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

Optimal. Leaf size=162 \[ \frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} (4 x+1)^{3/2}+\frac{95}{18} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{4543 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{36 \sqrt{2 x-5}}+\frac{1397 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{27 \sqrt{5-2 x}} \]

[Out]

(95*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/18 + (Sqrt[2 - 3*x]*Sqrt[-5 + 2*
x]*(1 + 4*x)^(3/2))/4 + (1397*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2
- 3*x])/Sqrt[11]], -1/2])/(27*Sqrt[5 - 2*x]) - (4543*Sqrt[11/6]*Sqrt[5 - 2*x]*E
llipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(36*Sqrt[-5 + 2*x])

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Rubi [A]  time = 0.398209, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} (4 x+1)^{3/2}+\frac{95}{18} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{4543 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{36 \sqrt{2 x-5}}+\frac{1397 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{27 \sqrt{5-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(95*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/18 + (Sqrt[2 - 3*x]*Sqrt[-5 + 2*
x]*(1 + 4*x)^(3/2))/4 + (1397*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2
- 3*x])/Sqrt[11]], -1/2])/(27*Sqrt[5 - 2*x]) - (4543*Sqrt[11/6]*Sqrt[5 - 2*x]*E
llipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(36*Sqrt[-5 + 2*x])

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Rubi in Sympy [A]  time = 37.3826, size = 196, normalized size = 1.21 \[ \frac{\sqrt{- 3 x + 2} \sqrt{2 x - 5} \left (4 x + 1\right )^{\frac{3}{2}}}{4} + \frac{95 \sqrt{- 3 x + 2} \sqrt{2 x - 5} \sqrt{4 x + 1}}{18} + \frac{1397 \sqrt{11} \sqrt{\frac{12 x}{11} + \frac{3}{11}} \sqrt{2 x - 5} E\left (\operatorname{asin}{\left (\frac{2 \sqrt{11} \sqrt{- 3 x + 2}}{11} \right )}\middle | - \frac{1}{2}\right )}{27 \sqrt{- \frac{6 x}{11} + \frac{15}{11}} \sqrt{4 x + 1}} - \frac{49973 \sqrt{11} \sqrt{- \frac{12 x}{11} + \frac{8}{11}} \sqrt{- \frac{4 x}{11} + \frac{10}{11}} F\left (\operatorname{asin}{\left (\frac{\sqrt{11} \sqrt{4 x + 1}}{11} \right )}\middle | 3\right )}{144 \sqrt{- 3 x + 2} \sqrt{2 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-3*x + 2)*sqrt(2*x - 5)*(4*x + 1)**(3/2)/4 + 95*sqrt(-3*x + 2)*sqrt(2*x - 5
)*sqrt(4*x + 1)/18 + 1397*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)/(27*sqrt(-6*x/11 + 15/11)*sqrt(4*x +
1)) - 49973*sqrt(11)*sqrt(-12*x/11 + 8/11)*sqrt(-4*x/11 + 10/11)*elliptic_f(asin
(sqrt(11)*sqrt(4*x + 1)/11), 3)/(144*sqrt(-3*x + 2)*sqrt(2*x - 5))

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Mathematica [A]  time = 0.270931, size = 120, normalized size = 0.74 \[ \frac{6 \sqrt{2-3 x} \sqrt{4 x+1} \left (72 x^2+218 x-995\right )-4543 \sqrt{66} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )+5588 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{216 \sqrt{2 x-5}} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-995 + 218*x + 72*x^2) + 5588*Sqrt[66]*Sqrt[5 -
2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] - 4543*Sqrt[66]*Sqrt[5 - 2
*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(216*Sqrt[-5 + 2*x])

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Maple [A]  time = 0.016, size = 151, normalized size = 0.9 \[ -{\frac{1}{5184\,{x}^{3}-15120\,{x}^{2}+4536\,x+2160}\sqrt{2-3\,x}\sqrt{-5+2\,x}\sqrt{1+4\,x} \left ( 13629\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticF} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -11176\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{1+4\,x}{\it EllipticE} \left ( 2/11\,\sqrt{2-3\,x}\sqrt{11},i/2\sqrt{2} \right ) -5184\,{x}^{4}-13536\,{x}^{3}+79044\,{x}^{2}-27234\,x-11940 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/216*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(13629*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))
-11176*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))-5184*x^4-13536*x^3+79044*x^2-27234*x-11940)/(24*x
^3-70*x^2+21*x+10)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((5*x + 7)*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)*(2-3*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. \[ \int \frac{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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