3.30 \(\int \frac{1+a x}{x^5 \sqrt{a x} \sqrt{1-a x}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{544 a^4 \sqrt{1-a x}}{315 \sqrt{a x}}-\frac{272 a^4 \sqrt{1-a x}}{315 (a x)^{3/2}}-\frac{68 a^4 \sqrt{1-a x}}{105 (a x)^{5/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}} \]

[Out]

(-2*a^4*Sqrt[1 - a*x])/(9*(a*x)^(9/2)) - (34*a^4*Sqrt[1 - a*x])/(63*(a*x)^(7/2))
- (68*a^4*Sqrt[1 - a*x])/(105*(a*x)^(5/2)) - (272*a^4*Sqrt[1 - a*x])/(315*(a*x)
^(3/2)) - (544*a^4*Sqrt[1 - a*x])/(315*Sqrt[a*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.146977, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{544 a^4 \sqrt{1-a x}}{315 \sqrt{a x}}-\frac{272 a^4 \sqrt{1-a x}}{315 (a x)^{3/2}}-\frac{68 a^4 \sqrt{1-a x}}{105 (a x)^{5/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^4*Sqrt[1 - a*x])/(9*(a*x)^(9/2)) - (34*a^4*Sqrt[1 - a*x])/(63*(a*x)^(7/2))
- (68*a^4*Sqrt[1 - a*x])/(105*(a*x)^(5/2)) - (272*a^4*Sqrt[1 - a*x])/(315*(a*x)
^(3/2)) - (544*a^4*Sqrt[1 - a*x])/(315*Sqrt[a*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.6351, size = 110, normalized size = 0.91 \[ - \frac{544 a^{4} \sqrt{- a x + 1}}{315 \sqrt{a x}} - \frac{272 a^{4} \sqrt{- a x + 1}}{315 \left (a x\right )^{\frac{3}{2}}} - \frac{68 a^{4} \sqrt{- a x + 1}}{105 \left (a x\right )^{\frac{5}{2}}} - \frac{34 a^{4} \sqrt{- a x + 1}}{63 \left (a x\right )^{\frac{7}{2}}} - \frac{2 a^{4} \sqrt{- a x + 1}}{9 \left (a x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-544*a**4*sqrt(-a*x + 1)/(315*sqrt(a*x)) - 272*a**4*sqrt(-a*x + 1)/(315*(a*x)**(
3/2)) - 68*a**4*sqrt(-a*x + 1)/(105*(a*x)**(5/2)) - 34*a**4*sqrt(-a*x + 1)/(63*(
a*x)**(7/2)) - 2*a**4*sqrt(-a*x + 1)/(9*(a*x)**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0451752, size = 53, normalized size = 0.44 \[ -\frac{2 \sqrt{-a x (a x-1)} \left (272 a^4 x^4+136 a^3 x^3+102 a^2 x^2+85 a x+35\right )}{315 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-(a*x*(-1 + a*x))]*(35 + 85*a*x + 102*a^2*x^2 + 136*a^3*x^3 + 272*a^4*x
^4))/(315*a*x^5)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 49, normalized size = 0.4 \[ -{\frac{544\,{a}^{4}{x}^{4}+272\,{a}^{3}{x}^{3}+204\,{a}^{2}{x}^{2}+170\,ax+70}{315\,{x}^{4}}\sqrt{-ax+1}{\frac{1}{\sqrt{ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(272*a^4*x^4+136*a^3*x^3+102*a^2*x^2+85*a*x+35)/x^4/(a*x)^(1/2)*(-a*x+1)^
(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.223, size = 69, normalized size = 0.57 \[ -\frac{2 \,{\left (272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right )} \sqrt{a x} \sqrt{-a x + 1}}{315 \, a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(272*a^4*x^4 + 136*a^3*x^3 + 102*a^2*x^2 + 85*a*x + 35)*sqrt(a*x)*sqrt(-a
*x + 1)/(a*x^5)

