3.2597 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=79 \[ \frac{676 \sqrt{5 x+3}}{17787 \sqrt{1-2 x}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2}}-\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)) + (676*Sqrt[3 + 5*x])/(17787*Sqrt[1 - 2*
x]) - (18*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi [A]  time = 0.166291, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{676 \sqrt{5 x+3}}{17787 \sqrt{1-2 x}}+\frac{4 \sqrt{5 x+3}}{231 (1-2 x)^{3/2}}-\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)) + (676*Sqrt[3 + 5*x])/(17787*Sqrt[1 - 2*
x]) - (18*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi in Sympy [A]  time = 14.7174, size = 73, normalized size = 0.92 \[ - \frac{18 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} + \frac{676 \sqrt{5 x + 3}}{17787 \sqrt{- 2 x + 1}} + \frac{4 \sqrt{5 x + 3}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-18*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/343 + 676*sqrt(5*x +
3)/(17787*sqrt(-2*x + 1)) + 4*sqrt(5*x + 3)/(231*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.141967, size = 65, normalized size = 0.82 \[ \frac{8 (123-169 x) \sqrt{5 x+3}}{17787 (1-2 x)^{3/2}}-\frac{9 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(8*(123 - 169*x)*Sqrt[3 + 5*x])/(17787*(1 - 2*x)^(3/2)) - (9*ArcTan[(-20 - 37*x)
/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Maple [B]  time = 0.022, size = 154, normalized size = 2. \[{\frac{1}{124509\, \left ( -1+2\,x \right ) ^{2}} \left ( 13068\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-13068\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3267\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -9464\,x\sqrt{-10\,{x}^{2}-x+3}+6888\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/124509*(13068*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-1
3068*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+3267*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-9464*x*(-10*x^2-x+3)^(1/2)+688
8*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(5*x + 3)*(3*x + 2)*(-2*x + 1)^(5/2)), x)

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Fricas [A]  time = 0.230043, size = 107, normalized size = 1.35 \[ -\frac{\sqrt{7}{\left (8 \, \sqrt{7}{\left (169 \, x - 123\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3267 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{124509 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/124509*sqrt(7)*(8*sqrt(7)*(169*x - 123)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3267*(
4*x^2 - 4*x + 1)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1)))
)/(4*x^2 - 4*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right ) \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((-2*x + 1)**(5/2)*(3*x + 2)*sqrt(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.260467, size = 153, normalized size = 1.94 \[ \frac{9}{3430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{8 \,{\left (169 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1122 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{444675 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/3430*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)
))) - 8/444675*(169*sqrt(5)*(5*x + 3) - 1122*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1)^2