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

Optimal. Leaf size=279 \[ -\frac{4 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{195 \sqrt{2 x-5}}+\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{39 \sqrt{5 x+7}}+\frac{2 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{5 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{69 \sqrt{\frac{2}{341}} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{-\frac{5-2 x}{4 x+1}} (4 x+1) \Pi \left (\frac{78}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{22}{39}} \sqrt{5 x+7}}{\sqrt{4 x+1}}\right )|\frac{39}{62}\right )}{25 \sqrt{2-3 x} \sqrt{2 x-5}} \]

[Out]

(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(39*Sqrt[7 + 5*x]) - (4*Sqrt[2 -
3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(195*Sqrt[-5 + 2*x]) + (2*Sqrt[11/39]*Sqrt[2 -
3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqr
t[-5 + 2*x]], -23/39])/(5*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (69*Sqrt[2/
341]*Sqrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[-((5 - 2*x)/(1 + 4*x))]*(1 + 4*x)*Ellipti
cPi[78/55, ArcSin[(Sqrt[22/39]*Sqrt[7 + 5*x])/Sqrt[1 + 4*x]], 39/62])/(25*Sqrt[2
- 3*x]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.626157, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189 \[ -\frac{4 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{195 \sqrt{2 x-5}}+\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{39 \sqrt{5 x+7}}+\frac{2 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{5 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{69 \sqrt{\frac{2}{341}} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{-\frac{5-2 x}{4 x+1}} (4 x+1) \Pi \left (\frac{78}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{22}{39}} \sqrt{5 x+7}}{\sqrt{4 x+1}}\right )|\frac{39}{62}\right )}{25 \sqrt{2-3 x} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(39*Sqrt[7 + 5*x]) - (4*Sqrt[2 -
3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(195*Sqrt[-5 + 2*x]) + (2*Sqrt[11/39]*Sqrt[2 -
3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqr
t[-5 + 2*x]], -23/39])/(5*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (69*Sqrt[2/
341]*Sqrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[-((5 - 2*x)/(1 + 4*x))]*(1 + 4*x)*Ellipti
cPi[78/55, ArcSin[(Sqrt[22/39]*Sqrt[7 + 5*x])/Sqrt[1 + 4*x]], 39/62])/(25*Sqrt[2
- 3*x]*Sqrt[-5 + 2*x])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 3.20499, size = 326, normalized size = 1.17 \[ \frac{\sqrt{2 x-5} \sqrt{4 x+1} \left (23 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-62 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-2 \sqrt{\frac{5 x+7}{3 x-2}} \left (39 \sqrt{682} (2-3 x)^2 \sqrt{\frac{4 x+1}{3 x-2}} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-961 \left (8 x^2-18 x-5\right )\right )\right )}{6045 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(-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]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] - 2*Sqrt
[(7 + 5*x)/(-2 + 3*x)]*(-961*(-5 - 18*x + 8*x^2) + 39*Sqrt[682]*(2 - 3*x)^2*Sqrt
[(1 + 4*x)/(-2 + 3*x)]*Sqrt[(-35 - 11*x + 10*x^2)/(2 - 3*x)^2]*EllipticPi[117/62
, ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62])))/(6045*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 = 924, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/10725*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)*(-5+2*x)^(1/2)*(880*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*EllipticF(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))+1104*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*EllipticPi(1/31*31^(
1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1
/2))-880*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))+440*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*Elli
pticF(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))+552*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*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x
)/(1+4*x))^(1/2),124/55,1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-440*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))+55*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)*EllipticF(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))+69*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)*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(1+4*x))^(1/2),124/55,1/39*
2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))-55*11^(1/2)*((7+5*x)/(1+4*x))^(1/2)*3^(1/2)*1
3^(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))-7590*
x^2+24035*x-12650)/(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. \[ \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{3}{2}} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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