3.2942 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{19 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{5 x+3}}{147 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{106 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}+\frac{19}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) - (8*Sqrt[3 + 5*x])/(147*S
qrt[1 - 2*x]*Sqrt[2 + 3*x]) - (19*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*Sqrt[2 + 3*x
]) + (19*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343 + (10
6*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])

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Rubi [A]  time = 0.338605, antiderivative size = 158, 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 \[ -\frac{19 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{5 x+3}}{147 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} \sqrt{3 x+2}}+\frac{106 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}+\frac{19}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) - (8*Sqrt[3 + 5*x])/(147*S
qrt[1 - 2*x]*Sqrt[2 + 3*x]) - (19*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*Sqrt[2 + 3*x
]) + (19*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/343 + (10
6*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])

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Rubi in Sympy [A]  time = 30.737, size = 143, normalized size = 0.91 \[ \frac{19 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1029} + \frac{106 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11319} + \frac{38 \sqrt{3 x + 2} \sqrt{5 x + 3}}{1029 \sqrt{- 2 x + 1}} - \frac{9 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} + \frac{11 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1029 + 106*sqrt(3
3)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/11319 + 38*sqrt(3*x + 2)*s
qrt(5*x + 3)/(1029*sqrt(-2*x + 1)) - 9*sqrt(5*x + 3)/(49*sqrt(-2*x + 1)*sqrt(3*x
+ 2)) + 11*sqrt(5*x + 3)/(21*(-2*x + 1)**(3/2)*sqrt(3*x + 2))

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Mathematica [A]  time = 0.226752, size = 100, normalized size = 0.63 \[ \frac{-\frac{2 \sqrt{5 x+3} \left (114 x^2-170 x-213\right )}{(1-2 x)^{3/2} \sqrt{3 x+2}}-\sqrt{2} \left (140 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+19 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{1029} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[3 + 5*x]*(-213 - 170*x + 114*x^2))/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]) - S
qrt[2]*(19*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 140*EllipticF[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/1029

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Maple [C]  time = 0.035, size = 276, normalized size = 1.8 \[{\frac{1}{ \left ( 15435\,{x}^{2}+19551\,x+6174 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 280\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+38\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-140\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -19\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1140\,{x}^{3}+1016\,{x}^{2}+3150\,x+1278 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1029*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(280*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)+38*2^(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))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-140
*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))-19*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))-1140*x^3+1016*x^2+3150*x+1278)/(15*x^2+19*x+6)/(-1+2*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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