3.1766 \(\int \left (a+\frac{b}{x}\right )^{5/2} x^{11/2} \, dx\)

Optimal. Leaf size=100 \[ -\frac{32 b^3 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{3003 a^4}+\frac{16 b^2 x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{429 a^3}-\frac{12 b x^{11/2} \left (a+\frac{b}{x}\right )^{7/2}}{143 a^2}+\frac{2 x^{13/2} \left (a+\frac{b}{x}\right )^{7/2}}{13 a} \]

[Out]

(-32*b^3*(a + b/x)^(7/2)*x^(7/2))/(3003*a^4) + (16*b^2*(a + b/x)^(7/2)*x^(9/2))/
(429*a^3) - (12*b*(a + b/x)^(7/2)*x^(11/2))/(143*a^2) + (2*(a + b/x)^(7/2)*x^(13
/2))/(13*a)

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Rubi [A]  time = 0.115908, 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 \[ -\frac{32 b^3 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{3003 a^4}+\frac{16 b^2 x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{429 a^3}-\frac{12 b x^{11/2} \left (a+\frac{b}{x}\right )^{7/2}}{143 a^2}+\frac{2 x^{13/2} \left (a+\frac{b}{x}\right )^{7/2}}{13 a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)*x^(11/2),x]

[Out]

(-32*b^3*(a + b/x)^(7/2)*x^(7/2))/(3003*a^4) + (16*b^2*(a + b/x)^(7/2)*x^(9/2))/
(429*a^3) - (12*b*(a + b/x)^(7/2)*x^(11/2))/(143*a^2) + (2*(a + b/x)^(7/2)*x^(13
/2))/(13*a)

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Rubi in Sympy [A]  time = 9.63752, size = 87, normalized size = 0.87 \[ \frac{2 x^{\frac{13}{2}} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{13 a} - \frac{12 b x^{\frac{11}{2}} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{143 a^{2}} + \frac{16 b^{2} x^{\frac{9}{2}} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{429 a^{3}} - \frac{32 b^{3} x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{3003 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(5/2)*x**(11/2),x)

[Out]

2*x**(13/2)*(a + b/x)**(7/2)/(13*a) - 12*b*x**(11/2)*(a + b/x)**(7/2)/(143*a**2)
+ 16*b**2*x**(9/2)*(a + b/x)**(7/2)/(429*a**3) - 32*b**3*x**(7/2)*(a + b/x)**(7
/2)/(3003*a**4)

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Mathematica [A]  time = 0.0644155, size = 60, normalized size = 0.6 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (231 a^3 x^3-126 a^2 b x^2+56 a b^2 x-16 b^3\right )}{3003 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2)*x^(11/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^3*(-16*b^3 + 56*a*b^2*x - 126*a^2*b*x^2 + 231
*a^3*x^3))/(3003*a^4)

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Maple [A]  time = 0.008, size = 55, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 231\,{a}^{3}{x}^{3}-126\,{a}^{2}b{x}^{2}+56\,a{b}^{2}x-16\,{b}^{3} \right ) }{3003\,{a}^{4}}{x}^{{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)*x^(11/2),x)

[Out]

2/3003*(a*x+b)*(231*a^3*x^3-126*a^2*b*x^2+56*a*b^2*x-16*b^3)*x^(5/2)*((a*x+b)/x)
^(5/2)/a^4

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Maxima [A]  time = 1.41766, size = 93, normalized size = 0.93 \[ \frac{2 \,{\left (231 \,{\left (a + \frac{b}{x}\right )}^{\frac{13}{2}} x^{\frac{13}{2}} - 819 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} b x^{\frac{11}{2}} + 1001 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} b^{2} x^{\frac{9}{2}} - 429 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b^{3} x^{\frac{7}{2}}\right )}}{3003 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)*x^(11/2),x, algorithm="maxima")

[Out]

2/3003*(231*(a + b/x)^(13/2)*x^(13/2) - 819*(a + b/x)^(11/2)*b*x^(11/2) + 1001*(
a + b/x)^(9/2)*b^2*x^(9/2) - 429*(a + b/x)^(7/2)*b^3*x^(7/2))/a^4

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Fricas [A]  time = 0.242532, size = 111, normalized size = 1.11 \[ \frac{2 \,{\left (231 \, a^{6} x^{6} + 567 \, a^{5} b x^{5} + 371 \, a^{4} b^{2} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{2} b^{4} x^{2} + 8 \, a b^{5} x - 16 \, b^{6}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3003 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)*x^(11/2),x, algorithm="fricas")

[Out]

2/3003*(231*a^6*x^6 + 567*a^5*b*x^5 + 371*a^4*b^2*x^4 + 5*a^3*b^3*x^3 - 6*a^2*b^
4*x^2 + 8*a*b^5*x - 16*b^6)*sqrt(x)*sqrt((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)**(5/2)*x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.243959, size = 309, normalized size = 3.09 \[ \frac{2}{315} \, b^{2}{\left (\frac{16 \, b^{\frac{9}{2}}}{a^{4}} + \frac{35 \,{\left (a x + b\right )}^{\frac{9}{2}} - 135 \,{\left (a x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{3}}{a^{4}}\right )}{\rm sign}\left (x\right ) - \frac{4}{3465} \, a b{\left (\frac{128 \, b^{\frac{11}{2}}}{a^{5}} - \frac{315 \,{\left (a x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (a x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (a x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{4}}{a^{5}}\right )}{\rm sign}\left (x\right ) + \frac{2}{9009} \, a^{2}{\left (\frac{256 \, b^{\frac{13}{2}}}{a^{6}} + \frac{693 \,{\left (a x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (a x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (a x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (a x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{5}}{a^{6}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^(5/2)*x^(11/2),x, algorithm="giac")

[Out]

2/315*b^2*(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) - 4/3465*a*b*(128*b^(1
1/2)/a^5 - (315*(a*x + b)^(11/2) - 1540*(a*x + b)^(9/2)*b + 2970*(a*x + b)^(7/2)
*b^2 - 2772*(a*x + b)^(5/2)*b^3 + 1155*(a*x + b)^(3/2)*b^4)/a^5)*sign(x) + 2/900
9*a^2*(256*b^(13/2)/a^6 + (693*(a*x + b)^(13/2) - 4095*(a*x + b)^(11/2)*b + 1001
0*(a*x + b)^(9/2)*b^2 - 12870*(a*x + b)^(7/2)*b^3 + 9009*(a*x + b)^(5/2)*b^4 - 3
003*(a*x + b)^(3/2)*b^5)/a^6)*sign(x)