3.2781 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{32}{175} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1972 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{4375}+\frac{106772 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{65625}-\frac{110014 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125}+\frac{53279 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2))/(5*Sqrt[3 + 5*x]) + (106772*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*Sqrt[3 + 5*x])/65625 - (1972*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/4375 - (32*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/175 + (53279*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/328125 - (110014*Sqr
t[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/328125

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Rubi [A]  time = 0.404995, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{32}{175} (3 x+2)^{3/2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{1972 (3 x+2)^{3/2} \sqrt{5 x+3} \sqrt{1-2 x}}{4375}+\frac{106772 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{65625}-\frac{110014 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125}+\frac{53279 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{328125} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2))/(5*Sqrt[3 + 5*x]) + (106772*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*Sqrt[3 + 5*x])/65625 - (1972*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 +
5*x])/4375 - (32*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/175 + (53279*Sq
rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/328125 - (110014*Sqr
t[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/328125

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Rubi in Sympy [A]  time = 42.2002, size = 172, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}}}{5 \sqrt{5 x + 3}} - \frac{32 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{175} + \frac{2958 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{4375} + \frac{3242 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{65625} + \frac{53279 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{984375} - \frac{1210154 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{11484375} \]

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)**(3/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**(3/2)/(5*sqrt(5*x + 3)) - 32*(-2*x + 1)**(3/2)*(
3*x + 2)**(3/2)*sqrt(5*x + 3)/175 + 2958*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*
x + 3)/4375 + 3242*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/65625 + 53279*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/984375 - 1210154*sqrt(35
)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/11484375

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Mathematica [A]  time = 0.434841, size = 107, normalized size = 0.56 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (22500 x^3-31350 x^2+9545 x+9168\right )}{\sqrt{5 x+3}}+1868510 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-53279 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{984375} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(9168 + 9545*x - 31350*x^2 + 22500*x^3))/Sqrt[3
+ 5*x] - 53279*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 186
8510*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/984375

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Maple [C]  time = 0.026, size = 174, normalized size = 0.9 \[ -{\frac{1}{29531250\,{x}^{3}[phone]\,{x}^{2}-6890625\,x-5906250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1868510\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -53279\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4050000\,{x}^{5}+4968000\,{x}^{4}+572400\,{x}^{3}-3817590\,{x}^{2}+297660\,x+550080 \right ) } \]

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)^(3/2),x)

[Out]

-1/984375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1868510*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))-53279*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
))-4050000*x^5+4968000*x^4+572400*x^3-3817590*x^2+297660*x+550080)/(30*x^3+23*x^
2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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