3.2962 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=218 \[ \frac{26062156 \sqrt{1-2 x} \sqrt{5 x+3}}{10168235 \sqrt{3 x+2}}+\frac{349904 \sqrt{1-2 x} \sqrt{5 x+3}}{1452605 (3 x+2)^{3/2}}-\frac{806 \sqrt{1-2 x} \sqrt{5 x+3}}{207515 (3 x+2)^{5/2}}+\frac{1336 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{837304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}}-\frac{26062156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}} \]

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (1336*Sqrt[3 + 5*x])/(
17787*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(207515
*(2 + 3*x)^(5/2)) + (349904*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1452605*(2 + 3*x)^(3/2
)) + (26062156*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10168235*Sqrt[2 + 3*x]) - (26062156
*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(924385*Sqrt[33]) - (837304*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(924385*Sqrt[33])

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Rubi [A]  time = 0.515687, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{26062156 \sqrt{1-2 x} \sqrt{5 x+3}}{10168235 \sqrt{3 x+2}}+\frac{349904 \sqrt{1-2 x} \sqrt{5 x+3}}{1452605 (3 x+2)^{3/2}}-\frac{806 \sqrt{1-2 x} \sqrt{5 x+3}}{207515 (3 x+2)^{5/2}}+\frac{1336 \sqrt{5 x+3}}{17787 \sqrt{1-2 x} (3 x+2)^{5/2}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac{837304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}}-\frac{26062156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{924385 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]),x]

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (1336*Sqrt[3 + 5*x])/(
17787*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) - (806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(207515
*(2 + 3*x)^(5/2)) + (349904*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1452605*(2 + 3*x)^(3/2
)) + (26062156*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10168235*Sqrt[2 + 3*x]) - (26062156
*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(924385*Sqrt[33]) - (837304*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(924385*Sqrt[33])

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Rubi in Sympy [A]  time = 45.4901, size = 201, normalized size = 0.92 \[ - \frac{26062156 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{30504705} - \frac{837304 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{32353475} - \frac{52124312 \sqrt{3 x + 2} \sqrt{5 x + 3}}{30504705 \sqrt{- 2 x + 1}} + \frac{768556 \sqrt{5 x + 3}}{132055 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} + \frac{10652 \sqrt{5 x + 3}}{18865 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{178 \sqrt{5 x + 3}}{2695 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{4 \sqrt{5 x + 3}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-2*x)**(5/2)/(2+3*x)**(7/2)/(3+5*x)**(1/2),x)

[Out]

-26062156*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/30504705 -
837304*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/32353475 -
52124312*sqrt(3*x + 2)*sqrt(5*x + 3)/(30504705*sqrt(-2*x + 1)) + 768556*sqrt(5*x
+ 3)/(132055*sqrt(-2*x + 1)*sqrt(3*x + 2)) + 10652*sqrt(5*x + 3)/(18865*sqrt(-2
*x + 1)*(3*x + 2)**(3/2)) + 178*sqrt(5*x + 3)/(2695*sqrt(-2*x + 1)*(3*x + 2)**(5
/2)) + 4*sqrt(5*x + 3)/(231*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2))

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Mathematica [A]  time = 0.433269, size = 107, normalized size = 0.49 \[ \frac{\frac{2 \sqrt{10 x+6} \left (1407356424 x^4+513206712 x^3-914077314 x^2-176797172 x+165071409\right )}{(1-2 x)^{3/2} (3 x+2)^{5/2}}-24493280 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{30504705 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]),x]

[Out]

((2*Sqrt[6 + 10*x]*(165071409 - 176797172*x - 914077314*x^2 + 513206712*x^3 + 14
07356424*x^4))/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + 52124312*EllipticE[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2] - 24493280*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2])/(30504705*Sqrt[2])

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Maple [C]  time = 0.038, size = 502, normalized size = 2.3 \[{\frac{2}{30504705\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 110219760\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-234559404\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-195466170\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-24493280\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-24493280\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-3030766434\,{x}^{3}-3626217802\,{x}^{2}[phone]\,x+495214227 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x)

[Out]

2/30504705*(1-2*x)^(1/2)*(110219760*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)-234559404*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)-195466170*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^2*(3+5*x)^(
1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-24493280*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)[phone]*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)-24493280*
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))[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*1
1^(1/2)*3^(1/2)*2^(1/2))[phone]*x^5+6788102832*x^4-3030766434*x^3-3626217802
*x^2+294965529*x+495214227)/(2+3*x)^(5/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*sqrt(5*x + 3)*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(1/(1-2*x)**(5/2)/(2+3*x)**(7/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(5*x + 3)*(3*x + 2)^(7/2)*(-2*x + 1)^(5/2)), x)