3.2666 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{36}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{13}{625} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1091 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125 \sqrt{33}}-\frac{1409 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (13*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/625 + (36*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/125
- (1409*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125 - (1
091*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3125*Sqrt[33])

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Rubi [A]  time = 0.335712, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{5 \sqrt{5 x+3}}+\frac{36}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{13}{625} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1091 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125 \sqrt{33}}-\frac{1409 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3125} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (13*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x])/625 + (36*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/125
- (1409*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3125 - (1
091*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3125*Sqrt[33])

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Rubi in Sympy [A]  time = 33.9458, size = 143, normalized size = 0.91 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{5 \sqrt{5 x + 3}} + \frac{36 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{125} + \frac{13 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{625} - \frac{1409 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9375} - \frac{1091 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{109375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(5*sqrt(5*x + 3)) + 36*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*sqrt(5*x + 3)/125 + 13*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/625
- 1409*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9375 - 1091*s
qrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/109375

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Mathematica [A]  time = 0.383844, size = 102, normalized size = 0.65 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (450 x^2+485 x+119\right )}{\sqrt{5 x+3}}+455 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2818 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{18750} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(119 + 485*x + 450*x^2))/Sqrt[3 + 5*x] + 2818*S
qrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 455*Sqrt[2]*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/18750

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Maple [C]  time = 0.026, size = 169, normalized size = 1.1 \[ -{\frac{1}{562500\,{x}^{3}+431250\,{x}^{2}-131250\,x-112500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 455\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2818\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -81000\,{x}^{4}-100800\,{x}^{3}-8970\,{x}^{2}+25530\,x+7140 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18750*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(455*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))+2818*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))-810
00*x^4-100800*x^3-8970*x^2+25530*x+7140)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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