3.2360 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ \frac{3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt{5 x+3}}+\frac{173 (1-2 x)^{3/2}}{28 (3 x+2) \sqrt{5 x+3}}-\frac{5709 \sqrt{1-2 x}}{28 \sqrt{5 x+3}}+\frac{5709 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]

[Out]

(-5709*Sqrt[1 - 2*x])/(28*Sqrt[3 + 5*x]) + (3*(1 - 2*x)^(5/2))/(14*(2 + 3*x)^2*S
qrt[3 + 5*x]) + (173*(1 - 2*x)^(3/2))/(28*(2 + 3*x)*Sqrt[3 + 5*x]) + (5709*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4*Sqrt[7])

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Rubi [A]  time = 0.172553, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 (1-2 x)^{5/2}}{14 (3 x+2)^2 \sqrt{5 x+3}}+\frac{173 (1-2 x)^{3/2}}{28 (3 x+2) \sqrt{5 x+3}}-\frac{5709 \sqrt{1-2 x}}{28 \sqrt{5 x+3}}+\frac{5709 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-5709*Sqrt[1 - 2*x])/(28*Sqrt[3 + 5*x]) + (3*(1 - 2*x)^(5/2))/(14*(2 + 3*x)^2*S
qrt[3 + 5*x]) + (173*(1 - 2*x)^(3/2))/(28*(2 + 3*x)*Sqrt[3 + 5*x]) + (5709*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4*Sqrt[7])

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Rubi in Sympy [A]  time = 13.3027, size = 110, normalized size = 0.96 \[ - \frac{10 \left (- 2 x + 1\right )^{\frac{5}{2}}}{11 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} - \frac{173 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{22 \left (3 x + 2\right )^{2}} - \frac{519 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4 \left (3 x + 2\right )} + \frac{5709 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*(-2*x + 1)**(5/2)/(11*(3*x + 2)**2*sqrt(5*x + 3)) - 173*(-2*x + 1)**(3/2)*sq
rt(5*x + 3)/(22*(3*x + 2)**2) - 519*sqrt(-2*x + 1)*sqrt(5*x + 3)/(4*(3*x + 2)) +
5709*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/28

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Mathematica [A]  time = 0.0878533, size = 77, normalized size = 0.67 \[ \frac{5709 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{8 \sqrt{7}}-\frac{\sqrt{1-2 x} \left (7485 x^2+9815 x+3212\right )}{4 (3 x+2)^2 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(3212 + 9815*x + 7485*x^2))/(4*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (570
9*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

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Maple [B]  time = 0.02, size = 202, normalized size = 1.8 \[ -{\frac{1}{56\, \left ( 2+3\,x \right ) ^{2}} \left ( 256905\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+496683\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+319704\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+104790\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+68508\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +137410\,x\sqrt{-10\,{x}^{2}-x+3}+44968\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/56*(256905*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+496
683*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+319704*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+104790*x^2*(-10*x^2-x+3)^
(1/2)+68508*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+137410*x*
(-10*x^2-x+3)^(1/2)+44968*(-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 = 1.50959, size = 193, normalized size = 1.68 \[ -\frac{5709}{56} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2495 \, x}{6 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2605}{12 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{49}{18 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{1127}{36 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5709/56*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2495/6*x/sq
rt(-10*x^2 - x + 3) - 2605/12/sqrt(-10*x^2 - x + 3) + 49/18/(9*sqrt(-10*x^2 - x
+ 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 1127/36/(3*sq
rt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.219877, size = 127, normalized size = 1.1 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (7485 \, x^{2} + 9815 \, x + 3212\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 5709 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{56 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/56*sqrt(7)*(2*sqrt(7)*(7485*x^2 + 9815*x + 3212)*sqrt(5*x + 3)*sqrt(-2*x + 1)
+ 5709*(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))))/(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)**(3/2)/(2+3*x)**3/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.314501, size = 427, normalized size = 3.71 \[ -\frac{5709}{560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{55 \,{\left (61 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 13384 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{2 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5709/560*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)))) - 11/2*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))) - 55/2*(61*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 + 13384*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)*sq
rt(-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