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 = 0.405623, 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 \[ -\frac{62 \sqrt{2 x-5} \sqrt{4 x+1}}{897 \sqrt{2-3 x} \sqrt{5 x+7}}-\frac{\sqrt{\frac{22}{31}} \sqrt{4 x+1} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{39 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}}+\frac{2 \sqrt{682} \sqrt{4 x+1} E\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{897 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}} \]

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[6
82]*Sqrt[1 + 4*x]*EllipticE[ArcTan[(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))])

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Rubi in Sympy [A]  time = 33.807, size = 396, normalized size = 6.6 \[ - \frac{22 \sqrt{\frac{- 156 x - 39}{- 110 x - 154}} \sqrt{\frac{117 x - 78}{55 x + 77}} \sqrt{2 x - 5} \sqrt{5 x + 7} \sqrt{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}}{1521 \sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}} + \frac{22 \sqrt{506} \sqrt{\frac{- 156 x - 39}{- 110 x - 154}} \sqrt{\frac{117 x - 78}{55 x + 77}} \left (5 x + 7\right ) \sqrt{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1} E\left (\operatorname{atan}{\left (\frac{\sqrt{506} \sqrt{2 x - 5}}{22 \sqrt{5 x + 7}} \right )}\middle | - \frac{39}{23}\right )}{34983 \sqrt{\frac{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}} \sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}} - \frac{22 \sqrt{506} \sqrt{\frac{- 156 x - 39}{- 110 x - 154}} \sqrt{\frac{117 x - 78}{55 x + 77}} \left (5 x + 7\right ) \sqrt{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1} F\left (\operatorname{atan}{\left (\frac{\sqrt{506} \sqrt{2 x - 5}}{22 \sqrt{5 x + 7}} \right )}\middle | - \frac{39}{23}\right )}{34983 \sqrt{\frac{\frac{31 \left (2 x - 5\right )}{11 \left (5 x + 7\right )} + 1}{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}} \sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{\frac{23 \left (2 x - 5\right )}{22 \left (5 x + 7\right )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-3*x)**(1/2)/(7+5*x)**(3/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

-22*sqrt((-156*x - 39)/(-110*x - 154))*sqrt((117*x - 78)/(55*x + 77))*sqrt(2*x -
5)*sqrt(5*x + 7)*sqrt(31*(2*x - 5)/(11*(5*x + 7)) + 1)/(1521*sqrt(-3*x + 2)*sqr
t(4*x + 1)*sqrt(23*(2*x - 5)/(22*(5*x + 7)) + 1)) + 22*sqrt(506)*sqrt((-156*x -
39)/(-110*x - 154))*sqrt((117*x - 78)/(55*x + 77))*(5*x + 7)*sqrt(31*(2*x - 5)/(
11*(5*x + 7)) + 1)*elliptic_e(atan(sqrt(506)*sqrt(2*x - 5)/(22*sqrt(5*x + 7))),
-39/23)/(34983*sqrt((31*(2*x - 5)/(11*(5*x + 7)) + 1)/(23*(2*x - 5)/(22*(5*x + 7
)) + 1))*sqrt(-3*x + 2)*sqrt(4*x + 1)*sqrt(23*(2*x - 5)/(22*(5*x + 7)) + 1)) - 2
2*sqrt(506)*sqrt((-156*x - 39)/(-110*x - 154))*sqrt((117*x - 78)/(55*x + 77))*(5
*x + 7)*sqrt(31*(2*x - 5)/(11*(5*x + 7)) + 1)*elliptic_f(atan(sqrt(506)*sqrt(2*x
- 5)/(22*sqrt(5*x + 7))), -39/23)/(34983*sqrt((31*(2*x - 5)/(11*(5*x + 7)) + 1)
/(23*(2*x - 5)/(22*(5*x + 7)) + 1))*sqrt(-3*x + 2)*sqrt(4*x + 1)*sqrt(23*(2*x -
5)/(22*(5*x + 7)) + 1))

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Mathematica [B]  time = 1.90847, size = 237, normalized size = 3.95 \[ \frac{\sqrt{2 x-5} \sqrt{4 x+1} \left (-1922 \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )-23 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+62 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )}{27807 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

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)*E
llipticE[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[Sqr
t[31/39]*Sqrt[(-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 = 0.036, size = 348, normalized size = 5.8 \[{\frac{2}{107640\,{x}^{4}-163254\,{x}^{3}-345345\,{x}^{2}+176709\,x+62790}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x}\sqrt{7+5\,x} \left ( 16\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}{x}^{2}{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +8\,\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}x{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}},1/39\,\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13} \right ) +\sqrt{11}\sqrt{{\frac{7+5\,x}{1+4\,x}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{-5+2\,x}{1+4\,x}}}\sqrt{{\frac{-2+3\,x}{1+4\,x}}}{\it EllipticE} \left ({\frac{\sqrt{31}\sqrt{11}}{31}\sqrt{{\frac{7+5\,x}{1+4\,x}}}},{\frac{\sqrt{2}\sqrt{3}\sqrt{31}\sqrt{13}}{39}} \right ) +138\,{x}^{2}-437\,x+230 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/897*(2-3*x)^(1/2)*(7+5*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(16*11^(1/2)*((7+
5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))
^(1/2)*x^2*EllipticE(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)
*3^(1/2)*31^(1/2)*13^(1/2))+8*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*
((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x))^(1/2)*x*EllipticE(1/31*31^(1/2)*11^(
1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))+11^(1/2)*((
7+5*x)/(1+4*x))^(1/2)*3^(1/2)*13^(1/2)*((-5+2*x)/(1+4*x))^(1/2)*((-2+3*x)/(1+4*x
))^(1/2)*EllipticE(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3
^(1/2)*31^(1/2)*13^(1/2))+138*x^2-437*x+230)/(120*x^4-182*x^3-385*x^2+197*x+70)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x + 2)/((5*x + 7)^(3/2)*sqrt(4*x + 1)*sqrt(2*x - 5)),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. \[{\rm integral}\left (\frac{\sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{3}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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