3.2761 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{40}{81} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2108 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1701}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{64628 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{8505}-\frac{310399 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{76545}-\frac{310399 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725}-\frac{25111 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725} \]

[Out]

(-310399*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/76545 + (64628*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/8505 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*Sq
rt[2 + 3*x]) - (2108*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/1701 - (40*(1
- 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/81 - (25111*Sqrt[11/3]*EllipticE[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/382725 - (310399*Sqrt[11/3]*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/382725

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Rubi [A]  time = 0.480322, antiderivative size = 222, 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 \[ -\frac{40}{81} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2108 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1701}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{64628 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{8505}-\frac{310399 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{76545}-\frac{310399 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725}-\frac{25111 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725} \]

Antiderivative was successfully verified.

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

[Out]

(-310399*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/76545 + (64628*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/8505 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*Sq
rt[2 + 3*x]) - (2108*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/1701 - (40*(1
- 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/81 - (25111*Sqrt[11/3]*EllipticE[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/382725 - (310399*Sqrt[11/3]*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/382725

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Rubi in Sympy [A]  time = 49.7676, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \sqrt{3 x + 2}} - \frac{40 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{81} + \frac{5270 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{1701} - \frac{3329 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1701} + \frac{19172 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{76545} - \frac{25111 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1148175} - \frac{3414389 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{13395375} \]

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)**(3/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/(3*sqrt(3*x + 2)) - 40*(-2*x + 1)**(3/2)*s
qrt(3*x + 2)*(5*x + 3)**(5/2)/81 + 5270*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*(5*x + 3
)**(3/2)/1701 - 3329*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/1701 + 19172*
sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/76545 - 25111*sqrt(33)*elliptic_e(asi
n(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1148175 - 3414389*sqrt(35)*elliptic_f(asin(
sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/13395375

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Mathematica [A]  time = 0.452249, size = 112, normalized size = 0.5 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (567000 x^4-386100 x^3-259650 x^2+245751 x+21964\right )}{\sqrt{3 x+2}}[phone] \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+50222 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2296350} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(21964 + 245751*x - 259650*x^2 - 386100*x^3 + 5
67000*x^4))/Sqrt[2 + 3*x] + 50222*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2] + 10192945*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2])/2296350

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Maple [C]  time = 0.026, size = 179, normalized size = 0.8 \[ -{\frac{1}{68890500\,{x}^{3}[phone]\,{x}^{2}-16074450\,x-13778100}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -170100000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +50222\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-100684800\,{x}^{3}-37330230\,{x}^{2}[phone]\,x+1976760 \right ) } \]

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)^(3/2),x)

[Out]

-1/2296350*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-170100000*x^6+10192945*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))+50222*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+140508000*x^4-100684800*x^3-37330230*x^2+214586
70*x+1976760)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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