3.2607 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{42230 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}+\frac{956}{17787 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{54 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 956/(17787*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
- (42230*Sqrt[1 - 2*x])/(195657*Sqrt[3 + 5*x]) + (54*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi [A]  time = 0.242879, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{42230 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}+\frac{956}{17787 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{54 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

4/(231*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 956/(17787*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
- (42230*Sqrt[1 - 2*x])/(195657*Sqrt[3 + 5*x]) + (54*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi in Sympy [A]  time = 21.6534, size = 94, normalized size = 0.93 \[ \frac{54 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} + \frac{16892 \sqrt{5 x + 3}}{195657 \sqrt{- 2 x + 1}} - \frac{1070}{2541 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

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)**(3/2),x)

[Out]

54*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/343 + 16892*sqrt(5*x +
3)/(195657*sqrt(-2*x + 1)) - 1070/(2541*sqrt(-2*x + 1)*sqrt(5*x + 3)) + 4/(231*
(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.158569, size = 70, normalized size = 0.69 \[ \frac{27 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{49 \sqrt{7}}-\frac{2 \left (84460 x^2-73944 x+14163\right )}{195657 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(14163 - 73944*x + 84460*x^2))/(195657*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (27*
ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Maple [B]  time = 0.023, size = 202, normalized size = 2. \[ -{\frac{1}{1369599\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 2156220\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-862488\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-754677\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1182440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+323433\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1035216\,x\sqrt{-10\,{x}^{2}-x+3}+198282\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1369599*(1-2*x)^(1/2)*(2156220*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^3-862488*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^2-754677*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+11824
40*x^2*(-10*x^2-x+3)^(1/2)+323433*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))-1035216*x*(-10*x^2-x+3)^(1/2)+198282*(-10*x^2-x+3)^(1/2))/(-1+2*x)^
2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227456, size = 127, normalized size = 1.26 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (84460 \, x^{2} - 73944 \, x + 14163\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 107811 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1369599 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1369599*sqrt(7)*(2*sqrt(7)*(84460*x^2 - 73944*x + 14163)*sqrt(5*x + 3)*sqrt(-
2*x + 1) + 107811*(20*x^3 - 8*x^2 - 7*x + 3)*arctan(1/14*sqrt(7)*(37*x + 20)/(sq
rt(5*x + 3)*sqrt(-2*x + 1))))/(20*x^3 - 8*x^2 - 7*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266328, size = 232, normalized size = 2.3 \[ -\frac{27}{3430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{25}{2662} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{8 \,{\left (548 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3399 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4891425 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/3430*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(2
2)))) - 25/2662*sqrt(10)*((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))) - 8/4891425*(548*sqrt(5)*(5
*x + 3) - 3399*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2