3.2963 \(\int \frac{(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{896 (3 x+2)^{7/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4439 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 \sqrt{5 x+3}}-\frac{932783 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{332750}-\frac{21713939 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1663750}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{378125 \sqrt{33}}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1512500 \sqrt{33}} \]

[Out]

(4439*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(19965*Sqrt[3 + 5*x]) - (896*(2 + 3*x)^(7/2
))/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(9/2))/(33*(1 - 2*x)^(3/2)*S
qrt[3 + 5*x]) - (21713939*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1663750 - (
932783*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/332750 - (1508889271*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1512500*Sqrt[33]) - (11346991*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(378125*Sqrt[33])

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Rubi [A]  time = 0.493724, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{896 (3 x+2)^{7/2}}{363 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4439 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 \sqrt{5 x+3}}-\frac{932783 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{332750}-\frac{21713939 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1663750}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{378125 \sqrt{33}}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1512500 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

(4439*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(19965*Sqrt[3 + 5*x]) - (896*(2 + 3*x)^(7/2
))/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(9/2))/(33*(1 - 2*x)^(3/2)*S
qrt[3 + 5*x]) - (21713939*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1663750 - (
932783*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/332750 - (1508889271*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1512500*Sqrt[33]) - (11346991*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(378125*Sqrt[33])

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Rubi in Sympy [A]  time = 46.2477, size = 201, normalized size = 0.92 \[ \frac{4439 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{19965 \sqrt{5 x + 3}} - \frac{932783 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{332750} - \frac{21713939 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1663750} - \frac{1508889271 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{49912500} - \frac{11346991 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{12478125} - \frac{896 \left (3 x + 2\right )^{\frac{7}{2}}}{363 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{7 \left (3 x + 2\right )^{\frac{9}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(11/2)/(1-2*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

4439*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(19965*sqrt(5*x + 3)) - 932783*sqrt(-2*x +
1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/332750 - 21713939*sqrt(-2*x + 1)*sqrt(3*x + 2)
*sqrt(5*x + 3)/1663750 - 1508889271*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x
+ 1)/7), 35/33)/49912500 - 11346991*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x
+ 1)/7), 35/33)/12478125 - 896*(3*x + 2)**(7/2)/(363*sqrt(-2*x + 1)*sqrt(5*x + 3
)) + 7*(3*x + 2)**(9/2)/(33*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.403644, size = 107, normalized size = 0.49 \[ \frac{-\frac{5 \sqrt{6 x+4} \left (48514950 x^4+286777260 x^3-1463754851 x^2-376752444 x+356556921\right )}{(1-2 x)^{3/2} \sqrt{5 x+3}}-759987865 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{24956250 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]

[Out]

((-5*Sqrt[4 + 6*x]*(356556921 - 376752444*x - 1463754851*x^2 + 286777260*x^3 + 4
8514950*x^4))/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 1508889271*EllipticE[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2] - 759987865*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2])/(24956250*Sqrt[2])

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Maple [C]  time = 0.037, size = 286, normalized size = 1.3 \[{\frac{1}{ \left ( 748687500\,{x}^{2}[phone]\,x+299475000 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1519975730\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3017778542\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-759987865\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1455448500\,{x}^{5}-9573616800\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-3161658750\,x-7131138420 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x)

[Out]

1/49912500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1519975730*2^(1/2)*Ellipti
cF(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)-3017778542*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)-759987865*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elli
pticF(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))+150888
9271*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))-1455448500*x^5-9573616800*x
^4+38177100330*x^3+40577670340*x^2-3161658750*x-7131138420)/(15*x^2+19*x+6)/(-1+
2*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(11/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(11/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(11/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)),x, algorithm="fricas")

[Out]

integral((243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*sqrt(3*x + 2)/((2
0*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*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((2+3*x)**(11/2)/(1-2*x)**(5/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(11/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(11/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)), x)