3.2290 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=93 \[ \frac{3 \sqrt{5 x+3} (1-2 x)^{3/2}}{14 (3 x+2)^2}+\frac{107 \sqrt{5 x+3} \sqrt{1-2 x}}{28 (3 x+2)}-\frac{1177 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

[Out]

(3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (107*Sqrt[1 - 2*x]*Sqrt[3 +
 5*x])/(28*(2 + 3*x)) - (1177*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28
*Sqrt[7])

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Rubi [A]  time = 0.125185, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 \sqrt{5 x+3} (1-2 x)^{3/2}}{14 (3 x+2)^2}+\frac{107 \sqrt{5 x+3} \sqrt{1-2 x}}{28 (3 x+2)}-\frac{1177 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (107*Sqrt[1 - 2*x]*Sqrt[3 +
 5*x])/(28*(2 + 3*x)) - (1177*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28
*Sqrt[7])

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Rubi in Sympy [A]  time = 10.1662, size = 83, normalized size = 0.89 \[ \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{14 \left (3 x + 2\right )^{2}} + \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{28 \left (3 x + 2\right )} - \frac{1177 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{196} \]

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)**(1/2),x)

[Out]

3*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(14*(3*x + 2)**2) + 107*sqrt(-2*x + 1)*sqrt(5*
x + 3)/(28*(3*x + 2)) - 1177*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3
)))/196

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Mathematica [A]  time = 0.0674659, size = 72, normalized size = 0.77 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} (309 x+220)}{28 (3 x+2)^2}-\frac{1177 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(220 + 309*x))/(28*(2 + 3*x)^2) - (1177*ArcTan[(-20
 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Maple [B]  time = 0.019, size = 154, normalized size = 1.7 \[{\frac{1}{392\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 10593\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+14124\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+4708\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +4326\,x\sqrt{-10\,{x}^{2}-x+3}+3080\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/392*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(10593*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^2+14124*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x+4708*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4326*
x*(-10*x^2-x+3)^(1/2)+3080*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 1.49978, size = 103, normalized size = 1.11 \[ \frac{1177}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{2 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{103 \, \sqrt{-10 \, x^{2} - x + 3}}{28 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1177/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/2*sqrt(-1
0*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 103/28*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.228697, size = 107, normalized size = 1.15 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (309 \, x + 220\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1177 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{392 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/392*sqrt(7)*(2*sqrt(7)*(309*x + 220)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 1177*(9*x^
2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(
9*x^2 + 12*x + 4)

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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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283803, size = 347, normalized size = 3.73 \[ \frac{11}{3920} \, \sqrt{5}{\left (107 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{280 \, \sqrt{2}{\left (173 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{29960 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{119840 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/3920*sqrt(5)*(107*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)))) + 280*sqrt(2)*(173*((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 + 29960*(sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 119840*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-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)