3.2771 \(\int \frac{(1-2 x)^{5/2} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{35} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}}{2625}+\frac{30922 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}-\frac{132824 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625}-\frac{408311 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]

[Out]

(30922*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/118125 + (326*(1 - 2*x)^(3/2)*
Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2625 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/35 - (408311*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5
90625 - (132824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/59
0625

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Rubi [A]  time = 0.335705, 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 \[ \frac{2}{35} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}}{2625}+\frac{30922 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}-\frac{132824 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625}-\frac{408311 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]

Antiderivative was successfully verified.

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

[Out]

(30922*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/118125 + (326*(1 - 2*x)^(3/2)*
Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2625 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/35 - (408311*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5
90625 - (132824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/59
0625

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Rubi in Sympy [A]  time = 33.8684, size = 143, normalized size = 0.89 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{35} + \frac{326 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{2625} + \frac{30922 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{118125} - \frac{408311 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1771875} - \frac{132824 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1771875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/35 + 326*(-2*x + 1)**(3/2)*sqrt(
3*x + 2)*sqrt(5*x + 3)/2625 + 30922*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1
18125 - 408311*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/17718
75 - 132824*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1771875

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Mathematica [A]  time = 0.254527, size = 102, normalized size = 0.64 \[ \frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (13500 x^2-28170 x+26171\right )+1783285 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+408311 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1771875} \]

Antiderivative was successfully verified.

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(26171 - 28170*x + 13500*x^2) + 40
8311*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1783285*Sqrt[2
]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/1771875

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Maple [C]  time = 0.02, size = 174, normalized size = 1.1 \[ -{\frac{1}{53156250\,{x}^{3}[phone]\,{x}^{2}-12403125\,x-10631250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1783285\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +408311\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -12150000\,{x}^{5}[phone]\,{x}^{4}-1281600\,{x}^{3}-21543690\,{x}^{2}+425310\,x+4710780 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1771875*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1783285*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))+408311*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))-12150000*x^5+16038000*x^4-1281600*x^3-21543690*x^2+425310*x+4710780)/(30*x^
3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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