3.2765 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{2776 \sqrt{1-2 x} (5 x+3)^{5/2}}{1701 (3 x+2)^{5/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{13316 \sqrt{1-2 x} (5 x+3)^{3/2}}{35721 (3 x+2)^{3/2}}-\frac{1241596 \sqrt{1-2 x} \sqrt{5 x+3}}{750141 \sqrt{3 x+2}}-\frac{1241596 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141}-\frac{100444 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141} \]

[Out]

(-1241596*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(750141*Sqrt[2 + 3*x]) - (13316*Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))/(35721*(2 + 3*x)^(3/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/
2))/(27*(2 + 3*x)^(9/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(567*(2 + 3*x)^
(7/2)) + (2776*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1701*(2 + 3*x)^(5/2)) - (100444*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/750141 - (1241596*S
qrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/750141

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Rubi [A]  time = 0.494021, 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 \[ \frac{2776 \sqrt{1-2 x} (5 x+3)^{5/2}}{1701 (3 x+2)^{5/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{13316 \sqrt{1-2 x} (5 x+3)^{3/2}}{35721 (3 x+2)^{3/2}}-\frac{1241596 \sqrt{1-2 x} \sqrt{5 x+3}}{750141 \sqrt{3 x+2}}-\frac{1241596 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141}-\frac{100444 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]

[Out]

(-1241596*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(750141*Sqrt[2 + 3*x]) - (13316*Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))/(35721*(2 + 3*x)^(3/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/
2))/(27*(2 + 3*x)^(9/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(567*(2 + 3*x)^
(7/2)) + (2776*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1701*(2 + 3*x)^(5/2)) - (100444*S
qrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/750141 - (1241596*S
qrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/750141

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Rubi in Sympy [A]  time = 48.9722, size = 201, normalized size = 0.91 \[ - \frac{9434 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{83349 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{370 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3969 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{33290 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{35721 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{191720 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{107163 \sqrt{3 x + 2}} - \frac{100444 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2250423} - \frac{1241596 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2250423} \]

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)**(11/2),x)

[Out]

-9434*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(83349*(3*x + 2)**(5/2)) - 370*(-2*x + 1)*
*(5/2)*(5*x + 3)**(3/2)/(3969*(3*x + 2)**(7/2)) - 2*(-2*x + 1)**(5/2)*(5*x + 3)*
*(5/2)/(27*(3*x + 2)**(9/2)) + 33290*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(35721*(3*x
+ 2)**(3/2)) + 191720*sqrt(-2*x + 1)*sqrt(5*x + 3)/(107163*sqrt(3*x + 2)) - 100
444*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2250423 - 124159
6*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2250423

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Mathematica [A]  time = 0.31405, size = 109, normalized size = 0.49 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (29072682 x^4+115002639 x^3+142557831 x^2+71920155 x+12903031\right )}{(3 x+2)^{9/2}}+\sqrt{2} \left (10192945 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+50222 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2250423} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12903031 + 71920155*x + 142557831*x^2 + 1150
02639*x^3 + 29072682*x^4))/(2 + 3*x)^(9/2) + Sqrt[2]*(50222*EllipticE[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2] + 10192945*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2])))/2250423

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Maple [C]  time = 0.03, size = 624, normalized size = 2.8 \[ -{\frac{2}{22504230\,{x}^{2}+2250423\,x-6751269} \left ( 825628545\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4067982\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4821312\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-872180460\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +803552\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3537297216\,{x}^{5}-4360088709\,{x}^{4}-1550254392\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+116127279 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

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)^(11/2),x)

[Out]

-2/2250423*(825628545*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+406798
2*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*Ellipti
cF(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)*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)
[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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*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*
(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+4821312*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)-872180460*x^6+163087120*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))+803552*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))-3537297216*x^
5-4360088709*x^4-1550254392*x^3+680169084*x^2+608572302*x+116127279)*(3+5*x)^(1/
2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

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)^(11/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2),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)/((24
3*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*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)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

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)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2), x)