3.2641 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{189 \sqrt{3 x+2}}-\frac{214}{189} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{494}{189} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(189*Sqrt[2 + 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(9*(2 + 3*x)^(3/2)) + (494*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt
[1 - 2*x]], 35/33])/189 - (214*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/189

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Rubi [A]  time = 0.258408, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}-\frac{214 \sqrt{1-2 x} \sqrt{5 x+3}}{189 \sqrt{3 x+2}}-\frac{214}{189} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{494}{189} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(189*Sqrt[2 + 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(9*(2 + 3*x)^(3/2)) + (494*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt
[1 - 2*x]], 35/33])/189 - (214*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/189

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Rubi in Sympy [A]  time = 23.8583, size = 114, normalized size = 0.88 \[ - \frac{214 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{189 \sqrt{3 x + 2}} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{494 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{567} - \frac{2354 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{6615} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-214*sqrt(-2*x + 1)*sqrt(5*x + 3)/(189*sqrt(3*x + 2)) - 2*sqrt(-2*x + 1)*(5*x +
3)**(3/2)/(9*(3*x + 2)**(3/2)) + 494*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x
+ 1)/7), 35/33)/567 - 2354*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11)
, 33/35)/6615

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Mathematica [A]  time = 0.296617, size = 97, normalized size = 0.75 \[ \frac{1}{567} \left (-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} (426 x+277)}{(3 x+2)^{3/2}}+4025 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-494 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(277 + 426*x))/(2 + 3*x)^(3/2) - 494*Sqrt[2]*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 4025*Sqrt[2]*EllipticF[ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/567

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Maple [C]  time = 0.027, size = 267, normalized size = 2.1 \[ -{\frac{1}{5670\,{x}^{2}+567\,x-1701} \left ( 12075\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1482\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+8050\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -988\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +25560\,{x}^{3}+19176\,{x}^{2}-6006\,x-4986 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/567*(12075*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)-1482*2^(1/2)*Ell
ipticE(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)+8050*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))-988*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))+25560*x^3+19176*
x^2-6006*x-4986)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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