3.2751 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{3 \sqrt{3 x+2}}-\frac{32}{63} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{2108 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{1575}+\frac{124724 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{14175}+\frac{124724 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{70875}-\frac{481339 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{70875} \]

[Out]

(124724*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/14175 - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/(3*Sqrt[2 + 3*x]) - (2108*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(
3/2))/1575 - (32*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/63 - (481339*Sqr
t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/70875 + (124724*Sqrt[
11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/70875

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Rubi [A]  time = 0.396581, 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 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{3 \sqrt{3 x+2}}-\frac{32}{63} \sqrt{3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{2108 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{1575}+\frac{124724 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{14175}+\frac{124724 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{70875}-\frac{481339 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{70875} \]

Antiderivative was successfully verified.

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

[Out]

(124724*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/14175 - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/(3*Sqrt[2 + 3*x]) - (2108*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(
3/2))/1575 - (32*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/63 - (481339*Sqr
t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/70875 + (124724*Sqrt[
11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/70875

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Rubi in Sympy [A]  time = 39.6748, size = 172, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \sqrt{3 x + 2}} - \frac{32 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{63} + \frac{1054 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{315} + \frac{20378 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{14175} - \frac{481339 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{212625} + \frac{124724 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{212625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/(3*sqrt(3*x + 2)) - 32*(-2*x + 1)**(3/2)*s
qrt(3*x + 2)*(5*x + 3)**(3/2)/63 + 1054*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x
+ 3)/315 + 20378*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/14175 - 481339*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/212625 + 124724*sqrt(33)
*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/212625

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Mathematica [A]  time = 0.33506, size = 107, normalized size = 0.56 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (13500 x^3-21690 x^2+14727 x+32033\right )}{\sqrt{3 x+2}}-2539285 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+481339 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{212625} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32033 + 14727*x - 21690*x^2 + 13500*x^3))/Sqrt
[2 + 3*x] + 481339*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] -
2539285*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/212625

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Maple [C]  time = 0.024, size = 174, normalized size = 0.9 \[{\frac{1}{6378750\,{x}^{3}+4890375\,{x}^{2}-1488375\,x-1275750}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2539285\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -481339\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4050000\,{x}^{5}-6102000\,{x}^{4}+2552400\,{x}^{3}[phone]\,{x}^{2}-364440\,x-2882970 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/212625*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(2539285*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))-481339*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^5-6102000*x^4+2552400*x^3+12003810*x^2-364440*x-2882970)/(30*x^3+23
*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^(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)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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