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

Optimal. Leaf size=193 \[ \frac{72479 \sqrt{\frac{11}{6}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{756 \sqrt{2 x-5}}+\frac{5}{28} \sqrt{2-3 x} (2 x-5)^{3/2} (4 x+1)^{3/2}+\frac{136}{105} \sqrt{2-3 x} \sqrt{2 x-5} (4 x+1)^{3/2}-\frac{20911 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{3780}-\frac{954811 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{22680 \sqrt{5-2 x}} \]

[Out]

(-20911*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/3780 + (136*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*(1 + 4*x)^(3/2))/
105 + (5*Sqrt[2 - 3*x]*(-5 + 2*x)^(3/2)*(1 + 4*x)^(3/2))/28 - (954811*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin
[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(22680*Sqrt[5 - 2*x]) + (72479*Sqrt[11/6]*Sqrt[5 - 2*x]*EllipticF[ArcSin[
Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(756*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.0774144, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {154, 158, 114, 113, 121, 119} \[ \frac{5}{28} \sqrt{2-3 x} (2 x-5)^{3/2} (4 x+1)^{3/2}+\frac{136}{105} \sqrt{2-3 x} \sqrt{2 x-5} (4 x+1)^{3/2}-\frac{20911 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{3780}+\frac{72479 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{756 \sqrt{2 x-5}}-\frac{954811 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{22680 \sqrt{5-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-20911*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/3780 + (136*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*(1 + 4*x)^(3/2))/
105 + (5*Sqrt[2 - 3*x]*(-5 + 2*x)^(3/2)*(1 + 4*x)^(3/2))/28 - (954811*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin
[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(22680*Sqrt[5 - 2*x]) + (72479*Sqrt[11/6]*Sqrt[5 - 2*x]*EllipticF[ArcSin[
Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(756*Sqrt[-5 + 2*x])

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

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 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x) \, dx &=\frac{5}{28} \sqrt{2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac{1}{28} \int \frac{\left (\frac{1249}{2}-1088 x\right ) \sqrt{-5+2 x} \sqrt{1+4 x}}{\sqrt{2-3 x}} \, dx\\ &=\frac{136}{105} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}+\frac{5}{28} \sqrt{2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac{1}{840} \int \frac{(38731-41822 x) \sqrt{1+4 x}}{\sqrt{2-3 x} \sqrt{-5+2 x}} \, dx\\ &=-\frac{20911 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{3780}+\frac{136}{105} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}+\frac{5}{28} \sqrt{2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac{\int \frac{-787710+1909622 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{7560}\\ &=-\frac{20911 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{3780}+\frac{136}{105} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}+\frac{5}{28} \sqrt{2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac{954811 \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx}{7560}+\frac{797269 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{1512}\\ &=-\frac{20911 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{3780}+\frac{136}{105} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}+\frac{5}{28} \sqrt{2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac{\left (72479 \sqrt{\frac{11}{2}} \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}{756 \sqrt{-5+2 x}}+\frac{\left (954811 \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}{7560 \sqrt{5-2 x}}\\ &=-\frac{20911 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{3780}+\frac{136}{105} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}+\frac{5}{28} \sqrt{2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac{954811 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{22680 \sqrt{5-2 x}}+\frac{72479 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{756 \sqrt{-5+2 x}}\\ \end{align*}

Mathematica [A]  time = 0.232161, size = 125, normalized size = 0.65 \[ \frac{724790 \sqrt{66} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )+24 \sqrt{2-3 x} \sqrt{4 x+1} \left (5400 x^3-6066 x^2-37975 x+48475\right )-954811 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{45360 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(48475 - 37975*x - 6066*x^2 + 5400*x^3) - 954811*Sqrt[66]*Sqrt[5 - 2*x]*Ellipt
icE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] + 724790*Sqrt[66]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1
+ 4*x]], 1/3])/(45360*Sqrt[-5 + 2*x])

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Maple [A]  time = 0.012, size = 150, normalized size = 0.8 \begin{align*}{\frac{1}{544320\,{x}^{3}-1587600\,{x}^{2}+476280\,x+226800}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 1087185\,\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 ) -954811\,\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 ) +777600\,{x}^{5}-1197504\,{x}^{4}-5234040\,{x}^{3}+9404484\,{x}^{2}-1997100\,x-1163400 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22680*(2-3*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(1087185*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*
EllipticF(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))-954811*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(4*x+1)^(1/2)*Ellipt
icE(2/11*(22-33*x)^(1/2),1/2*I*2^(1/2))+777600*x^5-1197504*x^4-5234040*x^3+9404484*x^2-1997100*x-1163400)/(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{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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