3.2742 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 \sqrt{3 x+2}}+\frac{196}{81} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{988}{405} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4418}{405} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + (10*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/(3*Sqrt[2 + 3*x]) + (196*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/
81 - (4418*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/405 + (
988*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/405

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Rubi [A]  time = 0.334035, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 \sqrt{3 x+2}}+\frac{196}{81} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{988}{405} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4418}{405} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + (10*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/(3*Sqrt[2 + 3*x]) + (196*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/
81 - (4418*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/405 + (
988*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/405

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Rubi in Sympy [A]  time = 32.0367, size = 143, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{9 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{10 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 \sqrt{3 x + 2}} + \frac{196 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{81} - \frac{4418 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1215} + \frac{10868 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{14175} \]

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)**(5/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(9*(3*x + 2)**(3/2)) + 10*(-2*x + 1)**(3/2)*s
qrt(5*x + 3)/(3*sqrt(3*x + 2)) + 196*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/
81 - 4418*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1215 + 108
68*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/14175

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Mathematica [A]  time = 0.318398, size = 102, normalized size = 0.64 \[ \frac{2 \left (\frac{15 \sqrt{1-2 x} \sqrt{5 x+3} \left (36 x^2+1077 x+653\right )}{(3 x+2)^{3/2}}-10360 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2209 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{1215} \]

Antiderivative was successfully verified.

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

[Out]

(2*((15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(653 + 1077*x + 36*x^2))/(2 + 3*x)^(3/2) + 2
209*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 10360*Sqrt[2]*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/1215

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Maple [C]  time = 0.027, size = 272, normalized size = 1.7 \[{\frac{2}{12150\,{x}^{2}+1215\,x-3645} \left ( 31080\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6627\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+20720\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4418\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +5400\,{x}^{4}+162090\,{x}^{3}+112485\,{x}^{2}-38670\,x-29385 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

2/1215*(31080*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)-6627*2^(1/2)*Ell
ipticE(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)+20720*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))-4418*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))+5400*x^4+16209
0*x^3+112485*x^2-38670*x-29385)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)
^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((9*x^2 + 12*x + 4)*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)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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