3.2791 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{178 (3 x+2)^{3/2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{572}{625} (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{8874 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3125}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2))/(15*(3 + 5*x)^(3/2)) - (178*(1 - 2*x)^(3/2)
*(2 + 3*x)^(3/2))/(75*Sqrt[3 + 5*x]) + (8874*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/3125 - (572*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/625 + (9206*Sqr
t[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625 - (7738*Sqrt[11/3
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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Rubi [A]  time = 0.400532, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{178 (3 x+2)^{3/2} (1-2 x)^{3/2}}{75 \sqrt{5 x+3}}-\frac{572}{625} (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{8874 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{3125}-\frac{7738 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}+\frac{9206 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2))/(15*(3 + 5*x)^(3/2)) - (178*(1 - 2*x)^(3/2)
*(2 + 3*x)^(3/2))/(75*Sqrt[3 + 5*x]) + (8874*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/3125 - (572*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/625 + (9206*Sqr
t[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625 - (7738*Sqrt[11/3
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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Rubi in Sympy [A]  time = 40.5084, size = 172, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{178 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2}}{825 \sqrt{5 x + 3}} - \frac{2836 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{20625} - \frac{1136 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{3125} + \frac{9206 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15625} - \frac{7738 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{46875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**(3/2)/(15*(5*x + 3)**(3/2)) - 178*(-2*x + 1)**(5
/2)*sqrt(3*x + 2)/(825*sqrt(5*x + 3)) - 2836*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqr
t(5*x + 3)/20625 - 1136*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/3125 + 9206*s
qrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15625 - 7738*sqrt(33)
*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/46875

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Mathematica [A]  time = 0.410056, size = 107, normalized size = 0.57 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (4500 x^3-9450 x^2-48650 x-25421\right )}{(5 x+3)^{3/2}}+155295 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-27618 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{46875} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-25421 - 48650*x - 9450*x^2 + 4500*x^3))/(3 +
5*x)^(3/2) - 27618*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] +
155295*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/46875

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Maple [C]  time = 0.028, size = 277, normalized size = 1.5 \[ -{\frac{1}{281250\,{x}^{2}+46875\,x-93750} \left ( 776475\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-138090\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+465885\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -82854\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -270000\,{x}^{5}+522000\,{x}^{4}+3103500\,{x}^{3}+1822760\,{x}^{2}-718790\,x-508420 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/46875*(776475*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-138090*2^(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))*
x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+465885*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))-82854*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticE(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))-270000*
x^5+522000*x^4+3103500*x^3+1822760*x^2-718790*x-508420)*(2+3*x)^(1/2)*(1-2*x)^(1
/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((12*x^3 - 4*x^2 - 5*x + 2)*sqrt(3*x + 2)*sqrt(-2*x + 1)/((25*x^2 + 30*x
+ 9)*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)**(5/2)*(2+3*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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