3.2773 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=129 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 \sqrt{3 x+2}}+\frac{428}{135} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{824}{675} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{8314}{675} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) + (428*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/135 - (8314*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/675 + (824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/675

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Rubi [A]  time = 0.257002, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 \sqrt{3 x+2}}+\frac{428}{135} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{824}{675} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{8314}{675} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) + (428*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/135 - (8314*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/675 + (824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/675

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Rubi in Sympy [A]  time = 25.7791, size = 114, normalized size = 0.88 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 \sqrt{3 x + 2}} + \frac{428 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{135} - \frac{8314 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2025} + \frac{9064 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{23625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(3*sqrt(3*x + 2)) + 428*sqrt(-2*x + 1)*sqrt(3
*x + 2)*sqrt(5*x + 3)/135 - 8314*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1
)/7), 35/33)/2025 + 9064*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 3
3/35)/23625

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Mathematica [A]  time = 0.295316, size = 97, normalized size = 0.75 \[ \frac{2 \left (\frac{15 \sqrt{1-2 x} \sqrt{5 x+3} (12 x+743)}{\sqrt{3 x+2}}-10955 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+4157 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{2025} \]

Antiderivative was successfully verified.

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

[Out]

(2*((15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(743 + 12*x))/Sqrt[2 + 3*x] + 4157*Sqrt[2]*E
llipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 10955*Sqrt[2]*EllipticF[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/2025

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Maple [C]  time = 0.026, size = 164, normalized size = 1.3 \[{\frac{2}{60750\,{x}^{3}+46575\,{x}^{2}-14175\,x-12150}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 10955\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4157\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1800\,{x}^{3}+111630\,{x}^{2}+10605\,x-33435 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2025*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(10955*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))-4157*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*E
llipticE(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))+180
0*x^3+111630*x^2+10605*x-33435)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/(sqrt(5*x + 3)*(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)/(2+3*x)**(3/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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