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

Optimal. Leaf size=205 \[ \frac{1}{7} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2+\frac{173}{60} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)+\frac{73207 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1080}-\frac{1679161 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{756 \sqrt{2 x-5}}+\frac{8198333 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{9072 \sqrt{5-2 x}} \]

[Out]

(73207*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/1080 + (173*Sqrt[2 - 3*x]*Sqr
t[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/60 + (Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 +
4*x]*(7 + 5*x)^2)/7 + (8198333*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt
[2 - 3*x])/Sqrt[11]], -1/2])/(9072*Sqrt[5 - 2*x]) - (1679161*Sqrt[11/6]*Sqrt[5 -
2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(756*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.589621, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ \frac{1}{7} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^2+\frac{173}{60} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)+\frac{73207 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1080}-\frac{1679161 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{756 \sqrt{2 x-5}}+\frac{8198333 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{9072 \sqrt{5-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(73207*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/1080 + (173*Sqrt[2 - 3*x]*Sqr
t[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/60 + (Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 +
4*x]*(7 + 5*x)^2)/7 + (8198333*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt
[2 - 3*x])/Sqrt[11]], -1/2])/(9072*Sqrt[5 - 2*x]) - (1679161*Sqrt[11/6]*Sqrt[5 -
2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(756*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 75.9344, size = 226, normalized size = 1.1 \[ - \frac{25 \left (- 3 x + 2\right )^{\frac{3}{2}} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{84} + \frac{136 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{21} + \frac{134699 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1}}{1512} + \frac{8198333 \sqrt{11} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{2 \sqrt{11} \sqrt{- 3 x + 2}}{11} \right )}\middle | - \frac{1}{2}\right )}{9072 \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{4 x + 1}} - \frac{18470771 \sqrt{11} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{11} \sqrt{4 x + 1}}{11} \right )}\middle | 3\right )}{3024 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-3*x + 2)**(3/2)*sqrt(2*x - 5)*(4*x + 1)**(3/2)/84 + 136*sqrt(-3*x + 2)*sqr
t(2*x - 5)*(4*x + 1)**(3/2)/21 + 134699*sqrt(-3*x + 2)*sqrt(2*x - 5)*sqrt(4*x +
1)/1512 + 8198333*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)/(9072*sqrt(-6*x/11 + 15/11)*sqrt(4*x + 1)) -
18470771*sqrt(11)*sqrt(-12*x/11 + 8/11)*sqrt(-4*x/11 + 10/11)*elliptic_f(asin(sq
rt(11)*sqrt(4*x + 1)/11), 3)/(3024*sqrt(-3*x + 2)*sqrt(2*x - 5))

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Mathematica [A]  time = 0.358238, size = 125, normalized size = 0.61 \[ \frac{12 \sqrt{2-3 x} \sqrt{4 x+1} \left (10800 x^3+46836 x^2+102592 x-717955\right )-6716644 \sqrt{66} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )+8198333 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{18144 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(12*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-717955 + 102592*x + 46836*x^2 + 10800*x^3) + 8
198333*Sqrt[66]*Sqrt[5 - 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] -
6716644*Sqrt[66]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3]
)/(18144*Sqrt[-5 + 2*x])

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Maple [A]  time = 0.018, size = 156, normalized size = 0.8 \[ -{\frac{1}{217728\,{x}^{3}-635040\,{x}^{2}+190512\,x+90720}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 10074966\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -8198333\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -777600\,{x}^{5}-3048192\,{x}^{4}-5851944\,{x}^{3}[phone]\,{x}^{2}-20307546\,x-8615460 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9072*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(10074966*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))-8198333*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))-777600*x^5-3048192*x^4-5851944*x^3+55332552
*x^2-20307546*x-8615460)/(24*x^3-70*x^2+21*x+10)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 70*x + 49)*sqrt(4*x + 1)*sqrt(-3*x + 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((7+5*x)**2*(2-3*x)**(1/2)*(1+4*x)**(1/2)/(-5+2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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