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

Optimal. Leaf size=68 \[ -\frac{16 b^2 (a+b x)^{3/2}}{105 a^3 x^{3/2}}+\frac{8 b (a+b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{2 (a+b x)^{3/2}}{7 a x^{7/2}} \]

[Out]

(-2*(a + b*x)^(3/2))/(7*a*x^(7/2)) + (8*b*(a + b*x)^(3/2))/(35*a^2*x^(5/2)) - (1
6*b^2*(a + b*x)^(3/2))/(105*a^3*x^(3/2))

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Rubi [A]  time = 0.042966, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{16 b^2 (a+b x)^{3/2}}{105 a^3 x^{3/2}}+\frac{8 b (a+b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{2 (a+b x)^{3/2}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/x^(9/2),x]

[Out]

(-2*(a + b*x)^(3/2))/(7*a*x^(7/2)) + (8*b*(a + b*x)^(3/2))/(35*a^2*x^(5/2)) - (1
6*b^2*(a + b*x)^(3/2))/(105*a^3*x^(3/2))

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Rubi in Sympy [A]  time = 5.72931, size = 63, normalized size = 0.93 \[ - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{7 a x^{\frac{7}{2}}} + \frac{8 b \left (a + b x\right )^{\frac{3}{2}}}{35 a^{2} x^{\frac{5}{2}}} - \frac{16 b^{2} \left (a + b x\right )^{\frac{3}{2}}}{105 a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a + b*x)**(3/2)/(7*a*x**(7/2)) + 8*b*(a + b*x)**(3/2)/(35*a**2*x**(5/2)) - 1
6*b**2*(a + b*x)**(3/2)/(105*a**3*x**(3/2))

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Mathematica [A]  time = 0.02273, size = 51, normalized size = 0.75 \[ -\frac{2 \sqrt{a+b x} \left (15 a^3+3 a^2 b x-4 a b^2 x^2+8 b^3 x^3\right )}{105 a^3 x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x]*(15*a^3 + 3*a^2*b*x - 4*a*b^2*x^2 + 8*b^3*x^3))/(105*a^3*x^(7/
2))

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Maple [A]  time = 0.007, size = 35, normalized size = 0.5 \[ -{\frac{16\,{b}^{2}{x}^{2}-24\,abx+30\,{a}^{2}}{105\,{a}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^(9/2),x)

[Out]

-2/105*(b*x+a)^(3/2)*(8*b^2*x^2-12*a*b*x+15*a^2)/x^(7/2)/a^3

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Maxima [A]  time = 1.33094, size = 62, normalized size = 0.91 \[ -\frac{2 \,{\left (\frac{35 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}}{x^{\frac{3}{2}}} - \frac{42 \,{\left (b x + a\right )}^{\frac{5}{2}} b}{x^{\frac{5}{2}}} + \frac{15 \,{\left (b x + a\right )}^{\frac{7}{2}}}{x^{\frac{7}{2}}}\right )}}{105 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(35*(b*x + a)^(3/2)*b^2/x^(3/2) - 42*(b*x + a)^(5/2)*b/x^(5/2) + 15*(b*x
+ a)^(7/2)/x^(7/2))/a^3

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Fricas [A]  time = 0.209236, size = 61, normalized size = 0.9 \[ -\frac{2 \,{\left (8 \, b^{3} x^{3} - 4 \, a b^{2} x^{2} + 3 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x + a}}{105 \, a^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(8*b^3*x^3 - 4*a*b^2*x^2 + 3*a^2*b*x + 15*a^3)*sqrt(b*x + a)/(a^3*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((b*x+a)**(1/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213154, size = 89, normalized size = 1.31 \[ \frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b x + a\right )}}{a^{4} b^{5}} - \frac{7}{a^{3} b^{5}}\right )} + \frac{35}{a^{2} b^{5}}\right )} b}{40320 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40320*(b*x + a)^(3/2)*(4*(b*x + a)*(2*(b*x + a)/(a^4*b^5) - 7/(a^3*b^5)) + 35/
(a^2*b^5))*b/(((b*x + a)*b - a*b)^(7/2)*abs(b))