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

Optimal. Leaf size=178 \[ \frac{35 \sqrt{b} (3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{35 (3 A b-a B)}{8 a^5 \sqrt{x}}-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3} \]

[Out]

(-35*(3*A*b - a*B))/(24*a^4*b*x^(3/2)) + (35*(3*A*b - a*B))/(8*a^5*Sqrt[x]) + (A
*b - a*B)/(3*a*b*x^(3/2)*(a + b*x)^3) + (3*A*b - a*B)/(4*a^2*b*x^(3/2)*(a + b*x)
^2) + (7*(3*A*b - a*B))/(8*a^3*b*x^(3/2)*(a + b*x)) + (35*Sqrt[b]*(3*A*b - a*B)*
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(11/2))

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Rubi [A]  time = 0.204836, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{35 \sqrt{b} (3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{35 (3 A b-a B)}{8 a^5 \sqrt{x}}-\frac{35 (3 A b-a B)}{24 a^4 b x^{3/2}}+\frac{7 (3 A b-a B)}{8 a^3 b x^{3/2} (a+b x)}+\frac{3 A b-a B}{4 a^2 b x^{3/2} (a+b x)^2}+\frac{A b-a B}{3 a b x^{3/2} (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-35*(3*A*b - a*B))/(24*a^4*b*x^(3/2)) + (35*(3*A*b - a*B))/(8*a^5*Sqrt[x]) + (A
*b - a*B)/(3*a*b*x^(3/2)*(a + b*x)^3) + (3*A*b - a*B)/(4*a^2*b*x^(3/2)*(a + b*x)
^2) + (7*(3*A*b - a*B))/(8*a^3*b*x^(3/2)*(a + b*x)) + (35*Sqrt[b]*(3*A*b - a*B)*
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(11/2))

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Rubi in Sympy [A]  time = 53.1624, size = 158, normalized size = 0.89 \[ \frac{A b - B a}{3 a b x^{\frac{3}{2}} \left (a + b x\right )^{3}} + \frac{3 A b - B a}{4 a^{2} b x^{\frac{3}{2}} \left (a + b x\right )^{2}} + \frac{7 \left (3 A b - B a\right )}{8 a^{3} b x^{\frac{3}{2}} \left (a + b x\right )} - \frac{35 \left (3 A b - B a\right )}{24 a^{4} b x^{\frac{3}{2}}} + \frac{35 \left (3 A b - B a\right )}{8 a^{5} \sqrt{x}} + \frac{35 \sqrt{b} \left (3 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 a^{\frac{11}{2}}} \]

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)**2,x)

[Out]

(A*b - B*a)/(3*a*b*x**(3/2)*(a + b*x)**3) + (3*A*b - B*a)/(4*a**2*b*x**(3/2)*(a
+ b*x)**2) + 7*(3*A*b - B*a)/(8*a**3*b*x**(3/2)*(a + b*x)) - 35*(3*A*b - B*a)/(2
4*a**4*b*x**(3/2)) + 35*(3*A*b - B*a)/(8*a**5*sqrt(x)) + 35*sqrt(b)*(3*A*b - B*a
)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**(11/2))

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Mathematica [A]  time = 0.260069, size = 130, normalized size = 0.73 \[ \frac{-16 a^4 (A+3 B x)+3 a^3 b x (48 A-77 B x)+7 a^2 b^2 x^2 (99 A-40 B x)-105 a b^3 x^3 (B x-8 A)+315 A b^4 x^4}{24 a^5 x^{3/2} (a+b x)^3}-\frac{35 \sqrt{b} (a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{8 a^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(315*A*b^4*x^4 + 3*a^3*b*x*(48*A - 77*B*x) + 7*a^2*b^2*x^2*(99*A - 40*B*x) - 105
*a*b^3*x^3*(-8*A + B*x) - 16*a^4*(A + 3*B*x))/(24*a^5*x^(3/2)*(a + b*x)^3) - (35
*Sqrt[b]*(-3*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(11/2))

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Maple [A]  time = 0.031, size = 190, normalized size = 1.1 \[ -{\frac{2\,A}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}}}+8\,{\frac{Ab}{\sqrt{x}{a}^{5}}}-2\,{\frac{B}{\sqrt{x}{a}^{4}}}+{\frac{41\,{b}^{4}A}{8\,{a}^{5} \left ( bx+a \right ) ^{3}}{x}^{{\frac{5}{2}}}}-{\frac{19\,B{b}^{3}}{8\,{a}^{4} \left ( bx+a \right ) ^{3}}{x}^{{\frac{5}{2}}}}+{\frac{35\,A{b}^{3}}{3\,{a}^{4} \left ( bx+a \right ) ^{3}}{x}^{{\frac{3}{2}}}}-{\frac{17\,{b}^{2}B}{3\,{a}^{3} \left ( bx+a \right ) ^{3}}{x}^{{\frac{3}{2}}}}+{\frac{55\,{b}^{2}A}{8\,{a}^{3} \left ( bx+a \right ) ^{3}}\sqrt{x}}-{\frac{29\,Bb}{8\,{a}^{2} \left ( bx+a \right ) ^{3}}\sqrt{x}}+{\frac{105\,{b}^{2}A}{8\,{a}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{35\,Bb}{8\,{a}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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)^2,x)

