3.2909 \(\int \frac{(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{5/2}}{605 \sqrt{5 x+3}}+\frac{10851 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{15125}+\frac{502941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{151250}+\frac{175111 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{68750}+\frac{2911577 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{34375} \]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(7/2))/(1
1*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (502941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/151250 + (10851*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/15125 + (291157
7*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/34375 + (175111*
Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/68750

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Rubi [A]  time = 0.407807, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{7 (3 x+2)^{7/2}}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^{5/2}}{605 \sqrt{5 x+3}}+\frac{10851 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{15125}+\frac{502941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{151250}+\frac{175111 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{68750}+\frac{2911577 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{34375} \]

Antiderivative was successfully verified.

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

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(7/2))/(1
1*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (502941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/151250 + (10851*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/15125 + (291157
7*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/34375 + (175111*
Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/68750

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Rubi in Sympy [A]  time = 40.977, size = 172, normalized size = 0.9 \[ - \frac{37 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{605 \sqrt{5 x + 3}} + \frac{10851 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15125} + \frac{502941 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{151250} + \frac{2911577 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{378125} + \frac{525333 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2406250} + \frac{7 \left (3 x + 2\right )^{\frac{7}{2}}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(605*sqrt(5*x + 3)) + 10851*sqrt(-2*x + 1)*(
3*x + 2)**(3/2)*sqrt(5*x + 3)/15125 + 502941*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5
*x + 3)/151250 + 2911577*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35
/33)/378125 + 525333*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35
)/2406250 + 7*(3*x + 2)**(7/2)/(11*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.372633, size = 132, normalized size = 0.69 \[ \frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (-490050 x^3-2188890 x^2+3684629 x+2892883\right )+5867645 \sqrt{2-4 x} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-11646308 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1512500 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(2892883 + 3684629*x - 2188890*x^2 - 490050*x^3)
- 11646308*Sqrt[2 - 4*x]*(3 + 5*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2] + 5867645*Sqrt[2 - 4*x]*(3 + 5*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2])/(1512500*Sqrt[1 - 2*x]*(3 + 5*x))

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Maple [C]  time = 0.035, size = 169, normalized size = 0.9 \[ -{\frac{1}{45375000\,{x}^{3}[phone]\,{x}^{2}-10587500\,x-9075000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 5867645\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -11646308\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -14701500\,{x}^{4}-75467700\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+57857660 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1512500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5867645*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))-11646308*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))-14701500*x^4-75467700*x^3+66761070*x^2+160479070*x+57857660)/(30*x^3+23*x
^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(3*x + 2)/((10*x^2 + 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)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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