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

Optimal. Leaf size=365 \[ \frac{\sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{4 \sqrt{2 x-5}}-\frac{39 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{8 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{179 \sqrt{\frac{11}{62}} \sqrt{2-3 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{16 \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}}-\frac{\sqrt{429} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{8 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{4117 \sqrt{2-3 x} \Pi \left (\frac{78}{55};\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{80 \sqrt{682} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}} \]

[Out]

(Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(4*Sqrt[-5 + 2*x]) - (Sqrt[429]*Sqrt
[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])
/Sqrt[-5 + 2*x]], -23/39])/(8*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (39*Sqr
t[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])],
-39/23])/(8*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (179*Sqrt[11/62]*Sqrt[2
- 3*x]*EllipticF[ArcTan[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(16
*Sqrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[1 + 4*x]) + (4117*Sqrt[2 - 3*x]*EllipticPi[78
/55, ArcTan[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(80*Sqrt[682]*S
qrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[1 + 4*x])

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Rubi [A]  time = 0.815693, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216 \[ \frac{\sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{4 \sqrt{2 x-5}}-\frac{39 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{8 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{179 \sqrt{\frac{11}{62}} \sqrt{2-3 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{16 \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}}-\frac{\sqrt{429} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{8 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{4117 \sqrt{2-3 x} \Pi \left (\frac{78}{55};\tan ^{-1}\left (\frac{\sqrt{\frac{22}{23}} \sqrt{5 x+7}}{\sqrt{2 x-5}}\right )|\frac{39}{62}\right )}{80 \sqrt{682} \sqrt{-\frac{2-3 x}{4 x+1}} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(4*Sqrt[-5 + 2*x]) - (Sqrt[429]*Sqrt
[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])
/Sqrt[-5 + 2*x]], -23/39])/(8*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (39*Sqr
t[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])],
-39/23])/(8*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) + (179*Sqrt[11/62]*Sqrt[2
- 3*x]*EllipticF[ArcTan[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(16
*Sqrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[1 + 4*x]) + (4117*Sqrt[2 - 3*x]*EllipticPi[78
/55, ArcTan[(Sqrt[22/23]*Sqrt[7 + 5*x])/Sqrt[-5 + 2*x]], 39/62])/(80*Sqrt[682]*S
qrt[-((2 - 3*x)/(1 + 4*x))]*Sqrt[1 + 4*x])

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Rubi in Sympy [A]  time = 85.8688, size = 593, normalized size = 1.62 \[ - \frac{\sqrt{897} \sqrt{\frac{- 66 x + 44}{- 22 x + 55}} \sqrt{\frac{110 x + 154}{- 46 x + 115}} \left (- 2 x + 5\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{897} \sqrt{4 x + 1}}{23 \sqrt{2 x - 5}} \right )}\middle | - \frac{23}{39}\right )}{16 \sqrt{- 3 x + 2} \sqrt{5 x + 7}} + \frac{31 \sqrt{897} \sqrt{\frac{- 66 x + 44}{- 22 x + 55}} \sqrt{\frac{110 x + 154}{- 46 x + 115}} \left (- 2 x + 5\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{897} \sqrt{4 x + 1}}{23 \sqrt{2 x - 5}} \right )}\middle | - \frac{23}{39}\right )}{312 \sqrt{- 3 x + 2} \sqrt{5 x + 7}} + \frac{1111 \sqrt{253} \sqrt{\frac{62 x - 155}{- 55 x - 77}} \sqrt{\frac{124 x + 31}{55 x + 77}} \left (5 x + 7\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{253} \sqrt{- 3 x + 2}}{11 \sqrt{5 x + 7}} \right )}\middle | - \frac{39}{23}\right )}{5704 \sqrt{2 x - 5} \sqrt{4 x + 1}} - \frac{179 \sqrt{2} \sqrt{\frac{- 55 x - 77}{62 x - 155}} \sqrt{\frac{- 44 x - 11}{22 x - 55}} \left (- 2 x + 5\right ) \sqrt{\frac{39 \left (- 3 x + 2\right )}{31 \left (2 x - 5\right )} + 1} F\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- 3 x + 2}}{\sqrt{2 x - 5}} \right )}\middle | \frac{23}{62}\right )}{32 \sqrt{\frac{\frac{39 \left (- 3 x + 2\right )}{31 \left (2 x - 5\right )} + 1}{\frac{2 \left (- 3 x + 2\right )}{2 x - 5} + 1}} \sqrt{4 x + 1} \sqrt{5 x + 7} \sqrt{\frac{2 \left (- 3 x + 2\right )}{2 x - 5} + 1}} + \frac{179 \sqrt{2} \sqrt{\frac{- 55 x - 77}{62 x - 155}} \sqrt{\frac{- 44 x - 11}{22 x - 55}} \left (- 2 x + 5\right ) \sqrt{\frac{39 \left (- 3 x + 2\right )}{31 \left (2 x - 5\right )} + 1} \Pi \left (\frac{2}{3}; \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- 3 x + 2}}{\sqrt{2 x - 5}} \right )}\middle | \frac{23}{62}\right )}{96 \sqrt{\frac{\frac{39 \left (- 3 x + 2\right )}{31 \left (2 x - 5\right )} + 1}{\frac{2 \left (- 3 x + 2\right )}{2 x - 5} + 1}} \sqrt{4 x + 1} \sqrt{5 x + 7} \sqrt{\frac{2 \left (- 3 x + 2\right )}{2 x - 5} + 1}} + \frac{\sqrt{- 3 x + 2} \sqrt{4 x + 1} \sqrt{5 x + 7}}{4 \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(897)*sqrt((-66*x + 44)/(-22*x + 55))*sqrt((110*x + 154)/(-46*x + 115))*(-2
*x + 5)*elliptic_e(asin(sqrt(897)*sqrt(4*x + 1)/(23*sqrt(2*x - 5))), -23/39)/(16
*sqrt(-3*x + 2)*sqrt(5*x + 7)) + 31*sqrt(897)*sqrt((-66*x + 44)/(-22*x + 55))*sq
rt((110*x + 154)/(-46*x + 115))*(-2*x + 5)*elliptic_f(asin(sqrt(897)*sqrt(4*x +
1)/(23*sqrt(2*x - 5))), -23/39)/(312*sqrt(-3*x + 2)*sqrt(5*x + 7)) + 1111*sqrt(2
53)*sqrt((62*x - 155)/(-55*x - 77))*sqrt((124*x + 31)/(55*x + 77))*(5*x + 7)*ell
iptic_f(asin(sqrt(253)*sqrt(-3*x + 2)/(11*sqrt(5*x + 7))), -39/23)/(5704*sqrt(2*
x - 5)*sqrt(4*x + 1)) - 179*sqrt(2)*sqrt((-55*x - 77)/(62*x - 155))*sqrt((-44*x
- 11)/(22*x - 55))*(-2*x + 5)*sqrt(39*(-3*x + 2)/(31*(2*x - 5)) + 1)*elliptic_f(
atan(sqrt(2)*sqrt(-3*x + 2)/sqrt(2*x - 5)), 23/62)/(32*sqrt((39*(-3*x + 2)/(31*(
2*x - 5)) + 1)/(2*(-3*x + 2)/(2*x - 5) + 1))*sqrt(4*x + 1)*sqrt(5*x + 7)*sqrt(2*
(-3*x + 2)/(2*x - 5) + 1)) + 179*sqrt(2)*sqrt((-55*x - 77)/(62*x - 155))*sqrt((-
44*x - 11)/(22*x - 55))*(-2*x + 5)*sqrt(39*(-3*x + 2)/(31*(2*x - 5)) + 1)*ellipt
ic_pi(2/3, atan(sqrt(2)*sqrt(-3*x + 2)/sqrt(2*x - 5)), 23/62)/(96*sqrt((39*(-3*x
+ 2)/(31*(2*x - 5)) + 1)/(2*(-3*x + 2)/(2*x - 5) + 1))*sqrt(4*x + 1)*sqrt(5*x +
7)*sqrt(2*(-3*x + 2)/(2*x - 5) + 1)) + sqrt(-3*x + 2)*sqrt(4*x + 1)*sqrt(5*x +
7)/(4*sqrt(2*x - 5))

