3.2721 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{32}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{5753 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}-\frac{47342 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (2719*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x]*Sqrt[3 + 5*x])/21875 + (2818*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5
*x])/4375 - (32*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 - (47342*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375 - (5753*Sqrt[11/3
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375

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Rubi [A]  time = 0.400805, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{32}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{5753 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}-\frac{47342 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (2719*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x]*Sqrt[3 + 5*x])/21875 + (2818*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5
*x])/4375 - (32*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 - (47342*Sqrt[1
1/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375 - (5753*Sqrt[11/3
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375

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Rubi in Sympy [A]  time = 39.6784, size = 172, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}}}{5 \sqrt{5 x + 3}} - \frac{32 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{175} + \frac{2818 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4375} + \frac{2719 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{21875} - \frac{47342 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{328125} - \frac{63283 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{3828125} \]

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)**(3/2),x)

[Out]

-2*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)/(5*sqrt(5*x + 3)) - 32*sqrt(-2*x + 1)*(3*x
+ 2)**(5/2)*sqrt(5*x + 3)/175 + 2818*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x +
3)/4375 + 2719*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/21875 - 47342*sqrt(33
)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/328125 - 63283*sqrt(35)*ell
iptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/3828125

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Mathematica [A]  time = 0.371305, size = 107, normalized size = 0.56 \[ \frac{-\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (22500 x^3+5400 x^2-22305 x-9697\right )}{\sqrt{5 x+3}}+95165 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+94684 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{656250} \]

Antiderivative was successfully verified.

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

[Out]

((-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-9697 - 22305*x + 5400*x^2 + 22500*x^3))/Sqrt
[3 + 5*x] + 94684*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 9
5165*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/656250

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Maple [C]  time = 0.026, size = 174, normalized size = 0.9 \[ -{\frac{1}{19687500\,{x}^{3}[phone]\,{x}^{2}-4593750\,x-3937500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 95165\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +94684\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4050000\,{x}^{5}+1647000\,{x}^{4}-5202900\,{x}^{3}-2738610\,{x}^{2}+1047390\,x+581820 \right ) } \]

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)^(3/2),x)

[Out]

-1/656250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(95165*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))+94684*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))
+4050000*x^5+1647000*x^4-5202900*x^3-2738610*x^2+1047390*x+581820)/(30*x^3+23*x^
2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)^(3/2),x, algorithm="fricas")

[Out]

integral(-(18*x^3 + 15*x^2 - 4*x - 4)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(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)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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