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

Optimal. Leaf size=189 \[ -\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{5 (5 x+7)}+\frac{152 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{125 \sqrt{2 x-5}}+\frac{6 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{25 \sqrt{5-2 x}}+\frac{26859 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{7750 \sqrt{11} \sqrt{2 x-5}} \]

[Out]

-(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(5*(7 + 5*x)) + (6*Sqrt[11]*Sqrt[-
5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(25*Sqrt[5 - 2*x])
+ (152*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3
])/(125*Sqrt[-5 + 2*x]) + (26859*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt
[2 - 3*x])/Sqrt[11]], -1/2])/(7750*Sqrt[11]*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.797165, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{\sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{5 (5 x+7)}+\frac{152 \sqrt{\frac{2}{33}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{125 \sqrt{2 x-5}}+\frac{6 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{25 \sqrt{5-2 x}}+\frac{26859 \sqrt{5-2 x} \Pi \left (\frac{55}{124};\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{7750 \sqrt{11} \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(5*(7 + 5*x)) + (6*Sqrt[11]*Sqrt[-
5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(25*Sqrt[5 - 2*x])
+ (152*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3
])/(125*Sqrt[-5 + 2*x]) + (26859*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt
[2 - 3*x])/Sqrt[11]], -1/2])/(7750*Sqrt[11]*Sqrt[-5 + 2*x])

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

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.751006, size = 132, normalized size = 0.7 \[ \frac{\sqrt{2 x-5} \left (\frac{3 \sqrt{11} \left (9424 F\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )+20460 E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )+26859 \Pi \left (\frac{55}{124};-\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )\right )}{\sqrt{5-2 x}}-\frac{51150 \sqrt{2-3 x} \sqrt{4 x+1}}{5 x+7}\right )}{255750} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-5 + 2*x]*((-51150*Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(7 + 5*x) + (3*Sqrt[11]*(2
0460*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 9424*EllipticF[ArcSin
[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 26859*EllipticPi[55/124, -ArcSin[(2*Sqrt[2
- 3*x])/Sqrt[11]], -1/2]))/Sqrt[5 - 2*x]))/255750

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Maple [B]  time = 0.038, size = 338, normalized size = 1.8 \[{\frac{1}{ \left ( 2046000\,{x}^{3}-5967500\,{x}^{2}+1790250\,x+852500 \right ) \left ( 7+5\,x \right ) }\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 47120\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) x+102300\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) x-134295\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ) x+65968\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) +143220\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -188013\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticPi} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},{\frac{55}{124}},i/2\sqrt{2} \right ) -409200\,{x}^{3}+1193500\,{x}^{2}-358050\,x-170500 \right ) } \]

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)^2,x)

[Out]

1/85250*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(47120*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)
)*x+102300*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))*x-134295*11^(1/2)*(2-3*x)^(1/2)*(5-2*x)^(1/2)
*(1+4*x)^(1/2)*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/124,1/2*I*2^(1/2))*x+65
968*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))+143220*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))-188013*11^(1/2)*(2-3*
x)^(1/2)*(5-2*x)^(1/2)*(1+4*x)^(1/2)*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/1
24,1/2*I*2^(1/2))-409200*x^3+1193500*x^2-358050*x-170500)/(24*x^3-70*x^2+21*x+10
)/(7+5*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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