3.2308 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{608185 \sqrt{1-2 x}}{924 \sqrt{5 x+3}}-\frac{6095 \sqrt{1-2 x}}{84 (5 x+3)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

[Out]

(-6095*Sqrt[1 - 2*x])/(84*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5
*x)^(3/2)) + (243*Sqrt[1 - 2*x])/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (608185*Sqrt[1
- 2*x])/(924*Sqrt[3 + 5*x]) - (126513*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(28*Sqrt[7])

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Rubi [A]  time = 0.312318, 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 \[ \frac{608185 \sqrt{1-2 x}}{924 \sqrt{5 x+3}}-\frac{6095 \sqrt{1-2 x}}{84 (5 x+3)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-6095*Sqrt[1 - 2*x])/(84*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5
*x)^(3/2)) + (243*Sqrt[1 - 2*x])/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (608185*Sqrt[1
- 2*x])/(924*Sqrt[3 + 5*x]) - (126513*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(28*Sqrt[7])

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Rubi in Sympy [A]  time = 28.9184, size = 124, normalized size = 0.91 \[ \frac{608185 \sqrt{- 2 x + 1}}{924 \sqrt{5 x + 3}} - \frac{6095 \sqrt{- 2 x + 1}}{84 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{243 \sqrt{- 2 x + 1}}{28 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{\sqrt{- 2 x + 1}}{2 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{126513 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{196} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

608185*sqrt(-2*x + 1)/(924*sqrt(5*x + 3)) - 6095*sqrt(-2*x + 1)/(84*(5*x + 3)**(
3/2)) + 243*sqrt(-2*x + 1)/(28*(3*x + 2)*(5*x + 3)**(3/2)) + sqrt(-2*x + 1)/(2*(
3*x + 2)**2*(5*x + 3)**(3/2)) - 126513*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sq
rt(5*x + 3)))/196

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Mathematica [A]  time = 0.10392, size = 82, normalized size = 0.6 \[ \frac{\sqrt{1-2 x} \left (27368325 x^3+52308690 x^2+33277877 x+7046540\right )}{924 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(7046540 + 33277877*x + 52308690*x^2 + 27368325*x^3))/(924*(2 + 3
*x)^2*(3 + 5*x)^(3/2)) - (126513*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 +
5*x])])/(56*Sqrt[7])

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Maple [B]  time = 0.022, size = 250, normalized size = 1.8 \[{\frac{1}{12936\, \left ( 2+3\,x \right ) ^{2}} \left ( 939359025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+732321660\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12936*(939359025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4+2379709530*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2258
636589*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+383156550*
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+732321660*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)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.51463, size = 232, normalized size = 1.69 \[ \frac{126513}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{608185 \, x}{462 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{635003}{924 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1985 \, x}{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{49}{18 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1645}{36 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{6433}{36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

126513/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 608185/46
2*x/sqrt(-10*x^2 - x + 3) + 635003/924/sqrt(-10*x^2 - x + 3) + 1985/6*x/(-10*x^2
- x + 3)^(3/2) + 49/18/(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)) + 1645/36/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-
10*x^2 - x + 3)^(3/2)) - 6433/36/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.221316, size = 147, normalized size = 1.07 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (27368325 \, x^{3} + 52308690 \, x^{2} + 33277877 \, x + 7046540\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4174929 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{12936 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12936*sqrt(7)*(2*sqrt(7)*(27368325*x^3 + 52308690*x^2 + 33277877*x + 7046540)*
sqrt(5*x + 3)*sqrt(-2*x + 1) + 4174929*(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 + 57
0*x^3 + 541*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)**(1/2)/(2+3*x)**3/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.364322, size = 509, normalized size = 3.72 \[ -\frac{1}{129360} \, \sqrt{5}{\left (1225 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 4174929 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 2910600 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{2744280 \, \sqrt{2}{\left (151 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{36120 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{144480 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/129360*sqrt(5)*(1225*sqrt(2)*((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 - 4174929*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)))) - 2910600*sq
rt(2)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))) - 2744280*sqrt(2)*(151*((sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2)))^3 + 36120*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 144480*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) - sqrt(22)))^2 +
280)^2)