_______________________________________________________________________________________

Sympy [A]  time = 150.567, size = 359, normalized size = 2.97 \[ a \left (\begin{cases} - \frac{32 a^{3} \sqrt{-1 + \frac{1}{a x}}}{35} - \frac{16 a^{2} \sqrt{-1 + \frac{1}{a x}}}{35 x} - \frac{12 a \sqrt{-1 + \frac{1}{a x}}}{35 x^{2}} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{7 x^{3}} & \text{for}\: \left |{\frac{1}{a x}}\right | > 1 \\- \frac{32 i a^{3} \sqrt{1 - \frac{1}{a x}}}{35} - \frac{16 i a^{2} \sqrt{1 - \frac{1}{a x}}}{35 x} - \frac{12 i a \sqrt{1 - \frac{1}{a x}}}{35 x^{2}} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{7 x^{3}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{256 a^{4} \sqrt{-1 + \frac{1}{a x}}}{315} - \frac{128 a^{3} \sqrt{-1 + \frac{1}{a x}}}{315 x} - \frac{32 a^{2} \sqrt{-1 + \frac{1}{a x}}}{105 x^{2}} - \frac{16 a \sqrt{-1 + \frac{1}{a x}}}{63 x^{3}} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{9 x^{4}} & \text{for}\: \left |{\frac{1}{a x}}\right | > 1 \\- \frac{256 i a^{4} \sqrt{1 - \frac{1}{a x}}}{315} - \frac{128 i a^{3} \sqrt{1 - \frac{1}{a x}}}{315 x} - \frac{32 i a^{2} \sqrt{1 - \frac{1}{a x}}}{105 x^{2}} - \frac{16 i a \sqrt{1 - \frac{1}{a x}}}{63 x^{3}} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{9 x^{4}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Piecewise((-32*a**3*sqrt(-1 + 1/(a*x))/35 - 16*a**2*sqrt(-1 + 1/(a*x))/(35*x)
- 12*a*sqrt(-1 + 1/(a*x))/(35*x**2) - 2*sqrt(-1 + 1/(a*x))/(7*x**3), Abs(1/(a*x)
) > 1), (-32*I*a**3*sqrt(1 - 1/(a*x))/35 - 16*I*a**2*sqrt(1 - 1/(a*x))/(35*x) -
12*I*a*sqrt(1 - 1/(a*x))/(35*x**2) - 2*I*sqrt(1 - 1/(a*x))/(7*x**3), True)) + Pi
ecewise((-256*a**4*sqrt(-1 + 1/(a*x))/315 - 128*a**3*sqrt(-1 + 1/(a*x))/(315*x)
- 32*a**2*sqrt(-1 + 1/(a*x))/(105*x**2) - 16*a*sqrt(-1 + 1/(a*x))/(63*x**3) - 2*
sqrt(-1 + 1/(a*x))/(9*x**4), Abs(1/(a*x)) > 1), (-256*I*a**4*sqrt(1 - 1/(a*x))/3
15 - 128*I*a**3*sqrt(1 - 1/(a*x))/(315*x) - 32*I*a**2*sqrt(1 - 1/(a*x))/(105*x**
2) - 16*I*a*sqrt(1 - 1/(a*x))/(63*x**3) - 2*I*sqrt(1 - 1/(a*x))/(9*x**4), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225855, size = 293, normalized size = 2.42 \[ -\frac{\frac{35 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{9}}{\left (a x\right )^{\frac{9}{2}}} + \frac{585 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac{7}{2}}} + \frac{4032 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac{5}{2}}} + \frac{17640 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac{3}{2}}} + \frac{83790 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}}{\sqrt{a x}} - \frac{{\left (35 \, a^{5} + \frac{585 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{2}}{x} + \frac{4032 \, a^{3}{\left (\sqrt{-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac{17640 \, a^{2}{\left (\sqrt{-a x + 1} - 1\right )}^{6}}{x^{3}} + \frac{83790 \, a{\left (\sqrt{-a x + 1} - 1\right )}^{8}}{x^{4}}\right )} \left (a x\right )^{\frac{9}{2}}}{{\left (\sqrt{-a x + 1} - 1\right )}^{9}}}{80640 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/80640*(35*a^5*(sqrt(-a*x + 1) - 1)^9/(a*x)^(9/2) + 585*a^5*(sqrt(-a*x + 1) -
1)^7/(a*x)^(7/2) + 4032*a^5*(sqrt(-a*x + 1) - 1)^5/(a*x)^(5/2) + 17640*a^5*(sqrt
(-a*x + 1) - 1)^3/(a*x)^(3/2) + 83790*a^5*(sqrt(-a*x + 1) - 1)/sqrt(a*x) - (35*a
^5 + 585*a^4*(sqrt(-a*x + 1) - 1)^2/x + 4032*a^3*(sqrt(-a*x + 1) - 1)^4/x^2 + 17
640*a^2*(sqrt(-a*x + 1) - 1)^6/x^3 + 83790*a*(sqrt(-a*x + 1) - 1)^8/x^4)*(a*x)^(
9/2)/(sqrt(-a*x + 1) - 1)^9)/a