3.2426 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ \frac{(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt{5 x+3}}+\frac{55 (1-2 x)^{3/2}}{4 (3 x+2) \sqrt{5 x+3}}-\frac{1815 \sqrt{1-2 x}}{4 \sqrt{5 x+3}}+\frac{1815}{4} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-1815*Sqrt[1 - 2*x])/(4*Sqrt[3 + 5*x]) + (1 - 2*x)^(5/2)/(2*(2 + 3*x)^2*Sqrt[3
+ 5*x]) + (55*(1 - 2*x)^(3/2))/(4*(2 + 3*x)*Sqrt[3 + 5*x]) + (1815*Sqrt[7]*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4

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Rubi [A]  time = 0.169128, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt{5 x+3}}+\frac{55 (1-2 x)^{3/2}}{4 (3 x+2) \sqrt{5 x+3}}-\frac{1815 \sqrt{1-2 x}}{4 \sqrt{5 x+3}}+\frac{1815}{4} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1815*Sqrt[1 - 2*x])/(4*Sqrt[3 + 5*x]) + (1 - 2*x)^(5/2)/(2*(2 + 3*x)^2*Sqrt[3
+ 5*x]) + (55*(1 - 2*x)^(3/2))/(4*(2 + 3*x)*Sqrt[3 + 5*x]) + (1815*Sqrt[7]*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4

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Rubi in Sympy [A]  time = 14.4653, size = 109, normalized size = 0.95 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{\left (3 x + 2\right )^{2} \sqrt{5 x + 3}} - \frac{35 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2 \left (3 x + 2\right )^{2}} - \frac{1155 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4 \left (3 x + 2\right )} + \frac{1815 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)/((3*x + 2)**2*sqrt(5*x + 3)) - 35*(-2*x + 1)**(3/2)*sqrt(5*
x + 3)/(2*(3*x + 2)**2) - 1155*sqrt(-2*x + 1)*sqrt(5*x + 3)/(4*(3*x + 2)) + 1815
*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/4

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Mathematica [A]  time = 0.0836055, size = 77, normalized size = 0.67 \[ \frac{1815}{8} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{\sqrt{1-2 x} \left (16657 x^2+21843 x+7148\right )}{4 (3 x+2)^2 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(7148 + 21843*x + 16657*x^2))/(4*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (1
815*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/8

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Maple [B]  time = 0.02, size = 202, normalized size = 1.8 \[ -{\frac{1}{8\, \left ( 2+3\,x \right ) ^{2}} \left ( 81675\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+157905\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+101640\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+33314\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+21780\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +43686\,x\sqrt{-10\,{x}^{2}-x+3}+14296\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(81675*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+15790
5*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+101640*7^(1/2)*
arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+33314*x^2*(-10*x^2-x+3)^(1/
2)+21780*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+43686*x*(-10
*x^2-x+3)^(1/2)+14296*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^2/(-10*x^2-x+3)
^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.51343, size = 193, normalized size = 1.68 \[ -\frac{1815}{8} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{16657 \, x}{18 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{52169}{108 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{54 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{833}{12 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1815/8*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 16657/18*x/s
qrt(-10*x^2 - x + 3) - 52169/108/sqrt(-10*x^2 - x + 3) + 343/54/(9*sqrt(-10*x^2
- x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 833/12/(3
*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.223557, size = 123, normalized size = 1.07 \[ -\frac{1815 \, \sqrt{7}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \,{\left (16657 \, x^{2} + 21843 \, x + 7148\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{8 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(1815*sqrt(7)*(45*x^3 + 87*x^2 + 56*x + 12)*arctan(1/14*sqrt(7)*(37*x + 20)
/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 2*(16657*x^2 + 21843*x + 7148)*sqrt(5*x + 3)*
sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)

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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/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.328858, size = 427, normalized size = 3.71 \[ -\frac{363}{16} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{121}{10} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{847 \,{\left (9 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1960 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{2 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-363/16*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)
*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22
)))) - 121/10*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 847/2*(9*sqrt(10)*((sqrt(2)
*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22)))^3 + 1960*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(
5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22)))^2 + 280)^2