3.2899 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{106558 \sqrt{1-2 x} \sqrt{5 x+3}}{1764735 \sqrt{3 x+2}}-\frac{106772 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 (3 x+2)^{3/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{5/2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{7/2}}-\frac{220028 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735}+\frac{106558 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735} \]

[Out]

(229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1029*(2 + 3*x)^(7/2)) - (37117*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(36015*(2 + 3*x)^(5/2)) - (106772*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2
52105*(2 + 3*x)^(3/2)) - (106558*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1764735*Sqrt[2 +
3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + (106558*Sqrt[11
/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1764735 - (220028*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1764735

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Rubi [A]  time = 0.518302, antiderivative size = 222, 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 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{106558 \sqrt{1-2 x} \sqrt{5 x+3}}{1764735 \sqrt{3 x+2}}-\frac{106772 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 (3 x+2)^{3/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{5/2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{7/2}}-\frac{220028 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735}+\frac{106558 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(9/2)),x]

[Out]

(229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1029*(2 + 3*x)^(7/2)) - (37117*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(36015*(2 + 3*x)^(5/2)) - (106772*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2
52105*(2 + 3*x)^(3/2)) - (106558*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1764735*Sqrt[2 +
3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + (106558*Sqrt[11
/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1764735 - (220028*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1764735

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Rubi in Sympy [A]  time = 45.6285, size = 201, normalized size = 0.91 \[ - \frac{106558 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1764735 \sqrt{3 x + 2}} - \frac{106772 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{252105 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{37117 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36015 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{229 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1029 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{106558 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5294205} - \frac{220028 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5294205} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(9/2),x)

[Out]

-106558*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1764735*sqrt(3*x + 2)) - 106772*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(252105*(3*x + 2)**(3/2)) - 37117*sqrt(-2*x + 1)*sqrt(5*x + 3
)/(36015*(3*x + 2)**(5/2)) + 229*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1029*(3*x + 2)**(
7/2)) + 106558*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/52942
05 - 220028*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5294205
+ 11*(5*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*(3*x + 2)**(7/2))

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Mathematica [A]  time = 0.308878, size = 109, normalized size = 0.49 \[ \frac{2 \left (\frac{3 \sqrt{5 x+3} \left (2877066 x^4+11042235 x^3+12020751 x^2+4889131 x+616327\right )}{\sqrt{1-2 x} (3 x+2)^{7/2}}+\sqrt{2} \left (1868510 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-53279 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{5294205} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(9/2)),x]

[Out]

(2*((3*Sqrt[3 + 5*x]*(616327 + 4889131*x + 12020751*x^2 + 11042235*x^3 + 2877066
*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + Sqrt[2]*(-53279*EllipticE[ArcSin[Sqrt[2
/11]*Sqrt[3 + 5*x]], -33/2] + 1868510*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2])))/5294205

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Maple [C]  time = 0.037, size = 505, normalized size = 2.3 \[ -{\frac{2}{52942050\,{x}^{2}+5294205\,x-15882615}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 50449770\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-1438533\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2877066\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1918044\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -426232\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+5546943 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(9/2),x)

[Out]

-2/5294205*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(50449770*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*(1-2*x)^(1/2)*(3+5*x)
^(1/2)*(2+3*x)^(1/2)-1438533*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)
[phone]*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-2877066*2^(1/2)*E
llipticE(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^2
*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*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)-1918044*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)+
14948080*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))-426232*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))[phone]*x^5+191527119*x^4+279691380*x^3+1815237
24*x^2+53247084*x+5546943)/(2+3*x)^(7/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((162*x^5 + 351*x^4 + 216*x^3 - 24*x
^2 - 64*x - 16)*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)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(3/2)), x)