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

Optimal. Leaf size=391 \[ -\frac{1}{9} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} (2-3 x)^{3/2}+\frac{23}{240} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} \sqrt{2-3 x}-\frac{13027 \sqrt{4 x+1} \sqrt{5 x+7} \sqrt{2-3 x}}{4800 \sqrt{2 x-5}}-\frac{1368371 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{43200 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{13027 \sqrt{\frac{143}{3}} \sqrt{\frac{5 x+7}{5-2 x}} \sqrt{2-3 x} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{3200 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{65750101 \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} (2-3 x) \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{216000 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

[Out]

(-13027*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(4800*Sqrt[-5 + 2*x]) + (23*S
qrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/240 - ((2 - 3*x)^(3/2)*
Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/9 + (13027*Sqrt[143/3]*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])/(3200*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (1368371*Sqrt
[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -
39/23])/(43200*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) - (65750101*(2 - 3*x)*S
qrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[
(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(216000*Sqrt[429]*Sqrt[-5 +
2*x]*Sqrt[1 + 4*x])

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Rubi [A]  time = 1.25774, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.297 \[ -\frac{1}{9} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} (2-3 x)^{3/2}+\frac{23}{240} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} \sqrt{2-3 x}-\frac{13027 \sqrt{4 x+1} \sqrt{5 x+7} \sqrt{2-3 x}}{4800 \sqrt{2 x-5}}-\frac{1368371 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{43200 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{13027 \sqrt{\frac{143}{3}} \sqrt{\frac{5 x+7}{5-2 x}} \sqrt{2-3 x} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{3200 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{65750101 \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} (2-3 x) \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{216000 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-13027*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(4800*Sqrt[-5 + 2*x]) + (23*S
qrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/240 - ((2 - 3*x)^(3/2)*
Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/9 + (13027*Sqrt[143/3]*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])/(3200*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (1368371*Sqrt
[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -
39/23])/(43200*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) - (65750101*(2 - 3*x)*S
qrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[
(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(216000*Sqrt[429]*Sqrt[-5 +
2*x]*Sqrt[1 + 4*x])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 3.63086, size = 340, normalized size = 0.87 \[ -\frac{\sqrt{2 x-5} \sqrt{4 x+1} \left (-4532324 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+7269066 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+\sqrt{\frac{5 x+7}{3 x-2}} \left (186 \left (1152000 x^5-2184000 x^4-7723240 x^3+7278862 x^2+17658613 x+3848705\right )-2120971 \sqrt{682} (2-3 x)^2 \sqrt{\frac{4 x+1}{3 x-2}} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )\right )}{5356800 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7269066*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] - 4532324*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
] + Sqrt[(7 + 5*x)/(-2 + 3*x)]*(186*(3848705 + 17658613*x + 7278862*x^2 - 772324
0*x^3 - 2184000*x^4 + 1152000*x^5) - 2120971*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])))/(5356800*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 = 0.026, size = 939, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/123552000*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)*(454813040*
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))-780517328*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*Ellip
ticPi(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))-2682519840*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)*1
1^(1/2)*((7+5*x)/(1+4*x))^(1/2),1/39*2^(1/2)*3^(1/2)*31^(1/2)*13^(1/2))[phone]
0*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))-390258664*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*Ellipti
cPi(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)*3
1^(1/2)*13^(1/2))-1341259920*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))[phone]*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))-48782333*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*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))-167657490*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))[phone]*x^5-9369360000
*x^4-33132699600*x^3+49668641880*x^2+55468893480*x-48037189200)/(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. \[ \int \sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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