3.2300 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{5 x+3}}+\frac{14101 \sqrt{1-2 x}}{392 (3 x+2) \sqrt{5 x+3}}+\frac{81 \sqrt{1-2 x}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 \sqrt{5 x+3}}+\frac{1463447 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(-639565*Sqrt[1 - 2*x])/(1176*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*Sqrt
[3 + 5*x]) + (81*Sqrt[1 - 2*x])/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (14101*Sqrt[1 -
2*x])/(392*(2 + 3*x)*Sqrt[3 + 5*x]) + (1463447*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq
rt[3 + 5*x])])/(392*Sqrt[7])

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Rubi [A]  time = 0.309476, 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 \[ -\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{5 x+3}}+\frac{14101 \sqrt{1-2 x}}{392 (3 x+2) \sqrt{5 x+3}}+\frac{81 \sqrt{1-2 x}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 \sqrt{5 x+3}}+\frac{1463447 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-639565*Sqrt[1 - 2*x])/(1176*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*Sqrt
[3 + 5*x]) + (81*Sqrt[1 - 2*x])/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (14101*Sqrt[1 -
2*x])/(392*(2 + 3*x)*Sqrt[3 + 5*x]) + (1463447*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq
rt[3 + 5*x])])/(392*Sqrt[7])

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Rubi in Sympy [A]  time = 28.877, size = 131, normalized size = 0.91 \[ - \frac{639565 \sqrt{- 2 x + 1}}{1176 \sqrt{5 x + 3}} + \frac{14101 \sqrt{- 2 x + 1}}{392 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{81 \sqrt{- 2 x + 1}}{28 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{\sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} + \frac{1463447 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2744} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-639565*sqrt(-2*x + 1)/(1176*sqrt(5*x + 3)) + 14101*sqrt(-2*x + 1)/(392*(3*x + 2
)*sqrt(5*x + 3)) + 81*sqrt(-2*x + 1)/(28*(3*x + 2)**2*sqrt(5*x + 3)) + sqrt(-2*x
+ 1)/(3*(3*x + 2)**3*sqrt(5*x + 3)) + 1463447*sqrt(7)*atan(sqrt(7)*sqrt(-2*x +
1)/(7*sqrt(5*x + 3)))/2744

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Mathematica [A]  time = 0.105584, size = 82, normalized size = 0.57 \[ \frac{1463447 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{14 \sqrt{1-2 x} \left (5756085 x^3+11385261 x^2+7502166 x+1646704\right )}{(3 x+2)^3 \sqrt{5 x+3}}}{5488} \]

Antiderivative was successfully verified.

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

[Out]

((-14*Sqrt[1 - 2*x]*(1646704 + 7502166*x + 11385261*x^2 + 5756085*x^3))/((2 + 3*
x)^3*Sqrt[3 + 5*x]) + 1463447*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt
[3 + 5*x])])/5488

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Maple [B]  time = 0.022, size = 250, normalized size = 1.7 \[ -{\frac{1}{5488\, \left ( 2+3\,x \right ) ^{3}} \left ( 197565345\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+159393654\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5488*(197565345*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4+513669897*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+50049
8874*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+80585190*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+159393654*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)[phone]*(-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 = 1.51741, size = 285, normalized size = 1.98 \[ -\frac{1463447}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{639565 \, x}{588 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{222589}{392 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{7}{9 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{235}{36 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{13777}{168 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^(3/2)*(3*x + 2)^4),x, algorithm="maxima")

[Out]

-1463447/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 639565
/588*x/sqrt(-10*x^2 - x + 3) - 222589/392/sqrt(-10*x^2 - x + 3) + 7/9/(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)) + 235/36/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x
^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 13777/168/(3*sqrt(-10*x^2 - x + 3)*x
+ 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.224344, size = 147, normalized size = 1.02 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (5756085 \, x^{3} + 11385261 \, x^{2} + 7502166 \, x + 1646704\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1463447 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{5488 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^(3/2)*(3*x + 2)^4),x, algorithm="fricas")

[Out]

-1/5488*sqrt(7)*(2*sqrt(7)*(5756085*x^3 + 11385261*x^2 + 7502166*x + 1646704)*sq
rt(5*x + 3)*sqrt(-2*x + 1) + 1463447*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*
arctan(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)**(1/2)/(2+3*x)**4/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.346987, size = 505, normalized size = 3.51 \[ -\frac{1}{54880} \, \sqrt{5}{\left (1463447 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + 686000 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{27720 \, \sqrt{2}{\left (11747 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 5216960 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{615675200 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{2462700800 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-2*x + 1)/((5*x + 3)^(3/2)*(3*x + 2)^4),x, algorithm="giac")

[Out]

-1/54880*sqrt(5)*(1463447*sqrt(70)*sqrt(2)*(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)))) + 686000*sqrt(2)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 27720*sqrt(
2)*(11747*((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)))^5 + 5216960*((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)))^3
+ 615675200*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 2462700800*sqrt
(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 +
280)^3)