3.781 \(\int \frac{A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=240 \[ \frac{231 \sqrt{b} (13 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{15/2}}+\frac{231 (13 A b-3 a B)}{128 a^7 \sqrt{x}}-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5} \]

[Out]

(-77*(13*A*b - 3*a*B))/(128*a^6*b*x^(3/2)) + (231*(13*A*b - 3*a*B))/(128*a^7*Sqr
t[x]) + (A*b - a*B)/(5*a*b*x^(3/2)*(a + b*x)^5) + (13*A*b - 3*a*B)/(40*a^2*b*x^(
3/2)*(a + b*x)^4) + (11*(13*A*b - 3*a*B))/(240*a^3*b*x^(3/2)*(a + b*x)^3) + (33*
(13*A*b - 3*a*B))/(320*a^4*b*x^(3/2)*(a + b*x)^2) + (231*(13*A*b - 3*a*B))/(640*
a^5*b*x^(3/2)*(a + b*x)) + (231*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]])/(128*a^(15/2))

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Rubi [A]  time = 0.288551, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{231 \sqrt{b} (13 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{15/2}}+\frac{231 (13 A b-3 a B)}{128 a^7 \sqrt{x}}-\frac{77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac{231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac{33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac{11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac{13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac{A b-a B}{5 a b x^{3/2} (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-77*(13*A*b - 3*a*B))/(128*a^6*b*x^(3/2)) + (231*(13*A*b - 3*a*B))/(128*a^7*Sqr
t[x]) + (A*b - a*B)/(5*a*b*x^(3/2)*(a + b*x)^5) + (13*A*b - 3*a*B)/(40*a^2*b*x^(
3/2)*(a + b*x)^4) + (11*(13*A*b - 3*a*B))/(240*a^3*b*x^(3/2)*(a + b*x)^3) + (33*
(13*A*b - 3*a*B))/(320*a^4*b*x^(3/2)*(a + b*x)^2) + (231*(13*A*b - 3*a*B))/(640*
a^5*b*x^(3/2)*(a + b*x)) + (231*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]])/(128*a^(15/2))

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Rubi in Sympy [A]  time = 76.5371, size = 219, normalized size = 0.91 \[ \frac{A b - B a}{5 a b x^{\frac{3}{2}} \left (a + b x\right )^{5}} + \frac{13 A b - 3 B a}{40 a^{2} b x^{\frac{3}{2}} \left (a + b x\right )^{4}} + \frac{11 \left (13 A b - 3 B a\right )}{240 a^{3} b x^{\frac{3}{2}} \left (a + b x\right )^{3}} + \frac{33 \left (13 A b - 3 B a\right )}{320 a^{4} b x^{\frac{3}{2}} \left (a + b x\right )^{2}} + \frac{231 \left (13 A b - 3 B a\right )}{640 a^{5} b x^{\frac{3}{2}} \left (a + b x\right )} - \frac{77 \left (13 A b - 3 B a\right )}{128 a^{6} b x^{\frac{3}{2}}} + \frac{231 \left (13 A b - 3 B a\right )}{128 a^{7} \sqrt{x}} + \frac{231 \sqrt{b} \left (13 A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 a^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

(A*b - B*a)/(5*a*b*x**(3/2)*(a + b*x)**5) + (13*A*b - 3*B*a)/(40*a**2*b*x**(3/2)
*(a + b*x)**4) + 11*(13*A*b - 3*B*a)/(240*a**3*b*x**(3/2)*(a + b*x)**3) + 33*(13
*A*b - 3*B*a)/(320*a**4*b*x**(3/2)*(a + b*x)**2) + 231*(13*A*b - 3*B*a)/(640*a**
5*b*x**(3/2)*(a + b*x)) - 77*(13*A*b - 3*B*a)/(128*a**6*b*x**(3/2)) + 231*(13*A*
b - 3*B*a)/(128*a**7*sqrt(x)) + 231*sqrt(b)*(13*A*b - 3*B*a)*atan(sqrt(b)*sqrt(x
)/sqrt(a))/(128*a**(15/2))

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Mathematica [A]  time = 0.386623, size = 171, normalized size = 0.71 \[ \frac{\frac{\sqrt{a} \left (-1280 a^6 (A+3 B x)+5 a^5 b x (3328 A-6369 B x)+55 a^4 b^2 x^2 (2509 A-1422 B x)+66 a^3 b^3 x^3 (5135 A-1344 B x)+462 a^2 b^4 x^4 (832 A-105 B x)+1155 a b^5 x^5 (182 A-9 B x)+45045 A b^6 x^6\right )}{x^{3/2} (a+b x)^5}+3465 \sqrt{b} (13 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{1920 a^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((Sqrt[a]*(45045*A*b^6*x^6 + 5*a^5*b*x*(3328*A - 6369*B*x) + 55*a^4*b^2*x^2*(250
9*A - 1422*B*x) + 66*a^3*b^3*x^3*(5135*A - 1344*B*x) + 462*a^2*b^4*x^4*(832*A -
105*B*x) + 1155*a*b^5*x^5*(182*A - 9*B*x) - 1280*a^6*(A + 3*B*x)))/(x^(3/2)*(a +
b*x)^5) + 3465*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(192
0*a^(15/2))

