3.2361 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{608185 \sqrt{1-2 x}}{504 \sqrt{5 x+3}}+\frac{13409 \sqrt{1-2 x}}{168 (3 x+2) \sqrt{5 x+3}}+\frac{77 \sqrt{1-2 x}}{12 (3 x+2)^2 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 \sqrt{5 x+3}}+\frac{463881 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

[Out]

(-608185*Sqrt[1 - 2*x])/(504*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*S
qrt[3 + 5*x]) + (77*Sqrt[1 - 2*x])/(12*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (13409*Sqrt[
1 - 2*x])/(168*(2 + 3*x)*Sqrt[3 + 5*x]) + (463881*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*
Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Rubi [A]  time = 0.321034, antiderivative size = 144, 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 \[ -\frac{608185 \sqrt{1-2 x}}{504 \sqrt{5 x+3}}+\frac{13409 \sqrt{1-2 x}}{168 (3 x+2) \sqrt{5 x+3}}+\frac{77 \sqrt{1-2 x}}{12 (3 x+2)^2 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 \sqrt{5 x+3}}+\frac{463881 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

(-608185*Sqrt[1 - 2*x])/(504*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*S
qrt[3 + 5*x]) + (77*Sqrt[1 - 2*x])/(12*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (13409*Sqrt[
1 - 2*x])/(168*(2 + 3*x)*Sqrt[3 + 5*x]) + (463881*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*
Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Rubi in Sympy [A]  time = 28.4812, size = 133, normalized size = 0.92 \[ - \frac{608185 \sqrt{- 2 x + 1}}{504 \sqrt{5 x + 3}} + \frac{13409 \sqrt{- 2 x + 1}}{168 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{77 \sqrt{- 2 x + 1}}{12 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{7 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} + \frac{463881 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{392} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(3/2),x)

[Out]

-608185*sqrt(-2*x + 1)/(504*sqrt(5*x + 3)) + 13409*sqrt(-2*x + 1)/(168*(3*x + 2)
*sqrt(5*x + 3)) + 77*sqrt(-2*x + 1)/(12*(3*x + 2)**2*sqrt(5*x + 3)) + 7*sqrt(-2*
x + 1)/(9*(3*x + 2)**3*sqrt(5*x + 3)) + 463881*sqrt(7)*atan(sqrt(7)*sqrt(-2*x +
1)/(7*sqrt(5*x + 3)))/392

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Mathematica [A]  time = 0.106583, size = 82, normalized size = 0.57 \[ \frac{1}{784} \left (463881 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{14 \sqrt{1-2 x} \left (1824555 x^3+3608883 x^2+2378026 x+521968\right )}{(3 x+2)^3 \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

((-14*Sqrt[1 - 2*x]*(521968 + 2378026*x + 3608883*x^2 + 1824555*x^3))/((2 + 3*x)
^3*Sqrt[3 + 5*x]) + 463881*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3
+ 5*x])])/784

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Maple [B]  time = 0.023, size = 250, normalized size = 1.7 \[ -{\frac{1}{784\, \left ( 2+3\,x \right ) ^{3}} \left ( 62623935\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+50524362\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+7307552\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2),x)

[Out]

-1/784*(62623935*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+
162822231*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+1586473
02*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+25543770*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+50524362*x^2*(-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))[phone]*x*(-10*x^2-x+3)^(1/2)+7307552*(-10*x^2-x+
3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.52729, size = 285, normalized size = 1.98 \[ -\frac{463881}{784} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{608185 \, x}{252 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{635003}{504 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{49}{27 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{1561}{108 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{4367}{24 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

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)^4),x, algorithm="maxima")

[Out]

-463881/784*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 608185/2
52*x/sqrt(-10*x^2 - x + 3) - 635003/504/sqrt(-10*x^2 - x + 3) + 49/27/(27*sqrt(-
10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x
+ 8*sqrt(-10*x^2 - x + 3)) + 1561/108/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10
*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 4367/24/(3*sqrt(-10*x^2 - x + 3)*x
+ 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.222182, size = 147, normalized size = 1.02 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1824555 \, x^{3} + 3608883 \, x^{2} + 2378026 \, x + 521968\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 463881 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{784 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

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)^4),x, algorithm="fricas")

[Out]

-1/784*sqrt(7)*(2*sqrt(7)*(1824555*x^3 + 3608883*x^2 + 2378026*x + 521968)*sqrt(
5*x + 3)*sqrt(-2*x + 1) + 463881*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*arct
an(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(135*x^4 + 351*x^3
+ 342*x^2 + 148*x + 24)

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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)**4/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.354879, size = 509, normalized size = 3.53 \[ -\frac{463881}{7840} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{55}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{11 \,{\left (33989 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 15023680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1769566400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{28 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

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)^4),x, algorithm="giac")

[Out]

-463881/7840*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 55/2*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))) - 11/28*(33989*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^5 + 15023680*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 +
1769566400*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) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280)^3