3.2772 \(\int \frac{(1-2 x)^{5/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{4}{75} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1088 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3375}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875} \]

[Out]

(-1088*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3375 - (4*(1 - 2*x)^(3/2)*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x])/75 + (53194*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/16875 - (34154*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/16875

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Rubi [A]  time = 0.260521, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{4}{75} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1088 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3375}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16875} \]

Antiderivative was successfully verified.

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

[Out]

(-1088*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3375 - (4*(1 - 2*x)^(3/2)*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x])/75 + (53194*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/16875 - (34154*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/16875

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Rubi in Sympy [A]  time = 25.854, size = 114, normalized size = 0.88 \[ - \frac{4 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{75} - \frac{1088 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{3375} + \frac{53194 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{50625} - \frac{34154 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{50625} \]

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]

-4*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/75 - 1088*sqrt(-2*x + 1)*sqrt(3
*x + 2)*sqrt(5*x + 3)/3375 + 53194*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x +
1)/7), 35/33)/50625 - 34154*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7)
, 35/33)/50625

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Mathematica [A]  time = 0.20934, size = 97, normalized size = 0.75 \[ \frac{60 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (90 x-317)+616735 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-53194 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{50625} \]

Antiderivative was successfully verified.

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

[Out]

(60*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-317 + 90*x) - 53194*Sqrt[2]*Elli
pticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 616735*Sqrt[2]*EllipticF[ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/50625

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Maple [C]  time = 0.02, size = 169, normalized size = 1.3 \[ -{\frac{1}{1518750\,{x}^{3}+1164375\,{x}^{2}-354375\,x-303750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 616735\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -53194\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -162000\,{x}^{4}+446400\,{x}^{3}+475260\,{x}^{2}-100740\,x-114120 \right ) } \]

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]

-1/50625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(616735*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))-53194*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))
-162000*x^4+446400*x^3+475260*x^2-100740*x-114120)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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