3.1775 \(\int \frac{x^{7/2}}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=126 \[ \frac{256 b^4 \sqrt{x} \sqrt{a+\frac{b}{x}}}{315 a^5}-\frac{128 b^3 x^{3/2} \sqrt{a+\frac{b}{x}}}{315 a^4}+\frac{32 b^2 x^{5/2} \sqrt{a+\frac{b}{x}}}{105 a^3}-\frac{16 b x^{7/2} \sqrt{a+\frac{b}{x}}}{63 a^2}+\frac{2 x^{9/2} \sqrt{a+\frac{b}{x}}}{9 a} \]

[Out]

(256*b^4*Sqrt[a + b/x]*Sqrt[x])/(315*a^5) - (128*b^3*Sqrt[a + b/x]*x^(3/2))/(315
*a^4) + (32*b^2*Sqrt[a + b/x]*x^(5/2))/(105*a^3) - (16*b*Sqrt[a + b/x]*x^(7/2))/
(63*a^2) + (2*Sqrt[a + b/x]*x^(9/2))/(9*a)

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Rubi [A]  time = 0.155256, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{256 b^4 \sqrt{x} \sqrt{a+\frac{b}{x}}}{315 a^5}-\frac{128 b^3 x^{3/2} \sqrt{a+\frac{b}{x}}}{315 a^4}+\frac{32 b^2 x^{5/2} \sqrt{a+\frac{b}{x}}}{105 a^3}-\frac{16 b x^{7/2} \sqrt{a+\frac{b}{x}}}{63 a^2}+\frac{2 x^{9/2} \sqrt{a+\frac{b}{x}}}{9 a} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/Sqrt[a + b/x],x]

[Out]

(256*b^4*Sqrt[a + b/x]*Sqrt[x])/(315*a^5) - (128*b^3*Sqrt[a + b/x]*x^(3/2))/(315
*a^4) + (32*b^2*Sqrt[a + b/x]*x^(5/2))/(105*a^3) - (16*b*Sqrt[a + b/x]*x^(7/2))/
(63*a^2) + (2*Sqrt[a + b/x]*x^(9/2))/(9*a)

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Rubi in Sympy [A]  time = 13.6008, size = 110, normalized size = 0.87 \[ \frac{2 x^{\frac{9}{2}} \sqrt{a + \frac{b}{x}}}{9 a} - \frac{16 b x^{\frac{7}{2}} \sqrt{a + \frac{b}{x}}}{63 a^{2}} + \frac{32 b^{2} x^{\frac{5}{2}} \sqrt{a + \frac{b}{x}}}{105 a^{3}} - \frac{128 b^{3} x^{\frac{3}{2}} \sqrt{a + \frac{b}{x}}}{315 a^{4}} + \frac{256 b^{4} \sqrt{x} \sqrt{a + \frac{b}{x}}}{315 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(9/2)*sqrt(a + b/x)/(9*a) - 16*b*x**(7/2)*sqrt(a + b/x)/(63*a**2) + 32*b**2
*x**(5/2)*sqrt(a + b/x)/(105*a**3) - 128*b**3*x**(3/2)*sqrt(a + b/x)/(315*a**4)
+ 256*b**4*sqrt(x)*sqrt(a + b/x)/(315*a**5)

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Mathematica [A]  time = 0.0571691, size = 64, normalized size = 0.51 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (35 a^4 x^4-40 a^3 b x^3+48 a^2 b^2 x^2-64 a b^3 x+128 b^4\right )}{315 a^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(128*b^4 - 64*a*b^3*x + 48*a^2*b^2*x^2 - 40*a^3*b*x^3 +
35*a^4*x^4))/(315*a^5)

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Maple [A]  time = 0.007, size = 66, normalized size = 0.5 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 35\,{x}^{4}{a}^{4}-40\,b{x}^{3}{a}^{3}+48\,{b}^{2}{x}^{2}{a}^{2}-64\,{b}^{3}xa+128\,{b}^{4} \right ) }{315\,{a}^{5}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(a+b/x)^(1/2),x)

[Out]

2/315*(a*x+b)*(35*a^4*x^4-40*a^3*b*x^3+48*a^2*b^2*x^2-64*a*b^3*x+128*b^4)/a^5/x^
(1/2)/((a*x+b)/x)^(1/2)

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Maxima [A]  time = 1.43875, size = 116, normalized size = 0.92 \[ \frac{2 \,{\left (35 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} x^{\frac{9}{2}} - 180 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b x^{\frac{7}{2}} + 378 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b^{2} x^{\frac{5}{2}} - 420 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{3} x^{\frac{3}{2}} + 315 \, \sqrt{a + \frac{b}{x}} b^{4} \sqrt{x}\right )}}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*(a + b/x)^(9/2)*x^(9/2) - 180*(a + b/x)^(7/2)*b*x^(7/2) + 378*(a + b/x
)^(5/2)*b^2*x^(5/2) - 420*(a + b/x)^(3/2)*b^3*x^(3/2) + 315*sqrt(a + b/x)*b^4*sq
rt(x))/a^5

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Fricas [A]  time = 0.234517, size = 81, normalized size = 0.64 \[ \frac{2 \,{\left (35 \, a^{4} x^{4} - 40 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 64 \, a b^{3} x + 128 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*a^4*x^4 - 40*a^3*b*x^3 + 48*a^2*b^2*x^2 - 64*a*b^3*x + 128*b^4)*sqrt(x
)*sqrt((a*x + b)/x)/a^5

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229347, size = 95, normalized size = 0.75 \[ -\frac{256 \, b^{\frac{9}{2}}}{315 \, a^{5}} + \frac{2 \,{\left (35 \,{\left (a x + b\right )}^{\frac{9}{2}} - 180 \,{\left (a x + b\right )}^{\frac{7}{2}} b + 378 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{2} - 420 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{3} + 315 \, \sqrt{a x + b} b^{4}\right )}}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-256/315*b^(9/2)/a^5 + 2/315*(35*(a*x + b)^(9/2) - 180*(a*x + b)^(7/2)*b + 378*(
a*x + b)^(5/2)*b^2 - 420*(a*x + b)^(3/2)*b^3 + 315*sqrt(a*x + b)*b^4)/a^5