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Maple [A]  time = 0.039, size = 266, normalized size = 1.1 \[ -{\frac{2\,A}{3\,{a}^{6}}{x}^{-{\frac{3}{2}}}}+12\,{\frac{Ab}{\sqrt{x}{a}^{7}}}-2\,{\frac{B}{\sqrt{x}{a}^{6}}}+{\frac{1467\,{b}^{6}A}{128\,{a}^{7} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}-{\frac{437\,B{b}^{5}}{128\,{a}^{6} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{9629\,A{b}^{5}}{192\,{a}^{6} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}-{\frac{977\,{b}^{4}B}{64\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}+{\frac{1253\,{b}^{4}A}{15\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{131\,B{b}^{3}}{5\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}+{\frac{12131\,A{b}^{3}}{192\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}-{\frac{1327\,{b}^{2}B}{64\,{a}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{2373\,{b}^{2}A}{128\,{a}^{3} \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{843\,Bb}{128\,{a}^{2} \left ( bx+a \right ) ^{5}}\sqrt{x}}+{\frac{3003\,{b}^{2}A}{128\,{a}^{7}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{693\,Bb}{128\,{a}^{6}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2/3*A/a^6/x^(3/2)+12/x^(1/2)/a^7*A*b-2/x^(1/2)/a^6*B+1467/128/a^7*b^6/(b*x+a)^5
*x^(9/2)*A-437/128/a^6*b^5/(b*x+a)^5*x^(9/2)*B+9629/192/a^6*b^5/(b*x+a)^5*A*x^(7
/2)-977/64/a^5*b^4/(b*x+a)^5*B*x^(7/2)+1253/15/a^5*b^4/(b*x+a)^5*x^(5/2)*A-131/5
/a^4*b^3/(b*x+a)^5*x^(5/2)*B+12131/192/a^4*b^3/(b*x+a)^5*x^(3/2)*A-1327/64/a^3*b
^2/(b*x+a)^5*x^(3/2)*B+2373/128/a^3*b^2/(b*x+a)^5*x^(1/2)*A-843/128/a^2*b/(b*x+a
)^5*x^(1/2)*B+3003/128/a^7*b^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-693/1
28/a^6*b/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^3*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.325867, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^3*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/3840*(2560*A*a^6 + 6930*(3*B*a*b^5 - 13*A*b^6)*x^6 + 32340*(3*B*a^2*b^4 - 13
*A*a*b^5)*x^5 + 59136*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 52140*(3*B*a^4*b^2 - 13
*A*a^3*b^3)*x^3 + 21230*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 3465*((3*B*a*b^5 - 13*A
*b^6)*x^6 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x
^4 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + (3
*B*a^6 - 13*A*a^5*b)*x)*sqrt(x)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a
)/(b*x + a)) + 2560*(3*B*a^6 - 13*A*a^5*b)*x)/((a^7*b^5*x^6 + 5*a^8*b^4*x^5 + 10
*a^9*b^3*x^4 + 10*a^10*b^2*x^3 + 5*a^11*b*x^2 + a^12*x)*sqrt(x)), -1/1920*(1280*
A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 +
29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^3
+ 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 - 3465*((3*B*a*b^5 - 13*A*b^6)*x^6 + 5*(
3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 10*(3*B*a^
4*b^2 - 13*A*a^3*b^3)*x^3 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + (3*B*a^6 - 13*A*a
^5*b)*x)*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 1280*(3*B*a^6 - 13*
A*a^5*b)*x)/((a^7*b^5*x^6 + 5*a^8*b^4*x^5 + 10*a^9*b^3*x^4 + 10*a^10*b^2*x^3 + 5
*a^11*b*x^2 + a^12*x)*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((B*x+A)/x**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279631, size = 243, normalized size = 1.01 \[ -\frac{231 \,{\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{7}} - \frac{2 \,{\left (3 \, B a x - 18 \, A b x + A a\right )}}{3 \, a^{7} x^{\frac{3}{2}}} - \frac{6555 \, B a b^{5} x^{\frac{9}{2}} - 22005 \, A b^{6} x^{\frac{9}{2}} + 29310 \, B a^{2} b^{4} x^{\frac{7}{2}} - 96290 \, A a b^{5} x^{\frac{7}{2}} + 50304 \, B a^{3} b^{3} x^{\frac{5}{2}} - 160384 \, A a^{2} b^{4} x^{\frac{5}{2}} + 39810 \, B a^{4} b^{2} x^{\frac{3}{2}} - 121310 \, A a^{3} b^{3} x^{\frac{3}{2}} + 12645 \, B a^{5} b \sqrt{x} - 35595 \, A a^{4} b^{2} \sqrt{x}}{1920 \,{\left (b x + a\right )}^{5} a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^3*x^(5/2)),x, algorithm="giac")

[Out]

-231/128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7) - 2/3*
(3*B*a*x - 18*A*b*x + A*a)/(a^7*x^(3/2)) - 1/1920*(6555*B*a*b^5*x^(9/2) - 22005*
A*b^6*x^(9/2) + 29310*B*a^2*b^4*x^(7/2) - 96290*A*a*b^5*x^(7/2) + 50304*B*a^3*b^
3*x^(5/2) - 160384*A*a^2*b^4*x^(5/2) + 39810*B*a^4*b^2*x^(3/2) - 121310*A*a^3*b^
3*x^(3/2) + 12645*B*a^5*b*sqrt(x) - 35595*A*a^4*b^2*sqrt(x))/((b*x + a)^5*a^7)