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

Optimal. Leaf size=60 \[ \frac{2 \sqrt{\frac{11}{39}} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{22}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{62}{39}\right )}{23 \sqrt{2 x-5}} \]

[Out]

(2*Sqrt[11/39]*Sqrt[5 - 2*x]*EllipticE[ArcSin[(Sqrt[39/22]*Sqrt[1 + 4*x])/Sqrt[7 + 5*x]], 62/39])/(23*Sqrt[-5
+ 2*x])

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Rubi [B]  time = 0.130884, antiderivative size = 195, normalized size of antiderivative = 3.25, number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {176, 422, 418, 492, 411} \[ -\frac{62 \sqrt{2 x-5} \sqrt{4 x+1}}{897 \sqrt{2-3 x} \sqrt{5 x+7}}-\frac{\sqrt{\frac{22}{31}} \sqrt{4 x+1} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{39 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}}+\frac{2 \sqrt{682} \sqrt{4 x+1} E\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{897 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-62*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(897*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]) + (2*Sqrt[682]*Sqrt[1 + 4*x]*EllipticE[Ar
cTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sqrt[7 + 5*x]], 39/62])/(897*Sqrt[2 - 3*x]*Sqrt[-((1 + 4*x)/(2 - 3*x))]) - (
Sqrt[22/31]*Sqrt[1 + 4*x]*EllipticF[ArcTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sqrt[7 + 5*x]], 39/62])/(39*Sqrt[2 - 3
*x]*Sqrt[-((1 + 4*x)/(2 - 3*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 422

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

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 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*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[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\sqrt{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}} \, dx &=\frac{\left (\sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{31 x^2}{11}}}{\sqrt{1+\frac{23 x^2}{22}}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{39 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}\\ &=\frac{\left (\sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{23 x^2}{22}} \sqrt{1+\frac{31 x^2}{11}}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{39 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}+\frac{\left (31 \sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{23 x^2}{22}} \sqrt{1+\frac{31 x^2}{11}}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{429 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}\\ &=-\frac{62 \sqrt{-5+2 x} \sqrt{1+4 x}}{897 \sqrt{2-3 x} \sqrt{7+5 x}}-\frac{\sqrt{\frac{22}{31}} \sqrt{1+4 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{39 \sqrt{2-3 x} \sqrt{-\frac{1+4 x}{2-3 x}}}-\frac{\left (62 \sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{23 x^2}{22}}}{\left (1+\frac{31 x^2}{11}\right )^{3/2}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{897 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}\\ &=-\frac{62 \sqrt{-5+2 x} \sqrt{1+4 x}}{897 \sqrt{2-3 x} \sqrt{7+5 x}}+\frac{2 \sqrt{682} \sqrt{1+4 x} E\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{897 \sqrt{2-3 x} \sqrt{-\frac{1+4 x}{2-3 x}}}-\frac{\sqrt{\frac{22}{31}} \sqrt{1+4 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{39 \sqrt{2-3 x} \sqrt{-\frac{1+4 x}{2-3 x}}}\\ \end{align*}

Mathematica [B]  time = 1.72862, size = 237, normalized size = 3.95 \[ \frac{\sqrt{2 x-5} \sqrt{4 x+1} \left (-23 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right ),\frac{39}{62}\right )-1922 \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )+62 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )}{27807 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(-1922*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2) + 62*Sqrt[682]*Sqrt[(-5 -
18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39
/62] - 23*Sqrt[682]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticF[ArcSin[Sqrt[31/39]*S
qrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62]))/(27807*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x
+ 8*x^2))

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Maple [B]  time = 0.025, size = 330, normalized size = 5.5 \begin{align*}{\frac{2}{107640\,{x}^{4}-163254\,{x}^{3}-345345\,{x}^{2}+176709\,x+62790}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1}\sqrt{7+5\,x} \left ( 16\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}{x}^{2}{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +8\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}x{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}{\it EllipticE} \left ({\frac{\sqrt{31}\sqrt{11}}{31}\sqrt{{\frac{7+5\,x}{4\,x+1}}}},{\frac{\sqrt{31}\sqrt{78}}{39}} \right ) +138\,{x}^{2}-437\,x+230 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/897*(2-3*x)^(1/2)*(7+5*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(16*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^
(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*x^2*EllipticE(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^
(1/2),1/39*31^(1/2)*78^(1/2))+8*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2
+3*x)/(4*x+1))^(1/2)*x*EllipticE(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))+11^(1/
2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*EllipticE(1/31*31
^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))+138*x^2-437*x+230)/(120*x^4-182*x^3-385*x^2+19
7*x+70)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{200 \, x^{4} + 110 \, x^{3} - 993 \, x^{2} - 1232 \, x - 245}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(200*x^4 + 110*x^3 - 993*x^2 - 1232*x - 245)
, 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((2-3*x)**(1/2)/(7+5*x)**(3/2)/(-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 \frac{\sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{3}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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