3.2745 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{63 (3 x+2)^{7/2}}+\frac{7810384 \sqrt{5 x+3} \sqrt{1-2 x}}{83349 \sqrt{3 x+2}}+\frac{112436 \sqrt{5 x+3} \sqrt{1-2 x}}{11907 (3 x+2)^{3/2}}+\frac{832 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}-\frac{234856 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349}-\frac{7810384 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349} \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (10*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/(63*(2 + 3*x)^(7/2)) + (832*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(567*(2 +
3*x)^(5/2)) + (112436*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(11907*(2 + 3*x)^(3/2)) + (78
10384*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(83349*Sqrt[2 + 3*x]) - (7810384*Sqrt[11/3]*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/83349 - (234856*Sqrt[11/3]*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/83349

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Rubi [A]  time = 0.501484, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{63 (3 x+2)^{7/2}}+\frac{7810384 \sqrt{5 x+3} \sqrt{1-2 x}}{83349 \sqrt{3 x+2}}+\frac{112436 \sqrt{5 x+3} \sqrt{1-2 x}}{11907 (3 x+2)^{3/2}}+\frac{832 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}-\frac{234856 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349}-\frac{7810384 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83349} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (10*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/(63*(2 + 3*x)^(7/2)) + (832*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(567*(2 +
3*x)^(5/2)) + (112436*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(11907*(2 + 3*x)^(3/2)) + (78
10384*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(83349*Sqrt[2 + 3*x]) - (7810384*Sqrt[11/3]*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/83349 - (234856*Sqrt[11/3]*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/83349

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Rubi in Sympy [A]  time = 45.5849, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{10 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{63 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{7810384 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{83349 \sqrt{3 x + 2}} + \frac{112436 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{11907 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{832 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{567 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{7810384 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{250047} - \frac{234856 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{250047} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(27*(3*x + 2)**(9/2)) + 10*(-2*x + 1)**(3/2)*
sqrt(5*x + 3)/(63*(3*x + 2)**(7/2)) + 7810384*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8334
9*sqrt(3*x + 2)) + 112436*sqrt(-2*x + 1)*sqrt(5*x + 3)/(11907*(3*x + 2)**(3/2))
+ 832*sqrt(-2*x + 1)*sqrt(5*x + 3)/(567*(3*x + 2)**(5/2)) - 7810384*sqrt(33)*ell
iptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/250047 - 234856*sqrt(33)*ellipti
c_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/250047

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Mathematica [A]  time = 0.378305, size = 111, normalized size = 0.5 \[ \frac{4 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (316320552 x^4+854146674 x^3+865270206 x^2+389804925 x+65886031\right )}{2 (3 x+2)^{9/2}}+\sqrt{2} \left (1952596 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-983815 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{250047} \]

Antiderivative was successfully verified.

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(65886031 + 389804925*x + 865270206*x^2 + 854
146674*x^3 + 316320552*x^4))/(2*(2 + 3*x)^(9/2)) + Sqrt[2]*(1952596*EllipticE[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 983815*EllipticF[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2])))/250047

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Maple [C]  time = 0.03, size = 624, normalized size = 2.8 \[{\frac{2}{2500470\,{x}^{2}+250047\,x-750141} \left ( 159378030\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-316320552\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-843521472\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-843521472\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-374898432\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -62483072\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-4641436149\,{x}^{2}-3310586232\,x-592974279 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/250047*(159378030*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)-31632055
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)*Elliptic
F(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)-843521472*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)-
843521472*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)-374898432*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)[phone]*x^6+31482080*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))-62483072*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(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]
76*x^5+25673661234*x^4+6602638302*x^3-4641436149*x^2-3310586232*x-592974279)*(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. \[ \int \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((243*x^5 + 810*x^4 + 10
80*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)**(1/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2),x, algorithm="giac")

[Out]

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