3.2713 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{35} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{62}{875} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{487 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{13125}-\frac{2281 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65625}-\frac{46159 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65625} \]

[Out]

(-487*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/13125 + (62*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/875 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]
)/35 - (46159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6562
5 - (2281*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/65625

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Rubi [A]  time = 0.321382, 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 \[ \frac{2}{35} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{62}{875} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{487 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{13125}-\frac{2281 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65625}-\frac{46159 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65625} \]

Antiderivative was successfully verified.

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

[Out]

(-487*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/13125 + (62*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/875 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]
)/35 - (46159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6562
5 - (2281*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/65625

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Rubi in Sympy [A]  time = 31.7769, size = 143, normalized size = 0.89 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{35} + \frac{62 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{875} - \frac{487 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{13125} - \frac{46159 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{196875} - \frac{25091 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2296875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/35 + 62*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*sqrt(5*x + 3)/875 - 487*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/13
125 - 46159*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/196875 -
25091*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2296875

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Mathematica [A]  time = 0.296766, size = 97, normalized size = 0.61 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (-4500 x^2+2040 x+2873\right )-17045 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+92318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{196875 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(2873 + 2040*x - 4500*x^2) + 92318
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 17045*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2])/(196875*Sqrt[2])

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Maple [C]  time = 0.017, size = 174, normalized size = 1.1 \[{\frac{1}{11812500\,{x}^{3}+9056250\,{x}^{2}-2756250\,x-2362500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 17045\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -92318\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4050000\,{x}^{5}-1269000\,{x}^{4}+4938300\,{x}^{3}+2363970\,{x}^{2}-970530\,x-517140 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/393750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(17045*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))-92318*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))-
4050000*x^5-1269000*x^4+4938300*x^3+2363970*x^2-970530*x-517140)/(30*x^3+23*x^2-
7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (6 \, x^{2} + x - 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(6*x^2 + x - 2)*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)**(3/2)*(2+3*x)**(3/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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