3.576 \(\int \frac{1}{x^{9/2} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=92 \[ \frac{32 b^3 \sqrt{a+b x}}{35 a^4 \sqrt{x}}-\frac{16 b^2 \sqrt{a+b x}}{35 a^3 x^{3/2}}+\frac{12 b \sqrt{a+b x}}{35 a^2 x^{5/2}}-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}} \]

[Out]

(-2*Sqrt[a + b*x])/(7*a*x^(7/2)) + (12*b*Sqrt[a + b*x])/(35*a^2*x^(5/2)) - (16*b
^2*Sqrt[a + b*x])/(35*a^3*x^(3/2)) + (32*b^3*Sqrt[a + b*x])/(35*a^4*Sqrt[x])

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Rubi [A]  time = 0.0686943, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{32 b^3 \sqrt{a+b x}}{35 a^4 \sqrt{x}}-\frac{16 b^2 \sqrt{a+b x}}{35 a^3 x^{3/2}}+\frac{12 b \sqrt{a+b x}}{35 a^2 x^{5/2}}-\frac{2 \sqrt{a+b x}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(9/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(7*a*x^(7/2)) + (12*b*Sqrt[a + b*x])/(35*a^2*x^(5/2)) - (16*b
^2*Sqrt[a + b*x])/(35*a^3*x^(3/2)) + (32*b^3*Sqrt[a + b*x])/(35*a^4*Sqrt[x])

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Rubi in Sympy [A]  time = 9.33687, size = 87, normalized size = 0.95 \[ - \frac{2 \sqrt{a + b x}}{7 a x^{\frac{7}{2}}} + \frac{12 b \sqrt{a + b x}}{35 a^{2} x^{\frac{5}{2}}} - \frac{16 b^{2} \sqrt{a + b x}}{35 a^{3} x^{\frac{3}{2}}} + \frac{32 b^{3} \sqrt{a + b x}}{35 a^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(9/2)/(b*x+a)**(1/2),x)

[Out]

-2*sqrt(a + b*x)/(7*a*x**(7/2)) + 12*b*sqrt(a + b*x)/(35*a**2*x**(5/2)) - 16*b**
2*sqrt(a + b*x)/(35*a**3*x**(3/2)) + 32*b**3*sqrt(a + b*x)/(35*a**4*sqrt(x))

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Mathematica [A]  time = 0.0281607, size = 51, normalized size = 0.55 \[ -\frac{2 \sqrt{a+b x} \left (5 a^3-6 a^2 b x+8 a b^2 x^2-16 b^3 x^3\right )}{35 a^4 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(9/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x]*(5*a^3 - 6*a^2*b*x + 8*a*b^2*x^2 - 16*b^3*x^3))/(35*a^4*x^(7/2
))

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Maple [A]  time = 0.007, size = 46, normalized size = 0.5 \[ -{\frac{-32\,{b}^{3}{x}^{3}+16\,a{b}^{2}{x}^{2}-12\,{a}^{2}bx+10\,{a}^{3}}{35\,{a}^{4}}\sqrt{bx+a}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(9/2)/(b*x+a)^(1/2),x)

[Out]

-2/35*(b*x+a)^(1/2)*(-16*b^3*x^3+8*a*b^2*x^2-6*a^2*b*x+5*a^3)/x^(7/2)/a^4

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Maxima [A]  time = 1.34327, size = 82, normalized size = 0.89 \[ \frac{2 \,{\left (\frac{35 \, \sqrt{b x + a} b^{3}}{\sqrt{x}} - \frac{35 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}}{x^{\frac{3}{2}}} + \frac{21 \,{\left (b x + a\right )}^{\frac{5}{2}} b}{x^{\frac{5}{2}}} - \frac{5 \,{\left (b x + a\right )}^{\frac{7}{2}}}{x^{\frac{7}{2}}}\right )}}{35 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*x^(9/2)),x, algorithm="maxima")

[Out]

2/35*(35*sqrt(b*x + a)*b^3/sqrt(x) - 35*(b*x + a)^(3/2)*b^2/x^(3/2) + 21*(b*x +
a)^(5/2)*b/x^(5/2) - 5*(b*x + a)^(7/2)/x^(7/2))/a^4

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Fricas [A]  time = 0.233639, size = 61, normalized size = 0.66 \[ \frac{2 \,{\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt{b x + a}}{35 \, a^{4} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*x^(9/2)),x, algorithm="fricas")

[Out]

2/35*(16*b^3*x^3 - 8*a*b^2*x^2 + 6*a^2*b*x - 5*a^3)*sqrt(b*x + a)/(a^4*x^(7/2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(9/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212779, size = 111, normalized size = 1.21 \[ -\frac{{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b x + a\right )}}{a^{4} b^{5}} - \frac{7}{a^{3} b^{5}}\right )} + \frac{35}{a^{2} b^{5}}\right )} - \frac{35}{a b^{5}}\right )} \sqrt{b x + a} b}{13440 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*x^(9/2)),x, algorithm="giac")

[Out]

-1/13440*(2*(b*x + a)*(4*(b*x + a)*(2*(b*x + a)/(a^4*b^5) - 7/(a^3*b^5)) + 35/(a
^2*b^5)) - 35/(a*b^5))*sqrt(b*x + a)*b/(((b*x + a)*b - a*b)^(7/2)*abs(b))