{2}{11}} \sqrt{5-2 x}\right ) \int \frac{1}{\sqrt{2-3 x} \sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{375 \sqrt{-5+2 x}}-\frac{\left (55614 \sqrt{\frac{3}{11}} \sqrt{5-2 x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (31-5 x^2\right ) \sqrt{\frac{11}{3}-\frac{4 x^2}{3}} \sqrt{1+\frac{2 x^2}{11}}} \, dx,x,\sqrt{2-3 x}\right )}{125 \sqrt{-5+2 x}}+\frac{\left (427 \sqrt{-5+2 x}\right ) \int \frac{\sqrt{\frac{15}{11}-\frac{6 x}{11}}}{\sqrt{2-3 x} \sqrt{\frac{3}{11}+\frac{12 x}{11}}} \, dx}{75 \sqrt{5-2 x}}\\ &=\frac{2}{15} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}-\frac{427 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{225 \sqrt{5-2 x}}-\frac{1253 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{375 \sqrt{-5+2 x}}-\frac{2691 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{125 \sqrt{11} \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.765967, size = 141, normalized size = 0.77 \[ \frac{\sqrt{2 x-5} \left (-3759 \sqrt{11} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right ),-\frac{1}{2}\right )+1650 \sqrt{2-3 x} \sqrt{5-2 x} \sqrt{4 x+1}-23485 \sqrt{11} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )-24219 \sqrt{11} \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{12375 \sqrt{5-2 x}} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x),x]
[Out]
(Sqrt[-5 + 2*x]*(1650*Sqrt[2 - 3*x]*Sqrt[5 - 2*x]*Sqrt[1 + 4*x] - 23485*Sqrt[11]*EllipticE[ArcSin[(2*Sqrt[2 -
3*x])/Sqrt[11]], -1/2] - 3759*Sqrt[11]*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] - 24219*Sqrt[11]*El
lipticPi[55/124, -ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(12375*Sqrt[5 - 2*x])
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Maple [A] time = 0.02, size = 183, normalized size = 1. \begin{align*} -{\frac{1}{297000\,{x}^{3}-866250\,{x}^{2}+259875\,x+123750}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 3759\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) +23485\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -24219\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticPi} \left ( 2/11\,\sqrt{22-33\,x},{\frac{55}{124}},i/2\sqrt{2} \right ) -39600\,{x}^{3}+115500\,{x}^{2}-34650\,x-16500 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2-3*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)/(7+5*x),x)
[Out]
-1/12375*(2-3*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(3759*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*El
lipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))+23485*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticE
(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))-24219*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*EllipticPi(2/11*
(22-33*x)^(1/2),55/124,1/2*I*2^(1/2))-39600*x^3+115500*x^2-34650*x-16500)/(24*x^3-70*x^2+21*x+10)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{5 \, x + 7}\,{d x} \end{align*}
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),x, algorithm="maxima")
[Out]
integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{5 \, x + 7}, x\right ) \end{align*}
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),x, algorithm="fricas")
[Out]
integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 - 3 x} \sqrt{2 x - 5} \sqrt{4 x + 1}}{5 x + 7}\, dx \end{align*}
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),x)
[Out]
Integral(sqrt(2 - 3*x)*sqrt(2*x - 5)*sqrt(4*x + 1)/(5*x + 7), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{5 \, x + 7}\,{d x} \end{align*}
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),x, algorithm="giac")
[Out]
integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7), x)