3.2885 \(\int \frac{\sqrt{2+3 x} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{\sqrt{3 x+2} (5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{10}{3} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{2}{3} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{133}{6} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3 + (Sqrt[2 + 3*x]*(3 + 5*x)^(3/2
))/Sqrt[1 - 2*x] + (133*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/6 + (2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3

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Rubi [A]  time = 0.246633, antiderivative size = 126, 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 \[ \frac{\sqrt{3 x+2} (5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{10}{3} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{2}{3} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{133}{6} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3 + (Sqrt[2 + 3*x]*(3 + 5*x)^(3/2
))/Sqrt[1 - 2*x] + (133*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/6 + (2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3

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Rubi in Sympy [A]  time = 24.5207, size = 110, normalized size = 0.87 \[ \frac{10 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{3} + \frac{133 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{18} + \frac{22 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{105} + \frac{\sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/3 + 133*sqrt(33)*elliptic_e(asin(s
qrt(21)*sqrt(-2*x + 1)/7), 35/33)/18 + 22*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt
(-2*x + 1)/11), 33/35)/105 + sqrt(3*x + 2)*(5*x + 3)**(3/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.185446, size = 105, normalized size = 0.83 \[ \frac{6 \sqrt{3 x+2} \sqrt{5 x+3} (19-5 x)+67 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-133 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{18 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(6*(19 - 5*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - 133*Sqrt[2 - 4*x]*EllipticE[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 67*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]], -33/2])/(18*Sqrt[1 - 2*x])

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Maple [C]  time = 0.023, size = 164, normalized size = 1.3 \[ -{\frac{1}{540\,{x}^{3}+414\,{x}^{2}-126\,x-108}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 67\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -133\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -450\,{x}^{3}+1140\,{x}^{2}+1986\,x+684 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(67*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))-133*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icE(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))-450*x^3+
1140*x^2+1986*x+684)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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