3.2368 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=104 \[ \frac{3 (1-2 x)^{5/2}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac{169 (1-2 x)^{3/2}}{21 (5 x+3)^{3/2}}+\frac{169 \sqrt{1-2 x}}{\sqrt{5 x+3}}-169 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-169*(1 - 2*x)^(3/2))/(21*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(5/2))/(7*(2 + 3*x)*(
3 + 5*x)^(3/2)) + (169*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] - 169*Sqrt[7]*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi [A]  time = 0.163692, antiderivative size = 104, 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 \[ \frac{3 (1-2 x)^{5/2}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac{169 (1-2 x)^{3/2}}{21 (5 x+3)^{3/2}}+\frac{169 \sqrt{1-2 x}}{\sqrt{5 x+3}}-169 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-169*(1 - 2*x)^(3/2))/(21*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(5/2))/(7*(2 + 3*x)*(
3 + 5*x)^(3/2)) + (169*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] - 169*Sqrt[7]*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi in Sympy [A]  time = 13.0054, size = 105, normalized size = 1.01 \[ - \frac{10 \left (- 2 x + 1\right )^{\frac{5}{2}}}{33 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{338 \left (- 2 x + 1\right )^{\frac{3}{2}}}{33 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{1183 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{11 \left (3 x + 2\right )} - 169 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*(-2*x + 1)**(5/2)/(33*(3*x + 2)*(5*x + 3)**(3/2)) + 338*(-2*x + 1)**(3/2)/(3
3*(3*x + 2)*sqrt(5*x + 3)) + 1183*sqrt(-2*x + 1)*sqrt(5*x + 3)/(11*(3*x + 2)) -
169*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))

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Mathematica [A]  time = 0.0904842, size = 77, normalized size = 0.74 \[ \frac{\sqrt{1-2 x} \left (7755 x^2+9652 x+2995\right )}{3 (3 x+2) (5 x+3)^{3/2}}-\frac{169}{2} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(2995 + 9652*x + 7755*x^2))/(3*(2 + 3*x)*(3 + 5*x)^(3/2)) - (169*
Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/2

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Maple [B]  time = 0.021, size = 202, normalized size = 1.9 \[{\frac{1}{12+18\,x} \left ( 38025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+70980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+44109\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+15510\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+9126\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +19304\,x\sqrt{-10\,{x}^{2}-x+3}+5990\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(38025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+70980*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+44109*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+15510*x^2*(-10*x^2-x+3)^(1/2)+
9126*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+19304*x*(-10*x^2
-x+3)^(1/2)+5990*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)/(-10*x^2-x+3)^(1/2)/
(3+5*x)^(3/2)

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Maxima [A]  time = 1.5047, size = 163, normalized size = 1.57 \[ \frac{169}{2} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1034 \, x}{3 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2699}{15 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3902 \, x}{45 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{343}{27 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{6343}{135 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

169/2*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1034/3*x/sqrt(
-10*x^2 - x + 3) + 2699/15/sqrt(-10*x^2 - x + 3) + 3902/45*x/(-10*x^2 - x + 3)^(
3/2) + 343/27/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 6343/1
35/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.219472, size = 123, normalized size = 1.18 \[ \frac{507 \, \sqrt{7}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \,{\left (7755 \, x^{2} + 9652 \, x + 2995\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(507*sqrt(7)*(75*x^3 + 140*x^2 + 87*x + 18)*arctan(1/14*sqrt(7)*(37*x + 20)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 2*(7755*x^2 + 9652*x + 2995)*sqrt(5*x + 3)*sqr
t(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.323356, size = 423, normalized size = 4.07 \[ -\frac{1}{240} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{169}{20} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{34}{5} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{462 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/240*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 + 169/20*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)))) + 34/5*sqrt(10)*((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))) + 462*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)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))^2 + 280)