3.2550 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{370 \sqrt{1-2 x}}{847 \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

[Out]

4/(77*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (370*Sqrt[1 - 2*x])/(847*Sqrt[3 + 5*x]) + (
18*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Rubi [A]  time = 0.165133, 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 \[ -\frac{370 \sqrt{1-2 x}}{847 \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

4/(77*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (370*Sqrt[1 - 2*x])/(847*Sqrt[3 + 5*x]) + (
18*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Rubi in Sympy [A]  time = 15.1956, size = 73, normalized size = 0.92 \[ - \frac{370 \sqrt{- 2 x + 1}}{847 \sqrt{5 x + 3}} + \frac{18 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} + \frac{4}{77 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-370*sqrt(-2*x + 1)/(847*sqrt(5*x + 3)) + 18*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)
/(7*sqrt(5*x + 3)))/49 + 4/(77*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.113767, size = 65, normalized size = 0.82 \[ \frac{2 (370 x-163)}{847 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{9 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-163 + 370*x))/(847*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (9*ArcTan[(-20 - 37*x)/(2
*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Maple [B]  time = 0.023, size = 154, normalized size = 2. \[ -{\frac{1}{-5929+11858\,x}\sqrt{1-2\,x} \left ( 10890\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1089\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-3267\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +5180\,x\sqrt{-10\,{x}^{2}-x+3}-2282\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5929*(1-2*x)^(1/2)*(10890*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))*x^2+1089*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-32
67*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+5180*x*(-10*x^2-x+
3)^(1/2)-2282*(-10*x^2-x+3)^(1/2))/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49764, size = 78, normalized size = 0.99 \[ -\frac{9}{49} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{740 \, x}{847 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{326}{847 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/49*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 740/847*x/sqrt
(-10*x^2 - x + 3) - 326/847/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.228698, size = 101, normalized size = 1.28 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (370 \, x - 163\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1089 \,{\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{5929 \,{\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5929*sqrt(7)*(2*sqrt(7)*(370*x - 163)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 1089*(10
*x^2 + x - 3)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(
10*x^2 + x - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.250956, size = 215, normalized size = 2.72 \[ -\frac{9}{490} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{5}{242} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{8 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4235 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/490*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)
))) - 5/242*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 8/4235*sqrt(5)*sqrt(5*x + 3)*
sqrt(-10*x + 5)/(2*x - 1)