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

Optimal. Leaf size=165 \[ -\frac{5}{12} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)-\frac{2135}{108} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}+\frac{2474201 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{216 \sqrt{66} \sqrt{2 x-5}}-\frac{487585 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{1296 \sqrt{5-2 x}} \]

[Out]

(-2135*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/108 - (5*Sqrt[2 - 3*x]*Sqrt[-
5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/12 - (487585*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE
[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(1296*Sqrt[5 - 2*x]) + (2474201*Sqrt
[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(216*Sqrt[66]*Sqrt[-
5 + 2*x])

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Rubi [A]  time = 0.489365, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5}{12} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)-\frac{2135}{108} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}+\frac{2474201 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{216 \sqrt{66} \sqrt{2 x-5}}-\frac{487585 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{1296 \sqrt{5-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2135*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/108 - (5*Sqrt[2 - 3*x]*Sqrt[-
5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/12 - (487585*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE
[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(1296*Sqrt[5 - 2*x]) + (2474201*Sqrt
[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(216*Sqrt[66]*Sqrt[-
5 + 2*x])

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Rubi in Sympy [A]  time = 96.7824, size = 197, normalized size = 1.19 \[ - \frac{25 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{48} - \frac{9575 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1}}{432} - \frac{487585 \sqrt{11} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{2 \sqrt{11} \sqrt{- 3 x + 2}}{11} \right )}\middle | - \frac{1}{2}\right )}{1296 \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{4 x + 1}} + \frac{2474201 \sqrt{11} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{11} \sqrt{4 x + 1}}{11} \right )}\middle | 3\right )}{864 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*sqrt(-3*x + 2)*sqrt(2*x - 5)*(4*x + 1)**(3/2)/48 - 9575*sqrt(-3*x + 2)*sqrt(
2*x - 5)*sqrt(4*x + 1)/432 - 487585*sqrt(11)*sqrt(12*x/11 + 3/11)*sqrt(2*x - 5)*
elliptic_e(asin(2*sqrt(11)*sqrt(-3*x + 2)/11), -1/2)/(1296*sqrt(-6*x/11 + 15/11)
*sqrt(4*x + 1)) + 2474201*sqrt(11)*sqrt(-12*x/11 + 8/11)*sqrt(-4*x/11 + 10/11)*e
lliptic_f(asin(sqrt(11)*sqrt(4*x + 1)/11), 3)/(864*sqrt(-3*x + 2)*sqrt(2*x - 5))

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Mathematica [A]  time = 0.322549, size = 120, normalized size = 0.73 \[ \frac{-6600 \sqrt{2-3 x} \sqrt{4 x+1} \left (18 x^2+151 x-490\right )+4948402 \sqrt{66} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )-5363435 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{28512 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(-6600*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-490 + 151*x + 18*x^2) - 5363435*Sqrt[66]*Sq
rt[5 - 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] + 4948402*Sqrt[66]*
Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(28512*Sqrt[-5 +
2*x])

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Maple [A]  time = 0.024, size = 151, normalized size = 0.9 \[{\frac{1}{342144\,{x}^{3}-997920\,{x}^{2}+299376\,x+142560}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 7422603\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -5363435\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -712800\,{x}^{4}-5682600\,{x}^{3}[phone]\,{x}^{2}-7088400\,x-3234000 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14256*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(7422603*11^(1/2)*(2-3*x)^(1/
2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticF(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/
2))-5363435*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticE(2/11*(2
-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))-712800*x^4-5682600*x^3+22014300*x^2-7088400*
x-3234000)/(24*x^3-70*x^2+21*x+10)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((5*x + 7)^3/(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. \[{\rm integral}\left (\frac{125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((125*x^3 + 525*x^2 + 735*x + 343)/(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*
x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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