3.29 \(\int \frac{1+a x}{x^4 \sqrt{a x} \sqrt{1-a x}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{208 a^3 \sqrt{1-a x}}{105 \sqrt{a x}}-\frac{104 a^3 \sqrt{1-a x}}{105 (a x)^{3/2}}-\frac{26 a^3 \sqrt{1-a x}}{35 (a x)^{5/2}}-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}} \]

[Out]

(-2*a^3*Sqrt[1 - a*x])/(7*(a*x)^(7/2)) - (26*a^3*Sqrt[1 - a*x])/(35*(a*x)^(5/2))
- (104*a^3*Sqrt[1 - a*x])/(105*(a*x)^(3/2)) - (208*a^3*Sqrt[1 - a*x])/(105*Sqrt
[a*x])

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Rubi [A]  time = 0.116592, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{208 a^3 \sqrt{1-a x}}{105 \sqrt{a x}}-\frac{104 a^3 \sqrt{1-a x}}{105 (a x)^{3/2}}-\frac{26 a^3 \sqrt{1-a x}}{35 (a x)^{5/2}}-\frac{2 a^3 \sqrt{1-a x}}{7 (a x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + a*x)/(x^4*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-2*a^3*Sqrt[1 - a*x])/(7*(a*x)^(7/2)) - (26*a^3*Sqrt[1 - a*x])/(35*(a*x)^(5/2))
- (104*a^3*Sqrt[1 - a*x])/(105*(a*x)^(3/2)) - (208*a^3*Sqrt[1 - a*x])/(105*Sqrt
[a*x])

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Rubi in Sympy [A]  time = 13.2971, size = 88, normalized size = 0.91 \[ - \frac{208 a^{3} \sqrt{- a x + 1}}{105 \sqrt{a x}} - \frac{104 a^{3} \sqrt{- a x + 1}}{105 \left (a x\right )^{\frac{3}{2}}} - \frac{26 a^{3} \sqrt{- a x + 1}}{35 \left (a x\right )^{\frac{5}{2}}} - \frac{2 a^{3} \sqrt{- a x + 1}}{7 \left (a x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x+1)/x**4/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

-208*a**3*sqrt(-a*x + 1)/(105*sqrt(a*x)) - 104*a**3*sqrt(-a*x + 1)/(105*(a*x)**(
3/2)) - 26*a**3*sqrt(-a*x + 1)/(35*(a*x)**(5/2)) - 2*a**3*sqrt(-a*x + 1)/(7*(a*x
)**(7/2))

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Mathematica [A]  time = 0.0425171, size = 45, normalized size = 0.46 \[ -\frac{2 \sqrt{-a x (a x-1)} \left (104 a^3 x^3+52 a^2 x^2+39 a x+15\right )}{105 a x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + a*x)/(x^4*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-2*Sqrt[-(a*x*(-1 + a*x))]*(15 + 39*a*x + 52*a^2*x^2 + 104*a^3*x^3))/(105*a*x^4
)

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Maple [A]  time = 0.008, size = 41, normalized size = 0.4 \[ -{\frac{208\,{a}^{3}{x}^{3}+104\,{a}^{2}{x}^{2}+78\,ax+30}{105\,{x}^{3}}\sqrt{-ax+1}{\frac{1}{\sqrt{ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x)

[Out]

-2/105*(104*a^3*x^3+52*a^2*x^2+39*a*x+15)/x^3/(a*x)^(1/2)*(-a*x+1)^(1/2)

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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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225531, size = 58, normalized size = 0.6 \[ -\frac{2 \,{\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt{a x} \sqrt{-a x + 1}}{105 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)/(sqrt(a*x)*sqrt(-a*x + 1)*x^4),x, algorithm="fricas")

[Out]

-2/105*(104*a^3*x^3 + 52*a^2*x^2 + 39*a*x + 15)*sqrt(a*x)*sqrt(-a*x + 1)/(a*x^4)

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Sympy [A]  time = 74.3046, size = 274, normalized size = 2.82 \[ a \left (\begin{cases} - \frac{16 a^{2} \sqrt{-1 + \frac{1}{a x}}}{15} - \frac{8 a \sqrt{-1 + \frac{1}{a x}}}{15 x} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{5 x^{2}} & \text{for}\: \left |{\frac{1}{a x}}\right | > 1 \\- \frac{16 i a^{2} \sqrt{1 - \frac{1}{a x}}}{15} - \frac{8 i a \sqrt{1 - \frac{1}{a x}}}{15 x} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{5 x^{2}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{32 a^{3} \sqrt{-1 + \frac{1}{a x}}}{35} - \frac{16 a^{2} \sqrt{-1 + \frac{1}{a x}}}{35 x} - \frac{12 a \sqrt{-1 + \frac{1}{a x}}}{35 x^{2}} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{7 x^{3}} & \text{for}\: \left |{\frac{1}{a x}}\right | > 1 \\- \frac{32 i a^{3} \sqrt{1 - \frac{1}{a x}}}{35} - \frac{16 i a^{2} \sqrt{1 - \frac{1}{a x}}}{35 x} - \frac{12 i a \sqrt{1 - \frac{1}{a x}}}{35 x^{2}} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{7 x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x+1)/x**4/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

a*Piecewise((-16*a**2*sqrt(-1 + 1/(a*x))/15 - 8*a*sqrt(-1 + 1/(a*x))/(15*x) - 2*
sqrt(-1 + 1/(a*x))/(5*x**2), Abs(1/(a*x)) > 1), (-16*I*a**2*sqrt(1 - 1/(a*x))/15
- 8*I*a*sqrt(1 - 1/(a*x))/(15*x) - 2*I*sqrt(1 - 1/(a*x))/(5*x**2), True)) + Pie
cewise((-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))

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GIAC/XCAS [A]  time = 0.225994, size = 236, normalized size = 2.43 \[ -\frac{\frac{15 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac{7}{2}}} + \frac{231 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac{5}{2}}} + \frac{1435 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac{3}{2}}} + \frac{7875 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}}{\sqrt{a x}} - \frac{{\left (15 \, a^{4} + \frac{231 \, a^{3}{\left (\sqrt{-a x + 1} - 1\right )}^{2}}{x} + \frac{1435 \, a^{2}{\left (\sqrt{-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac{7875 \, a{\left (\sqrt{-a x + 1} - 1\right )}^{6}}{x^{3}}\right )} \left (a x\right )^{\frac{7}{2}}}{{\left (\sqrt{-a x + 1} - 1\right )}^{7}}}{6720 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)/(sqrt(a*x)*sqrt(-a*x + 1)*x^4),x, algorithm="giac")

[Out]

-1/6720*(15*a^4*(sqrt(-a*x + 1) - 1)^7/(a*x)^(7/2) + 231*a^4*(sqrt(-a*x + 1) - 1
)^5/(a*x)^(5/2) + 1435*a^4*(sqrt(-a*x + 1) - 1)^3/(a*x)^(3/2) + 7875*a^4*(sqrt(-
a*x + 1) - 1)/sqrt(a*x) - (15*a^4 + 231*a^3*(sqrt(-a*x + 1) - 1)^2/x + 1435*a^2*
(sqrt(-a*x + 1) - 1)^4/x^2 + 7875*a*(sqrt(-a*x + 1) - 1)^6/x^3)*(a*x)^(7/2)/(sqr
t(-a*x + 1) - 1)^7)/a