3.1688 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{13/2}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{63 a^2}{4 b^5 \sqrt{x}}+\frac{21 a}{4 b^4 x^{3/2}}+\frac{9}{4 b^2 x^{5/2} (a x+b)}+\frac{1}{2 b x^{5/2} (a x+b)^2}-\frac{63}{20 b^3 x^{5/2}} \]

[Out]

-63/(20*b^3*x^(5/2)) + (21*a)/(4*b^4*x^(3/2)) - (63*a^2)/(4*b^5*Sqrt[x]) + 1/(2*
b*x^(5/2)*(b + a*x)^2) + 9/(4*b^2*x^(5/2)*(b + a*x)) - (63*a^(5/2)*ArcTan[(Sqrt[
a]*Sqrt[x])/Sqrt[b]])/(4*b^(11/2))

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Rubi [A]  time = 0.123342, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{63 a^2}{4 b^5 \sqrt{x}}+\frac{21 a}{4 b^4 x^{3/2}}+\frac{9}{4 b^2 x^{5/2} (a x+b)}+\frac{1}{2 b x^{5/2} (a x+b)^2}-\frac{63}{20 b^3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^(13/2)),x]

[Out]

-63/(20*b^3*x^(5/2)) + (21*a)/(4*b^4*x^(3/2)) - (63*a^2)/(4*b^5*Sqrt[x]) + 1/(2*
b*x^(5/2)*(b + a*x)^2) + 9/(4*b^2*x^(5/2)*(b + a*x)) - (63*a^(5/2)*ArcTan[(Sqrt[
a]*Sqrt[x])/Sqrt[b]])/(4*b^(11/2))

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Rubi in Sympy [A]  time = 22.2454, size = 104, normalized size = 0.95 \[ - \frac{63 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 b^{\frac{11}{2}}} - \frac{63 a^{2}}{4 b^{5} \sqrt{x}} + \frac{21 a}{4 b^{4} x^{\frac{3}{2}}} + \frac{1}{2 b x^{\frac{5}{2}} \left (a x + b\right )^{2}} + \frac{9}{4 b^{2} x^{\frac{5}{2}} \left (a x + b\right )} - \frac{63}{20 b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**(13/2),x)

[Out]

-63*a**(5/2)*atan(sqrt(a)*sqrt(x)/sqrt(b))/(4*b**(11/2)) - 63*a**2/(4*b**5*sqrt(
x)) + 21*a/(4*b**4*x**(3/2)) + 1/(2*b*x**(5/2)*(a*x + b)**2) + 9/(4*b**2*x**(5/2
)*(a*x + b)) - 63/(20*b**3*x**(5/2))

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Mathematica [A]  time = 0.0951834, size = 92, normalized size = 0.84 \[ -\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{315 a^4 x^4+525 a^3 b x^3+168 a^2 b^2 x^2-24 a b^3 x+8 b^4}{20 b^5 x^{5/2} (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^(13/2)),x]

[Out]

-(8*b^4 - 24*a*b^3*x + 168*a^2*b^2*x^2 + 525*a^3*b*x^3 + 315*a^4*x^4)/(20*b^5*x^
(5/2)*(b + a*x)^2) - (63*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*b^(11/2))

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Maple [A]  time = 0.023, size = 90, normalized size = 0.8 \[ -{\frac{2}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}-12\,{\frac{{a}^{2}}{{b}^{5}\sqrt{x}}}+2\,{\frac{a}{{b}^{4}{x}^{3/2}}}-{\frac{15\,{a}^{4}}{4\,{b}^{5} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,{a}^{3}}{4\,{b}^{4} \left ( ax+b \right ) ^{2}}\sqrt{x}}-{\frac{63\,{a}^{3}}{4\,{b}^{5}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^(13/2),x)

[Out]

-2/5/b^3/x^(5/2)-12*a^2/b^5/x^(1/2)+2*a/b^4/x^(3/2)-15/4/b^5*a^4/(a*x+b)^2*x^(3/
2)-17/4/b^4*a^3/(a*x+b)^2*x^(1/2)-63/4/b^5*a^3/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*b
)^(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(1/((a + b/x)^3*x^(13/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241508, size = 1, normalized size = 0.01 \[ \left [-\frac{630 \, a^{4} x^{4} + 1050 \, a^{3} b x^{3} + 336 \, a^{2} b^{2} x^{2} - 48 \, a b^{3} x + 16 \, b^{4} - 315 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt{x} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right )}{40 \,{\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )} \sqrt{x}}, -\frac{315 \, a^{4} x^{4} + 525 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} - 24 \, a b^{3} x + 8 \, b^{4} - 315 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt{x} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right )}{20 \,{\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(13/2)),x, algorithm="fricas")

[Out]

[-1/40*(630*a^4*x^4 + 1050*a^3*b*x^3 + 336*a^2*b^2*x^2 - 48*a*b^3*x + 16*b^4 - 3
15*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*sqrt(x)*sqrt(-a/b)*log((a*x - 2*b*sqrt(
x)*sqrt(-a/b) - b)/(a*x + b)))/((a^2*b^5*x^4 + 2*a*b^6*x^3 + b^7*x^2)*sqrt(x)),
-1/20*(315*a^4*x^4 + 525*a^3*b*x^3 + 168*a^2*b^2*x^2 - 24*a*b^3*x + 8*b^4 - 315*
(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*sqrt(x)*sqrt(a/b)*arctan(b*sqrt(a/b)/(a*sq
rt(x))))/((a^2*b^5*x^4 + 2*a*b^6*x^3 + b^7*x^2)*sqrt(x))]

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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/(a+b/x)**3/x**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220981, size = 108, normalized size = 0.98 \[ -\frac{63 \, a^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{5}} - \frac{15 \, a^{4} x^{\frac{3}{2}} + 17 \, a^{3} b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{5}} - \frac{2 \,{\left (30 \, a^{2} x^{2} - 5 \, a b x + b^{2}\right )}}{5 \, b^{5} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(13/2)),x, algorithm="giac")

[Out]

-63/4*a^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/4*(15*a^4*x^(3/2) + 17
*a^3*b*sqrt(x))/((a*x + b)^2*b^5) - 2/5*(30*a^2*x^2 - 5*a*b*x + b^2)/(b^5*x^(5/2
))