3.2746 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{99 (3 x+2)^{9/2}}+\frac{247408648 \sqrt{5 x+3} \sqrt{1-2 x}}{713097 \sqrt{3 x+2}}+\frac{3560432 \sqrt{5 x+3} \sqrt{1-2 x}}{101871 (3 x+2)^{3/2}}+\frac{76492 \sqrt{5 x+3} \sqrt{1-2 x}}{14553 (3 x+2)^{5/2}}+\frac{1900 \sqrt{5 x+3} \sqrt{1-2 x}}{2079 (3 x+2)^{7/2}}-\frac{7442032 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}}-\frac{247408648 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (10*(1 - 2*x)^(3/2)*S
qrt[3 + 5*x])/(99*(2 + 3*x)^(9/2)) + (1900*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2079*(2
+ 3*x)^(7/2)) + (76492*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14553*(2 + 3*x)^(5/2)) + (
3560432*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(101871*(2 + 3*x)^(3/2)) + (247408648*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x])/(713097*Sqrt[2 + 3*x]) - (247408648*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sqrt[33]) - (7442032*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sqrt[33])

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Rubi [A]  time = 0.591366, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{99 (3 x+2)^{9/2}}+\frac{247408648 \sqrt{5 x+3} \sqrt{1-2 x}}{713097 \sqrt{3 x+2}}+\frac{3560432 \sqrt{5 x+3} \sqrt{1-2 x}}{101871 (3 x+2)^{3/2}}+\frac{76492 \sqrt{5 x+3} \sqrt{1-2 x}}{14553 (3 x+2)^{5/2}}+\frac{1900 \sqrt{5 x+3} \sqrt{1-2 x}}{2079 (3 x+2)^{7/2}}-\frac{7442032 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}}-\frac{247408648 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(13/2),x]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (10*(1 - 2*x)^(3/2)*S
qrt[3 + 5*x])/(99*(2 + 3*x)^(9/2)) + (1900*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2079*(2
+ 3*x)^(7/2)) + (76492*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14553*(2 + 3*x)^(5/2)) + (
3560432*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(101871*(2 + 3*x)^(3/2)) + (247408648*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x])/(713097*Sqrt[2 + 3*x]) - (247408648*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sqrt[33]) - (7442032*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sqrt[33])

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Rubi in Sympy [A]  time = 53.8901, size = 230, normalized size = 0.92 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{33 \left (3 x + 2\right )^{\frac{11}{2}}} + \frac{10 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{99 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{247408648 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{713097 \sqrt{3 x + 2}} + \frac{3560432 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{101871 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{76492 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14553 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{1900 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2079 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{247408648 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2139291} - \frac{7442032 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2268945} \]

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)**(13/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(33*(3*x + 2)**(11/2)) + 10*(-2*x + 1)**(3/2)
*sqrt(5*x + 3)/(99*(3*x + 2)**(9/2)) + 247408648*sqrt(-2*x + 1)*sqrt(5*x + 3)/(7
13097*sqrt(3*x + 2)) + 3560432*sqrt(-2*x + 1)*sqrt(5*x + 3)/(101871*(3*x + 2)**(
3/2)) + 76492*sqrt(-2*x + 1)*sqrt(5*x + 3)/(14553*(3*x + 2)**(5/2)) + 1900*sqrt(
-2*x + 1)*sqrt(5*x + 3)/(2079*(3*x + 2)**(7/2)) - 247408648*sqrt(33)*elliptic_e(
asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2139291 - 7442032*sqrt(35)*elliptic_f(as
in(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2268945

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Mathematica [A]  time = 0.44067, size = 115, normalized size = 0.46 \[ \frac{\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (30060150732 x^5+101209884912 x^4+136342955970 x^3+91862628912 x^2+30956769477 x+4174268813\right )}{(3 x+2)^{11/2}}+32 \sqrt{2} \left (30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-15576890 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{8557164} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(13/2),x]

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4174268813 + 30956769477*x + 91862628912*x^2 +
136342955970*x^3 + 101209884912*x^4 + 30060150732*x^5))/(2 + 3*x)^(11/2) + 32*S
qrt[2]*(30926081*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 15576890*E
llipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/8557164

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Maple [C]  time = 0.03, size = 743, normalized size = 3. \[ \text{result too large to display} \]

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)^(13/2),x)

[Out]

2/2139291*(15140737080*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^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-30060
150732*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^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*E
llipticF(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)-100200502440*2^(1/2)*EllipticE(1/11*1
1^(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)*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2
+3*x)^(1/2)-133600669920*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)+448
61443200*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)-89067113280*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]*x^7+14953814400*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)-29689037760*2^(1/2)*EllipticE(1/11*1
1^(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+1993841920*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))-3958538368*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+2254018771062*x^4-22795632684*x^3-608665287387*x^2-266088118
854*x-37568419317)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(11/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((729*x^6 + 2916*x^5 + 4
860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*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)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)