3.2954 \(\int \frac{(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=187 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{910 \sqrt{5 x+3} (3 x+2)^{5/2}}{363 \sqrt{1-2 x}}-\frac{27271 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6050}-\frac{317384 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{663409 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13750 \sqrt{33}}-\frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27500 \sqrt{33}} \]

[Out]

(-317384*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15125 - (27271*Sqrt[1 - 2*x]
*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/6050 - (910*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(363*
Sqrt[1 - 2*x]) + (7*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (44109
377*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(27500*Sqrt[33]) - (66340
9*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(13750*Sqrt[33])

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Rubi [A]  time = 0.413924, antiderivative size = 187, 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 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac{910 \sqrt{5 x+3} (3 x+2)^{5/2}}{363 \sqrt{1-2 x}}-\frac{27271 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6050}-\frac{317384 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{663409 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13750 \sqrt{33}}-\frac{44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27500 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-317384*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15125 - (27271*Sqrt[1 - 2*x]
*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/6050 - (910*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/(363*
Sqrt[1 - 2*x]) + (7*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (44109
377*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(27500*Sqrt[33]) - (66340
9*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(13750*Sqrt[33])

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Rubi in Sympy [A]  time = 38.6434, size = 172, normalized size = 0.92 \[ - \frac{27271 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6050} - \frac{317384 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{15125} - \frac{44109377 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{907500} - \frac{663409 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{481250} - \frac{910 \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27271*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/6050 - 317384*sqrt(-2*x + 1
)*sqrt(3*x + 2)*sqrt(5*x + 3)/15125 - 44109377*sqrt(33)*elliptic_e(asin(sqrt(21)
*sqrt(-2*x + 1)/7), 35/33)/907500 - 663409*sqrt(35)*elliptic_f(asin(sqrt(55)*sqr
t(-2*x + 1)/11), 33/35)/481250 - 910*(3*x + 2)**(5/2)*sqrt(5*x + 3)/(363*sqrt(-2
*x + 1)) + 7*(3*x + 2)**(7/2)*sqrt(5*x + 3)/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.358094, size = 125, normalized size = 0.67 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (294030 x^3+1528956 x^2-9445541 x+3478434\right )-22216880 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{907500 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(3478434 - 9445541*x + 1528956*x^2 + 294030*x^3
) + 44109377*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2] - 22216880*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])/(907500*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 286, normalized size = 1.5 \[{\frac{1}{907500\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 44433760\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-88218754\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-22216880\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -44104500\,{x}^{5}-285209100\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-94170000\,x-208706040 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/907500*(44433760*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*1
1^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-88218754*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)-22216880*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*1
1^(1/2)*3^(1/2)*2^(1/2))[phone]*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)
)-44104500*x^5-285209100*x^4+1108687710*x^3+1181150330*x^2-94170000*x-208706040)
*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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