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Mathematica [A]  time = 2.19492, size = 347, normalized size = 0.95 \[ -\frac{-1265 \sqrt{341} \sqrt{\frac{3 x-2}{4 x+1}} \sqrt{\frac{5 x+7}{4 x+1}} \left (8 x^2-18 x-5\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right )|\frac{39}{62}\right )+6820 \sqrt{341} \sqrt{\frac{3 x-2}{4 x+1}} \sqrt{\frac{5 x+7}{4 x+1}} \left (8 x^2-18 x-5\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right )|\frac{39}{62}\right )+\sqrt{\frac{2 x-5}{4 x+1}} \left (4117 \sqrt{341} \sqrt{\frac{3 x-2}{4 x+1}} \sqrt{\frac{10 x^2-11 x-35}{(4 x+1)^2}} (4 x+1)^2 \Pi \left (\frac{78}{55};\sin ^{-1}\left (\sqrt{\frac{22}{39}} \sqrt{\frac{5 x+7}{4 x+1}}\right )|\frac{39}{62}\right )+13640 \sqrt{2} \left (30 x^3-53 x^2-83 x+70\right )\right )}{27280 \sqrt{2-3 x} \sqrt{4 x-10} \sqrt{\frac{2 x-5}{4 x+1}} \sqrt{4 x+1} \sqrt{5 x+7}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(6820*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*Sqrt[(7 + 5*x)/(1 + 4*x)]*(-5 - 18*x
+ 8*x^2)*EllipticE[ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 39/62] - 1265
*Sqrt[341]*Sqrt[(-2 + 3*x)/(1 + 4*x)]*Sqrt[(7 + 5*x)/(1 + 4*x)]*(-5 - 18*x + 8*x
^2)*EllipticF[ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 39/62] + Sqrt[(-5 +
2*x)/(1 + 4*x)]*(13640*Sqrt[2]*(70 - 83*x - 53*x^2 + 30*x^3) + 4117*Sqrt[341]*S
qrt[(-2 + 3*x)/(1 + 4*x)]*(1 + 4*x)^2*Sqrt[(-35 - 11*x + 10*x^2)/(1 + 4*x)^2]*El
lipticPi[78/55, ArcSin[Sqrt[22/39]*Sqrt[(7 + 5*x)/(1 + 4*x)]], 39/62]))/(27280*S
qrt[2 - 3*x]*Sqrt[-10 + 4*x]*Sqrt[(-5 + 2*x)/(1 + 4*x)]*Sqrt[1 + 4*x]*Sqrt[7 + 5
*x])

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Maple [B]  time = 0.031, size = 929, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/34320*(7+5*x)^(1/2)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(20240*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))+65872*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*3
1^(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))+68640*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))+10120*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*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))+32936*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))+34320*1
1^(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))+1265*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))+
4117*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))+4290*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)*Ellip
ticE(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))+514800*x^3-909480*x^2-1424280*x+1201200)/(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. \[ \int \frac{\sqrt{5 \, x + 7} \sqrt{-3 \, x + 2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 3 x + 2} \sqrt{5 x + 7}}{\sqrt{2 x - 5} \sqrt{4 x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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