3.2730 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{106 \sqrt{1-2 x} (3 x+2)^{5/2}}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{1558}{625} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3125}-\frac{8366 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}+\frac{1973 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (106*Sqrt[1 - 2*x]*(
2 + 3*x)^(5/2))/(25*Sqrt[3 + 5*x]) + (2264*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
5*x])/3125 + (1558*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/625 + (1973*Sqrt
[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625 - (8366*Sqrt[3/1
1]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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Rubi [A]  time = 0.412031, antiderivative size = 191, 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 \[ -\frac{106 \sqrt{1-2 x} (3 x+2)^{5/2}}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{1558}{625} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3125}-\frac{8366 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}+\frac{1973 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (106*Sqrt[1 - 2*x]*(
2 + 3*x)^(5/2))/(25*Sqrt[3 + 5*x]) + (2264*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 +
5*x])/3125 + (1558*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/625 + (1973*Sqrt
[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625 - (8366*Sqrt[3/1
1]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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Rubi in Sympy [A]  time = 39.1957, size = 172, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{106 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}}}{275 \sqrt{5 x + 3}} - \frac{1412 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6875} + \frac{2264 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{3125} + \frac{1973 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{46875} - \frac{8366 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{171875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)/(15*(5*x + 3)**(3/2)) - 106*(-2*x + 1)**(3
/2)*(3*x + 2)**(3/2)/(275*sqrt(5*x + 3)) - 1412*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*
sqrt(5*x + 3)/6875 + 2264*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/3125 + 1973
*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/46875 - 8366*sqrt(3
3)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/171875

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Mathematica [A]  time = 0.466886, size = 107, normalized size = 0.56 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (-6750 x^3+1050 x^2+2975 x-106\right )}{(5 x+3)^{3/2}}+39620 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1973 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{46875} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-106 + 2975*x + 1050*x^2 - 6750*x^3))/(3 + 5*x
)^(3/2) - 1973*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3962
0*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/46875

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Maple [C]  time = 0.03, size = 277, normalized size = 1.5 \[ -{\frac{1}{281250\,{x}^{2}+46875\,x-93750} \left ( 198100\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9865\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+118860\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5919\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +405000\,{x}^{5}+4500\,{x}^{4}-324000\,{x}^{3}-2390\,{x}^{2}+60560\,x-2120 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/46875*(198100*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)-9865*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)+118860*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))-5919*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elliptic
E(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))+405000*x^5
+4500*x^4-324000*x^3-2390*x^2+60560*x-2120)*(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(18*x^3 + 15*x^2 - 4*x - 4)*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)**(3/2)*(2+3*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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