3.2760 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{33} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{9698 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{93555}-\frac{146963 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{467775}-\frac{1654421 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4209975}-\frac{1654421 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1913625 \sqrt{33}}-\frac{146222113 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3827250 \sqrt{33}} \]

[Out]

(-1654421*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4209975 - (146963*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/467775 + (9698*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(5/2))/93555 + (74*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/891 +
(2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/33 - (146222113*EllipticE[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3827250*Sqrt[33]) - (1654421*EllipticF[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1913625*Sqrt[33])

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Rubi [A]  time = 0.485073, 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 \[ \frac{2}{33} (1-2 x)^{5/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{74}{891} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{9698 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{93555}-\frac{146963 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{467775}-\frac{1654421 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{4209975}-\frac{1654421 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1913625 \sqrt{33}}-\frac{146222113 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3827250 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-1654421*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4209975 - (146963*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/467775 + (9698*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(5/2))/93555 + (74*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/891 +
(2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/33 - (146222113*EllipticE[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3827250*Sqrt[33]) - (1654421*EllipticF[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1913625*Sqrt[33])

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Rubi in Sympy [A]  time = 49.5068, size = 201, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{33} - \frac{185 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{891} + \frac{3698 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{18711} + \frac{119732 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{467775} - \frac{1654421 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{4209975} - \frac{146222113 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{126299250} - \frac{1654421 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{63149625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*sqrt(3*x + 2)*(5*x + 3)**(5/2)/33 - 185*(-2*x + 1)**(5/2)*sq
rt(3*x + 2)*(5*x + 3)**(3/2)/891 + 3698*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*(5*x + 3
)**(3/2)/18711 + 119732*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/467775 - 1
654421*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/4209975 - 146222113*sqrt(33)*e
lliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/126299250 - 1654421*sqrt(33)*e
lliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/63149625

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Mathematica [A]  time = 0.381976, size = 107, normalized size = 0.49 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (25515000 x^4-12379500 x^3-16381350 x^2+9143865 x+3748468\right )-91626220 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{63149625 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(3748468 + 9143865*x - 16381350*x^
2 - 12379500*x^3 + 25515000*x^4) + 146222113*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2] - 91626220*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(
63149625*Sqrt[2])

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Maple [C]  time = 0.017, size = 184, normalized size = 0.8 \[{\frac{1}{3788977500\,{x}^{3}[phone]\,{x}^{2}-884094750\,x-757795500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 22963500000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -146222113\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -28643220000\,{x}^{5}-5066658000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-2433073980\,x-674724240 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/126299250*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(22963500000*x^7+646380000
0*x^6+91626220*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))-146222113*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))-28643220000*x^5-5066658000*x^4+15351281
550*x^3+3614874270*x^2-2433073980*x-674724240)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt
(3*x + 2), 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((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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