[Out]

-2/3*A/a^4/x^(3/2)+8/x^(1/2)/a^5*A*b-2/x^(1/2)/a^4*B+41/8/a^5*b^4/(b*x+a)^3*x^(5
/2)*A-19/8/a^4*b^3/(b*x+a)^3*x^(5/2)*B+35/3/a^4*b^3/(b*x+a)^3*A*x^(3/2)-17/3/a^3
*b^2/(b*x+a)^3*B*x^(3/2)+55/8/a^3*b^2/(b*x+a)^3*x^(1/2)*A-29/8/a^2*b/(b*x+a)^3*x
^(1/2)*B+105/8/a^5*b^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-35/8/a^4*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)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.330493, size = 1, normalized size = 0.01 \[ \left [-\frac{32 \, A a^{4} + 210 \,{\left (B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 560 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} + 462 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} + 105 \,{\left ({\left (B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 3 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} + 3 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} - 3 \, A a^{3} b\right )} x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 96 \,{\left (B a^{4} - 3 \, A a^{3} b\right )} x}{48 \,{\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )} \sqrt{x}}, -\frac{16 \, A a^{4} + 105 \,{\left (B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 280 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} + 231 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} - 105 \,{\left ({\left (B a b^{3} - 3 \, A b^{4}\right )} x^{4} + 3 \,{\left (B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{3} + 3 \,{\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} - 3 \, A a^{3} b\right )} x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 48 \,{\left (B a^{4} - 3 \, A a^{3} b\right )} x}{24 \,{\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )} \sqrt{x}}\right ] \]

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)^2*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/48*(32*A*a^4 + 210*(B*a*b^3 - 3*A*b^4)*x^4 + 560*(B*a^2*b^2 - 3*A*a*b^3)*x^3
+ 462*(B*a^3*b - 3*A*a^2*b^2)*x^2 + 105*((B*a*b^3 - 3*A*b^4)*x^4 + 3*(B*a^2*b^2
- 3*A*a*b^3)*x^3 + 3*(B*a^3*b - 3*A*a^2*b^2)*x^2 + (B*a^4 - 3*A*a^3*b)*x)*sqrt(
x)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 96*(B*a^4 - 3*
A*a^3*b)*x)/((a^5*b^3*x^4 + 3*a^6*b^2*x^3 + 3*a^7*b*x^2 + a^8*x)*sqrt(x)), -1/24
*(16*A*a^4 + 105*(B*a*b^3 - 3*A*b^4)*x^4 + 280*(B*a^2*b^2 - 3*A*a*b^3)*x^3 + 231
*(B*a^3*b - 3*A*a^2*b^2)*x^2 - 105*((B*a*b^3 - 3*A*b^4)*x^4 + 3*(B*a^2*b^2 - 3*A
*a*b^3)*x^3 + 3*(B*a^3*b - 3*A*a^2*b^2)*x^2 + (B*a^4 - 3*A*a^3*b)*x)*sqrt(x)*sqr
t(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 48*(B*a^4 - 3*A*a^3*b)*x)/((a^5*b^3*x^4
+ 3*a^6*b^2*x^3 + 3*a^7*b*x^2 + a^8*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27447, size = 184, normalized size = 1.03 \[ -\frac{35 \,{\left (B a b - 3 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} - \frac{105 \, B a b^{3} x^{4} - 315 \, A b^{4} x^{4} + 280 \, B a^{2} b^{2} x^{3} - 840 \, A a b^{3} x^{3} + 231 \, B a^{3} b x^{2} - 693 \, A a^{2} b^{2} x^{2} + 48 \, B a^{4} x - 144 \, A a^{3} b x + 16 \, A a^{4}}{24 \,{\left (b x^{\frac{3}{2}} + a \sqrt{x}\right )}^{3} a^{5}} \]

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)^2*x^(5/2)),x, algorithm="giac")

[Out]

-35/8*(B*a*b - 3*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/24*(105*
B*a*b^3*x^4 - 315*A*b^4*x^4 + 280*B*a^2*b^2*x^3 - 840*A*a*b^3*x^3 + 231*B*a^3*b*
x^2 - 693*A*a^2*b^2*x^2 + 48*B*a^4*x - 144*A*a^3*b*x + 16*A*a^4)/((b*x^(3/2) + a
*sqrt(x))^3*a^5)