3.2369 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{17825 \sqrt{1-2 x}}{12 \sqrt{5 x+3}}-\frac{655 \sqrt{1-2 x}}{4 (5 x+3)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (3 x+2) (5 x+3)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{40787 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]

[Out]

(-655*Sqrt[1 - 2*x])/(4*(3 + 5*x)^(3/2)) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 +
5*x)^(3/2)) + (235*Sqrt[1 - 2*x])/(12*(2 + 3*x)*(3 + 5*x)^(3/2)) + (17825*Sqrt[
1 - 2*x])/(12*Sqrt[3 + 5*x]) - (40787*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x
])])/(4*Sqrt[7])

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Rubi [A]  time = 0.329656, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{17825 \sqrt{1-2 x}}{12 \sqrt{5 x+3}}-\frac{655 \sqrt{1-2 x}}{4 (5 x+3)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (3 x+2) (5 x+3)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{40787 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-655*Sqrt[1 - 2*x])/(4*(3 + 5*x)^(3/2)) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 +
5*x)^(3/2)) + (235*Sqrt[1 - 2*x])/(12*(2 + 3*x)*(3 + 5*x)^(3/2)) + (17825*Sqrt[
1 - 2*x])/(12*Sqrt[3 + 5*x]) - (40787*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x
])])/(4*Sqrt[7])

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Rubi in Sympy [A]  time = 28.1091, size = 126, normalized size = 0.92 \[ \frac{17825 \sqrt{- 2 x + 1}}{12 \sqrt{5 x + 3}} - \frac{655 \sqrt{- 2 x + 1}}{4 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{235 \sqrt{- 2 x + 1}}{12 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \sqrt{- 2 x + 1}}{6 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{40787 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{28} \]

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)**(5/2),x)

[Out]

17825*sqrt(-2*x + 1)/(12*sqrt(5*x + 3)) - 655*sqrt(-2*x + 1)/(4*(5*x + 3)**(3/2)
) + 235*sqrt(-2*x + 1)/(12*(3*x + 2)*(5*x + 3)**(3/2)) + 7*sqrt(-2*x + 1)/(6*(3*
x + 2)**2*(5*x + 3)**(3/2)) - 40787*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(
5*x + 3)))/28

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Mathematica [A]  time = 0.10272, size = 82, normalized size = 0.6 \[ \frac{\sqrt{1-2 x} \left (802125 x^3+1533090 x^2+975325 x+206524\right )}{12 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{40787 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(206524 + 975325*x + 1533090*x^2 + 802125*x^3))/(12*(2 + 3*x)^2*(
3 + 5*x)^(3/2)) - (40787*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/
(8*Sqrt[7])

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Maple [B]  time = 0.021, size = 250, normalized size = 1.8 \[{\frac{1}{168\, \left ( 2+3\,x \right ) ^{2}} \left ( 27531225\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+21463260\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4404996\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+2891336\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*(27531225*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+6
9745770*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+66197301*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+11229750*x^3*(-10
*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x+21463260*x^2*(-10*x^2-x+3)^(1/2)+4404996*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)+2891336*(-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)^(3/2)

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Maxima [A]  time = 1.49418, size = 232, normalized size = 1.69 \[ \frac{40787}{56} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{17825 \, x}{6 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{18611}{12 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{13439 \, x}{18 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{343}{54 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{11123}{108 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{1613}{4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

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)^3),x, algorithm="maxima")

[Out]

40787/56*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 17825/6*x/s
qrt(-10*x^2 - x + 3) + 18611/12/sqrt(-10*x^2 - x + 3) + 13439/18*x/(-10*x^2 - x
+ 3)^(3/2) + 343/54/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*
x + 4*(-10*x^2 - x + 3)^(3/2)) + 11123/108/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10
*x^2 - x + 3)^(3/2)) - 1613/4/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.222908, size = 147, normalized size = 1.07 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (802125 \, x^{3} + 1533090 \, x^{2} + 975325 \, x + 206524\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 122361 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{168 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

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)^3),x, algorithm="fricas")

[Out]

1/168*sqrt(7)*(2*sqrt(7)*(802125*x^3 + 1533090*x^2 + 975325*x + 206524)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) + 122361*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(
1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(225*x^4 + 570*x^3 + 5
41*x^2 + 228*x + 36)

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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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.352459, size = 509, normalized size = 3.72 \[ -\frac{1}{48} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{40787}{560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{101}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{165 \,{\left (89 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 21224 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{2 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*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 + 40787/560*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)))) + 101/2*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))) + 165/2*(89*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 21224*sq
rt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sq
rt(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) - sqrt(22)))^2 + 280)^2