3.2912 \(\int \frac{(2+3 x)^{3/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{37 \sqrt{1-2 x} \sqrt{3 x+2}}{121 \sqrt{5 x+3}}+\frac{7 \sqrt{3 x+2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{2}{55} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{37}{55} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(7*Sqrt[2 + 3*x])/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (37*Sqrt[1 - 2*x]*Sqrt[2 +
3*x])/(121*Sqrt[3 + 5*x]) + (37*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/55 - (2*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3
3])/55

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Rubi [A]  time = 0.259615, 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 \[ -\frac{37 \sqrt{1-2 x} \sqrt{3 x+2}}{121 \sqrt{5 x+3}}+\frac{7 \sqrt{3 x+2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{2}{55} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{37}{55} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[2 + 3*x])/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (37*Sqrt[1 - 2*x]*Sqrt[2 +
3*x])/(121*Sqrt[3 + 5*x]) + (37*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/55 - (2*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3
3])/55

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Rubi in Sympy [A]  time = 24.7184, size = 114, normalized size = 0.88 \[ - \frac{37 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{121 \sqrt{5 x + 3}} + \frac{37 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{605} - \frac{6 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1925} + \frac{7 \sqrt{3 x + 2}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*sqrt(-2*x + 1)*sqrt(3*x + 2)/(121*sqrt(5*x + 3)) + 37*sqrt(33)*elliptic_e(as
in(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/605 - 6*sqrt(35)*elliptic_f(asin(sqrt(55)*
sqrt(-2*x + 1)/11), 33/35)/1925 + 7*sqrt(3*x + 2)/(11*sqrt(-2*x + 1)*sqrt(5*x +
3))

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Mathematica [A]  time = 0.205113, size = 122, normalized size = 0.95 \[ \frac{10 \sqrt{3 x+2} \sqrt{5 x+3} (37 x+20)+70 \sqrt{2-4 x} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-37 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{605 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(20 + 37*x) - 37*Sqrt[2 - 4*x]*(3 + 5*x)*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 70*Sqrt[2 - 4*x]*(3 + 5*x)*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(605*Sqrt[1 - 2*x]*(3 + 5*x))

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Maple [C]  time = 0.028, size = 159, normalized size = 1.2 \[ -{\frac{1}{18150\,{x}^{3}+13915\,{x}^{2}-4235\,x-3630}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -37\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1110\,{x}^{2}+1340\,x+400 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/605*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(70*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))-37*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))+1110*x^2
+1340*x+400)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3*x + 2)^(3/2)/((10*x^2 + x - 3)*sqrt(5*x + 3)*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((2+3*x)**(3/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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