3.2560 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{31030 \sqrt{1-2 x}}{27951 \sqrt{5 x+3}}-\frac{410 \sqrt{1-2 x}}{2541 (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{54 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

[Out]

4/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (410*Sqrt[1 - 2*x])/(2541*(3 + 5*x)^(3/2)
) + (31030*Sqrt[1 - 2*x])/(27951*Sqrt[3 + 5*x]) - (54*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Rubi [A]  time = 0.235072, 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 \[ \frac{31030 \sqrt{1-2 x}}{27951 \sqrt{5 x+3}}-\frac{410 \sqrt{1-2 x}}{2541 (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{54 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

4/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (410*Sqrt[1 - 2*x])/(2541*(3 + 5*x)^(3/2)
) + (31030*Sqrt[1 - 2*x])/(27951*Sqrt[3 + 5*x]) - (54*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])

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Rubi in Sympy [A]  time = 22.1333, size = 94, normalized size = 0.93 \[ \frac{31030 \sqrt{- 2 x + 1}}{27951 \sqrt{5 x + 3}} - \frac{410 \sqrt{- 2 x + 1}}{2541 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{54 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} + \frac{4}{77 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

31030*sqrt(-2*x + 1)/(27951*sqrt(5*x + 3)) - 410*sqrt(-2*x + 1)/(2541*(5*x + 3)*
*(3/2)) - 54*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/49 + 4/(77*s
qrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.135746, size = 70, normalized size = 0.69 \[ -\frac{2 \left (155150 x^2+11005 x-45016\right )}{27951 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{27 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-45016 + 11005*x + 155150*x^2))/(27951*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (27
*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 = 202, normalized size = 2. \[{\frac{1}{-195657+391314\,x}\sqrt{1-2\,x} \left ( 5390550\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+3773385\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-1293732\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2172100\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-970299\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +154070\,x\sqrt{-10\,{x}^{2}-x+3}-630224\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/195657*(1-2*x)^(1/2)*(5390550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))*x^3+3773385*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^2-1293732*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+21721
00*x^2*(-10*x^2-x+3)^(1/2)-970299*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))+154070*x*(-10*x^2-x+3)^(1/2)-630224*(-10*x^2-x+3)^(1/2))/(-1+2*x)/(
-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236475, size = 127, normalized size = 1.26 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (155150 \, x^{2} + 11005 \, x - 45016\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 107811 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{195657 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/195657*sqrt(7)*(2*sqrt(7)*(155150*x^2 + 11005*x - 45016)*sqrt(5*x + 3)*sqrt(-2
*x + 1) + 107811*(50*x^3 + 35*x^2 - 12*x - 9)*arctan(1/14*sqrt(7)*(37*x + 20)/(s
qrt(5*x + 3)*sqrt(-2*x + 1))))/(50*x^3 + 35*x^2 - 12*x - 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260523, size = 297, normalized size = 2.94 \[ -\frac{5}{63888} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{27}{490} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{145}{2662} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{16 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{46585 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/63888*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)))^3 + 27/490*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)))) + 145/2662*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))) - 16/46585*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)