3.2740 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{21} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{118}{525} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{4282 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{7875}+\frac{11806 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375}-\frac{86741 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375} \]

[Out]

(4282*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/7875 + (118*(1 - 2*x)^(3/2)*Sqr
t[2 + 3*x]*Sqrt[3 + 5*x])/525 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/
21 - (86741*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/39375
+ (11806*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/39375

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Rubi [A]  time = 0.328943, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{21} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{118}{525} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{4282 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{7875}+\frac{11806 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375}-\frac{86741 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{39375} \]

Antiderivative was successfully verified.

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

[Out]

(4282*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/7875 + (118*(1 - 2*x)^(3/2)*Sqr
t[2 + 3*x]*Sqrt[3 + 5*x])/525 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/
21 - (86741*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/39375
+ (11806*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/39375

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Rubi in Sympy [A]  time = 32.0948, size = 143, normalized size = 0.89 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{21} + \frac{118 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{525} + \frac{4282 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{7875} - \frac{86741 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{118125} + \frac{11806 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{118125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/21 + 118*(-2*x + 1)**(3/2)*sqrt(
3*x + 2)*sqrt(5*x + 3)/525 + 4282*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/787
5 - 86741*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/118125 + 1
1806*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/118125

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Mathematica [A]  time = 0.168412, size = 102, normalized size = 0.64 \[ \frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1500 x^2-3270 x+3401\right )-281540 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+86741 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{118125} \]

Antiderivative was successfully verified.

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(3401 - 3270*x + 1500*x^2) + 86741
*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 281540*Sqrt[2]*Ell
ipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/118125

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Maple [C]  time = 0.017, size = 174, normalized size = 1.1 \[{\frac{1}{3543750\,{x}^{3}+2716875\,{x}^{2}-826875\,x-708750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 281540\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -86741\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1350000\,{x}^{5}-1908000\,{x}^{4}+489600\,{x}^{3}+2763390\,{x}^{2}-125610\,x-612180 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/118125*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(281540*2^(1/2)*(3+5*x)^(1/2)
*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I
*11^(1/2)*3^(1/2)*2^(1/2))-86741*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/
2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))
+1350000*x^5-1908000*x^4+489600*x^3+2763390*x^2-125610*x-612180)/(30*x^3+23*x^2-
7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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