3.2638 \(\int \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{35} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{31}{525} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2252 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4725}-\frac{2252 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625}-\frac{148831 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250} \]

[Out]

(-2252*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4725 - (31*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/525 + (2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/
35 - (148831*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47250
- (2252*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/23625

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Rubi [A]  time = 0.333773, 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 \[ \frac{2}{35} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{31}{525} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{2252 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4725}-\frac{2252 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{23625}-\frac{148831 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250} \]

Antiderivative was successfully verified.

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

[Out]

(-2252*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4725 - (31*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/525 + (2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/
35 - (148831*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47250
- (2252*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/23625

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Rubi in Sympy [A]  time = 31.3404, size = 143, normalized size = 0.89 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{21} - \frac{41 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{315} - \frac{2129 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{4725} - \frac{148831 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{141750} - \frac{2252 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{70875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/21 - 41*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*sqrt(5*x + 3)/315 - 2129*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/4
725 - 148831*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/141750
- 2252*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/70875

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Mathematica [A]  time = 0.293005, size = 97, normalized size = 0.61 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (6750 x^2+6705 x-659\right )-74515 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+148831 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{70875 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-659 + 6705*x + 6750*x^2) + 14883
1*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 74515*EllipticF[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2])/(70875*Sqrt[2])

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Maple [C]  time = 0.015, size = 174, normalized size = 1.1 \[{\frac{1}{4252500\,{x}^{3}+3260250\,{x}^{2}-992250\,x-850500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 74515\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -148831\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +6075000\,{x}^{5}[phone]\,{x}^{4}+2615850\,{x}^{3}-3077760\,{x}^{2}-1068510\,x+118620 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/141750*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(74515*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))-148831*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))
+6075000*x^5+10692000*x^4+2615850*x^3-3077760*x^2-1068510*x+118620)/(30*x^3+23*x
^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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