3.2818 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{21 \sqrt{3 x+2}}-\frac{205}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{41}{189} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{974}{189} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-205*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/189 + (2*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2))/(21*Sqrt[2 + 3*x]) - (974*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/189 - (41*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/189

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Rubi [A]  time = 0.247893, 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 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{21 \sqrt{3 x+2}}-\frac{205}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{41}{189} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{974}{189} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-205*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/189 + (2*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2))/(21*Sqrt[2 + 3*x]) - (974*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/189 - (41*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/189

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Rubi in Sympy [A]  time = 24.8275, size = 114, normalized size = 0.88 \[ - \frac{205 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{189} + \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \sqrt{3 x + 2}} - \frac{974 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{567} - \frac{451 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{6615} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-205*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/189 + 2*sqrt(-2*x + 1)*(5*x + 3)
**(3/2)/(21*sqrt(3*x + 2)) - 974*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1
)/7), 35/33)/567 - 451*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/
35)/6615

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Mathematica [A]  time = 0.221964, size = 97, normalized size = 0.75 \[ \frac{-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} (525 x+356)}{\sqrt{3 x+2}}-595 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1948 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1134} \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(356 + 525*x))/Sqrt[2 + 3*x] + 1948*Sqrt[2]*Ell
ipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 595*Sqrt[2]*EllipticF[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2])/1134

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Maple [C]  time = 0.025, size = 164, normalized size = 1.3 \[{\frac{1}{34020\,{x}^{3}+26082\,{x}^{2}-7938\,x-6804}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 595\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1948\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -31500\,{x}^{3}-24510\,{x}^{2}+7314\,x+6408 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1134*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(595*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))-1948*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))-31500
*x^3-24510*x^2+7314*x+6408)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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