3.2684 \(\int (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{35} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2625}-\frac{2657 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{23625}-\frac{2657 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125}-\frac{118898 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125} \]

[Out]

(-2657*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/23625 + (194*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x]*(3 + 5*x)^(3/2))/2625 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3
/2))/35 - (118898*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/
118125 - (2657*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/118
125

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Rubi [A]  time = 0.326339, 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 \[ \frac{2}{35} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{194 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2625}-\frac{2657 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{23625}-\frac{2657 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125}-\frac{118898 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{118125} \]

Antiderivative was successfully verified.

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

[Out]

(-2657*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/23625 + (194*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x]*(3 + 5*x)^(3/2))/2625 + (2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3
/2))/35 - (118898*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/
118125 - (2657*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/118
125

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Rubi in Sympy [A]  time = 31.5468, size = 143, normalized size = 0.89 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{21} - \frac{107 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{525} + \frac{6946 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{23625} - \frac{118898 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{354375} - \frac{2657 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{354375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/21 - 107*(-2*x + 1)**(3/2)*sq
rt(3*x + 2)*sqrt(5*x + 3)/525 + 6946*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/
23625 - 118898*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/35437
5 - 2657*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/354375

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Mathematica [A]  time = 0.294373, size = 97, normalized size = 0.61 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (-13500 x^2+7380 x+6631\right )-150115 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+237796 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{354375 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(6631 + 7380*x - 13500*x^2) + 2377
96*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 150115*EllipticF[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(354375*Sqrt[2])

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Maple [C]  time = 0.016, size = 174, normalized size = 1.1 \[{\frac{1}{21262500\,{x}^{3}[phone]\,{x}^{2}-4961250\,x-4252500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 150115\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -237796\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -12150000\,{x}^{5}-2673000\,{x}^{4}[phone]\,{x}^{3}+5455590\,{x}^{2}-2720910\,x-1193580 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/708750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(150115*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))-237796*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^5-2673000*x^4+13895100*x^3+5455590*x^2-2720910*x-1193580)/(30*x^3+2
3*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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