3.2723 \(\int \frac{(1-2 x)^{3/2} \sqrt{2+3 x}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{16}{75} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{178}{375} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{458}{375} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (16*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/75 + (458*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/375 - (178*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/375

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Rubi [A]  time = 0.256816, 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 \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{16}{75} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{178}{375} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{458}{375} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (16*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/75 + (458*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/375 - (178*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/375

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Rubi in Sympy [A]  time = 24.9127, size = 114, normalized size = 0.88 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}}{5 \sqrt{5 x + 3}} - \frac{16 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{75} + \frac{458 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1125} - \frac{178 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1125} \]

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)**(3/2),x)

[Out]

-2*(-2*x + 1)**(3/2)*sqrt(3*x + 2)/(5*sqrt(5*x + 3)) - 16*sqrt(-2*x + 1)*sqrt(3*
x + 2)*sqrt(5*x + 3)/75 + 458*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7
), 35/33)/1125 - 178*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)
/1125

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Mathematica [A]  time = 0.261138, size = 97, normalized size = 0.75 \[ \frac{-\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} (10 x+39)}{\sqrt{5 x+3}}+3395 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-458 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1125} \]

Antiderivative was successfully verified.

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

[Out]

((-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(39 + 10*x))/Sqrt[3 + 5*x] - 458*Sqrt[2]*Ellip
ticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3395*Sqrt[2]*EllipticF[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2])/1125

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Maple [C]  time = 0.024, size = 164, normalized size = 1.3 \[ -{\frac{1}{33750\,{x}^{3}+25875\,{x}^{2}-7875\,x-6750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 3395\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -458\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1800\,{x}^{3}+7320\,{x}^{2}+570\,x-2340 \right ) } \]

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)^(3/2),x)

[Out]

-1/1125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(3395*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))-458*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))+1800
*x^3+7320*x^2+570*x-2340)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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