3.2649 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{4}{3} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{3}{5} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (4*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3
+ 5*x)^(3/2))/3 - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3*x]) - (3*Sqrt
[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi [A]  time = 0.331435, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{4}{3} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{3}{5} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (4*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3
+ 5*x)^(3/2))/3 - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3*x]) - (3*Sqrt
[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi in Sympy [A]  time = 31.6586, size = 138, normalized size = 0.88 \[ \frac{4 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{3} - \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \sqrt{3 x + 2}} - \frac{3 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5} - \frac{\sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/3 - sqrt(-2*x + 1)*sqrt(3*x + 2)
*sqrt(5*x + 3) - 2*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(3*sqrt(3*x + 2)) - 3*sqrt(33
)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5 - sqrt(33)*elliptic_f(asi
n(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15

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Mathematica [A]  time = 0.240574, size = 112, normalized size = 0.72 \[ \frac{10 \sqrt{1-2 x} x \sqrt{3 x+2} \sqrt{5 x+3} (10 x+7)+15 \sqrt{2} (3 x+2) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+18 \sqrt{2} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{90 x+60} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*x*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(7 + 10*x) + 18*Sqrt[2]*(2 + 3*x
)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 15*Sqrt[2]*(2 + 3*x)*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(60 + 90*x)

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Maple [C]  time = 0.024, size = 168, normalized size = 1.1 \[ -{\frac{1}{900\,{x}^{3}+690\,{x}^{2}-210\,x-180}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 15\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +18\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1000\,{x}^{4}-800\,{x}^{3}+230\,{x}^{2}+210\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(15*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))+18*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(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))-1000*x^4-
800*x^3+230*x^2+210*x)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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