3.2790 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{284}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{22866 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{32836 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}+\frac{49321 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (62*(1 - 2*x)^(3/2)*
(2 + 3*x)^(5/2))/(15*Sqrt[3 + 5*x]) + (33778*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/21875 + (22866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/4375 - (284*
Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 + (49321*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375 - (32836*Sqrt[11/3]*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375

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Rubi [A]  time = 0.485122, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{284}{175} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{22866 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{4375}+\frac{33778 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{21875}-\frac{32836 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375}+\frac{49321 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (62*(1 - 2*x)^(3/2)*
(2 + 3*x)^(5/2))/(15*Sqrt[3 + 5*x]) + (33778*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x])/21875 + (22866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/4375 - (284*
Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 + (49321*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375 - (32836*Sqrt[11/3]*EllipticF[Arc
Sin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/109375

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Rubi in Sympy [A]  time = 49.4968, size = 201, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{62 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{3}{2}}}{165 \sqrt{5 x + 3}} - \frac{1132 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{5775} + \frac{2976 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{4375} - \frac{942 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{21875} + \frac{49321 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{328125} - \frac{361196 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{3828125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**(5/2)/(15*(5*x + 3)**(3/2)) - 62*(-2*x + 1)**(5/
2)*(3*x + 2)**(3/2)/(165*sqrt(5*x + 3)) - 1132*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2
)*sqrt(5*x + 3)/5775 + 2976*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3)/4375 -
942*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/21875 + 49321*sqrt(33)*elliptic_
e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/328125 - 361196*sqrt(35)*elliptic_f(as
in(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/3828125

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Mathematica [A]  time = 0.441126, size = 112, normalized size = 0.5 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (67500 x^4-47250 x^3-41025 x^2-23425 x-19087\right )}{(5 x+3)^{3/2}}+591115 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-49321 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{328125} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-19087 - 23425*x - 41025*x^2 - 47250*x^3 + 675
00*x^4))/(3 + 5*x)^(3/2) - 49321*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2] + 591115*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])
/328125

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Maple [C]  time = 0.029, size = 282, normalized size = 1.3 \[ -{\frac{1}{1968750\,{x}^{2}+328125\,x-656250} \left ( 2955575\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-246605\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4050000\,{x}^{6}+1773345\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -147963\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2160000\,{x}^{5}+4284000\,{x}^{4}+870750\,{x}^{3}+558970\,{x}^{2}-277630\,x-381740 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/328125*(2955575*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*1
1^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-246605*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+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-4050000*x^6+1773345*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))-147963*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))+2160000*x^5+4284000*x^4+870750*x^3+558970*x^2-277630*x-381740)*(2+3*x)^
(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*sqrt(3*x + 2)*sqrt(-2*x + 1)/((25*
x^2 + 30*x + 9)*sqrt(5*x + 3)), 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)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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