3.1750 \(\int \sqrt{a+\frac{b}{x}} x^{7/2} \, dx\)

Optimal. Leaf size=100 \[ -\frac{32 b^3 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{315 a^4}+\frac{16 b^2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{4 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{21 a^2}+\frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{3/2}}{9 a} \]

[Out]

(-32*b^3*(a + b/x)^(3/2)*x^(3/2))/(315*a^4) + (16*b^2*(a + b/x)^(3/2)*x^(5/2))/(
105*a^3) - (4*b*(a + b/x)^(3/2)*x^(7/2))/(21*a^2) + (2*(a + b/x)^(3/2)*x^(9/2))/
(9*a)

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Rubi [A]  time = 0.116207, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{32 b^3 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{315 a^4}+\frac{16 b^2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{105 a^3}-\frac{4 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}}{21 a^2}+\frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{3/2}}{9 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]*x^(7/2),x]

[Out]

(-32*b^3*(a + b/x)^(3/2)*x^(3/2))/(315*a^4) + (16*b^2*(a + b/x)^(3/2)*x^(5/2))/(
105*a^3) - (4*b*(a + b/x)^(3/2)*x^(7/2))/(21*a^2) + (2*(a + b/x)^(3/2)*x^(9/2))/
(9*a)

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Rubi in Sympy [A]  time = 10.2459, size = 87, normalized size = 0.87 \[ \frac{2 x^{\frac{9}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{9 a} - \frac{4 b x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{21 a^{2}} + \frac{16 b^{2} x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{105 a^{3}} - \frac{32 b^{3} x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{315 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(1/2)*x**(7/2),x)

[Out]

2*x**(9/2)*(a + b/x)**(3/2)/(9*a) - 4*b*x**(7/2)*(a + b/x)**(3/2)/(21*a**2) + 16
*b**2*x**(5/2)*(a + b/x)**(3/2)/(105*a**3) - 32*b**3*x**(3/2)*(a + b/x)**(3/2)/(
315*a**4)

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Mathematica [A]  time = 0.0463412, size = 64, normalized size = 0.64 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (35 a^4 x^4+5 a^3 b x^3-6 a^2 b^2 x^2+8 a b^3 x-16 b^4\right )}{315 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]*x^(7/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-16*b^4 + 8*a*b^3*x - 6*a^2*b^2*x^2 + 5*a^3*b*x^3 + 35
*a^4*x^4))/(315*a^4)

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Maple [A]  time = 0.008, size = 55, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 35\,{a}^{3}{x}^{3}-30\,{a}^{2}b{x}^{2}+24\,a{b}^{2}x-16\,{b}^{3} \right ) }{315\,{a}^{4}}\sqrt{x}\sqrt{{\frac{ax+b}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)*x^(7/2),x)

[Out]

2/315*(a*x+b)*(35*a^3*x^3-30*a^2*b*x^2+24*a*b^2*x-16*b^3)*x^(1/2)*((a*x+b)/x)^(1
/2)/a^4

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Maxima [A]  time = 1.43662, size = 93, normalized size = 0.93 \[ \frac{2 \,{\left (35 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} x^{\frac{9}{2}} - 135 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b x^{\frac{7}{2}} + 189 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b^{2} x^{\frac{5}{2}} - 105 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{3} x^{\frac{3}{2}}\right )}}{315 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)*x^(7/2),x, algorithm="maxima")

[Out]

2/315*(35*(a + b/x)^(9/2)*x^(9/2) - 135*(a + b/x)^(7/2)*b*x^(7/2) + 189*(a + b/x
)^(5/2)*b^2*x^(5/2) - 105*(a + b/x)^(3/2)*b^3*x^(3/2))/a^4

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Fricas [A]  time = 0.243547, size = 81, normalized size = 0.81 \[ \frac{2 \,{\left (35 \, a^{4} x^{4} + 5 \, a^{3} b x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a b^{3} x - 16 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{315 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)*x^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*a^4*x^4 + 5*a^3*b*x^3 - 6*a^2*b^2*x^2 + 8*a*b^3*x - 16*b^4)*sqrt(x)*sq
rt((a*x + b)/x)/a^4

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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((a+b/x)**(1/2)*x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228357, size = 82, normalized size = 0.82 \[ \frac{2}{315} \,{\left (\frac{16 \, b^{\frac{9}{2}}}{a^{4}} + \frac{35 \,{\left (a x + b\right )}^{\frac{9}{2}} - 135 \,{\left (a x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{3}}{a^{4}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)*x^(7/2),x, algorithm="giac")

[Out]

2/315*(16*b^(9/2)/a^4 + (35*(a*x + b)^(9/2) - 135*(a*x + b)^(7/2)*b + 189*(a*x +
b)^(5/2)*b^2 - 105*(a*x + b)^(3/2)*b^3)/a^4)*sign(x)