3.2690 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{42623864 \sqrt{5 x+3} \sqrt{1-2 x}}{972405 \sqrt{3 x+2}}+\frac{613276 \sqrt{5 x+3} \sqrt{1-2 x}}{138915 (3 x+2)^{3/2}}+\frac{13136 \sqrt{5 x+3} \sqrt{1-2 x}}{19845 (3 x+2)^{5/2}}+\frac{82 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{7/2}}-\frac{1282376 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{972405}-\frac{42623864 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{972405} \]

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (82*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(567*(2 + 3*x)^(7/2)) + (13136*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19845*(2
+ 3*x)^(5/2)) + (613276*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*(2 + 3*x)^(3/2)) +
(42623864*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(972405*Sqrt[2 + 3*x]) - (42623864*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/972405 - (1282376*Sqrt[
11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/972405

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Rubi [A]  time = 0.505436, 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 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{42623864 \sqrt{5 x+3} \sqrt{1-2 x}}{972405 \sqrt{3 x+2}}+\frac{613276 \sqrt{5 x+3} \sqrt{1-2 x}}{138915 (3 x+2)^{3/2}}+\frac{13136 \sqrt{5 x+3} \sqrt{1-2 x}}{19845 (3 x+2)^{5/2}}+\frac{82 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{7/2}}-\frac{1282376 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{972405}-\frac{42623864 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{972405} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (82*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(567*(2 + 3*x)^(7/2)) + (13136*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19845*(2
+ 3*x)^(5/2)) + (613276*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*(2 + 3*x)^(3/2)) +
(42623864*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(972405*Sqrt[2 + 3*x]) - (42623864*Sqrt[
11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/972405 - (1282376*Sqrt[
11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/972405

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Rubi in Sympy [A]  time = 45.2764, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{42623864 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{972405 \sqrt{3 x + 2}} + \frac{613276 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{138915 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{13136 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{19845 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{82 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{567 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{42623864 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2917215} - \frac{14106136 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{34034175} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(27*(3*x + 2)**(9/2)) + 42623864*sqrt(-2*x +
1)*sqrt(5*x + 3)/(972405*sqrt(3*x + 2)) + 613276*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1
38915*(3*x + 2)**(3/2)) + 13136*sqrt(-2*x + 1)*sqrt(5*x + 3)/(19845*(3*x + 2)**(
5/2)) + 82*sqrt(-2*x + 1)*sqrt(5*x + 3)/(567*(3*x + 2)**(7/2)) - 42623864*sqrt(3
3)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2917215 - 14106136*sqrt(35
)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/34034175

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Mathematica [A]  time = 0.364187, size = 111, normalized size = 0.5 \[ \frac{4 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (1726266492 x^4+4661331894 x^3+4722182964 x^2+2127363207 x+359554583\right )}{2 (3 x+2)^{9/2}}+\sqrt{2} \left (10655966 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5366165 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2917215} \]

Antiderivative was successfully verified.

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(359554583 + 2127363207*x + 4722182964*x^2 +
4661331894*x^3 + 1726266492*x^4))/(2*(2 + 3*x)^(9/2)) + Sqrt[2]*(10655966*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5366165*EllipticF[ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]], -33/2])))/2917215

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Maple [C]  time = 0.03, size = 624, normalized size = 2.8 \[{\frac{2}{29172150\,{x}^{2}+2917215\,x-8751645} \left ( 869318730\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1726266492\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-4603377312\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4603377312\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2045945472\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -340990912\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-25330919565\,{x}^{2}-18067605114\,x-3235991247 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2917215*(869318730*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)-1726266
492*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)*Ellip
ticF(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)-4603377312*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)-4603377312*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)-2045945472*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+171717280*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))-340990912*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+140113086174*x^4+36035458056*x^3-25330919565*x^2-18067605114*
x-3235991247)*(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. \[ \int \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(5*x + 3)*(-2*x + 1)^(3/2)/((243*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)**(3/2)*(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. \[ \int \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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