3.105 \(\int \frac{A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=117 \[ \frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{b x (11 A b-7 a B)}{8 a^4 \left (a+b x^2\right )}+\frac{3 A b-a B}{a^4 x}+\frac{b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}-\frac{A}{3 a^3 x^3} \]

[Out]

-A/(3*a^3*x^3) + (3*A*b - a*B)/(a^4*x) + (b*(A*b - a*B)*x)/(4*a^3*(a + b*x^2)^2)
+ (b*(11*A*b - 7*a*B)*x)/(8*a^4*(a + b*x^2)) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTa
n[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))

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Rubi [A]  time = 0.432281, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{b x (11 A b-7 a B)}{8 a^4 \left (a+b x^2\right )}+\frac{3 A b-a B}{a^4 x}+\frac{b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}-\frac{A}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^4*(a + b*x^2)^3),x]

[Out]

-A/(3*a^3*x^3) + (3*A*b - a*B)/(a^4*x) + (b*(A*b - a*B)*x)/(4*a^3*(a + b*x^2)^2)
+ (b*(11*A*b - 7*a*B)*x)/(8*a^4*(a + b*x^2)) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTa
n[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))

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Rubi in Sympy [A]  time = 63.2241, size = 109, normalized size = 0.93 \[ - \frac{A}{3 a^{3} x^{3}} + \frac{b x \left (A b - B a\right )}{4 a^{3} \left (a + b x^{2}\right )^{2}} + \frac{b x \left (11 A b - 7 B a\right )}{8 a^{4} \left (a + b x^{2}\right )} + \frac{3 A b - B a}{a^{4} x} + \frac{5 \sqrt{b} \left (7 A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**4/(b*x**2+a)**3,x)

[Out]

-A/(3*a**3*x**3) + b*x*(A*b - B*a)/(4*a**3*(a + b*x**2)**2) + b*x*(11*A*b - 7*B*
a)/(8*a**4*(a + b*x**2)) + (3*A*b - B*a)/(a**4*x) + 5*sqrt(b)*(7*A*b - 3*B*a)*at
an(sqrt(b)*x/sqrt(a))/(8*a**(9/2))

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Mathematica [A]  time = 0.148157, size = 116, normalized size = 0.99 \[ \frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{-8 a^3 \left (A+3 B x^2\right )+a^2 b x^2 \left (56 A-75 B x^2\right )+5 a b^2 x^4 \left (35 A-9 B x^2\right )+105 A b^3 x^6}{24 a^4 x^3 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^3),x]

[Out]

(105*A*b^3*x^6 + a^2*b*x^2*(56*A - 75*B*x^2) + 5*a*b^2*x^4*(35*A - 9*B*x^2) - 8*
a^3*(A + 3*B*x^2))/(24*a^4*x^3*(a + b*x^2)^2) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTa
n[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))

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Maple [A]  time = 0.02, size = 152, normalized size = 1.3 \[ -{\frac{A}{3\,{a}^{3}{x}^{3}}}+3\,{\frac{Ab}{x{a}^{4}}}-{\frac{B}{x{a}^{3}}}+{\frac{11\,{b}^{3}A{x}^{3}}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{b}^{2}B{x}^{3}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,Ax{b}^{2}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bBx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{b}^{2}A}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,Bb}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^4/(b*x^2+a)^3,x)

[Out]

-1/3*A/a^3/x^3+3/x/a^4*A*b-1/x/a^3*B+11/8/a^4*b^3/(b*x^2+a)^2*A*x^3-7/8/a^3*b^2/
(b*x^2+a)^2*B*x^3+13/8/a^3*b^2/(b*x^2+a)^2*A*x-9/8/a^2*b/(b*x^2+a)^2*B*x+35/8/a^
4*b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*A-15/8/a^3*b/(a*b)^(1/2)*arctan(x*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^2 + A)/((b*x^2 + a)^3*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2464, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 50 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 16 \, A a^{3} + 16 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{48 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}, -\frac{15 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 25 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 8 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{24 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^3*x^4),x, algorithm="fricas")

[Out]

[-1/48*(30*(3*B*a*b^2 - 7*A*b^3)*x^6 + 50*(3*B*a^2*b - 7*A*a*b^2)*x^4 + 16*A*a^3
+ 16*(3*B*a^3 - 7*A*a^2*b)*x^2 + 15*((3*B*a*b^2 - 7*A*b^3)*x^7 + 2*(3*B*a^2*b -
7*A*a*b^2)*x^5 + (3*B*a^3 - 7*A*a^2*b)*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(
-b/a) - a)/(b*x^2 + a)))/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3), -1/24*(15*(3*B*a
*b^2 - 7*A*b^3)*x^6 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^4 + 8*A*a^3 + 8*(3*B*a^3 - 7*
A*a^2*b)*x^2 + 15*((3*B*a*b^2 - 7*A*b^3)*x^7 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^5 + (
3*B*a^3 - 7*A*a^2*b)*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^4*b^2*x^7 + 2*
a^5*b*x^5 + a^6*x^3)]

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Sympy [A]  time = 4.74115, size = 226, normalized size = 1.93 \[ \frac{5 \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log{\left (- \frac{5 a^{5} \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} - \frac{5 \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log{\left (\frac{5 a^{5} \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} - \frac{8 A a^{3} + x^{6} \left (- 105 A b^{3} + 45 B a b^{2}\right ) + x^{4} \left (- 175 A a b^{2} + 75 B a^{2} b\right ) + x^{2} \left (- 56 A a^{2} b + 24 B a^{3}\right )}{24 a^{6} x^{3} + 48 a^{5} b x^{5} + 24 a^{4} b^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**4/(b*x**2+a)**3,x)

[Out]

5*sqrt(-b/a**9)*(-7*A*b + 3*B*a)*log(-5*a**5*sqrt(-b/a**9)*(-7*A*b + 3*B*a)/(-35
*A*b**2 + 15*B*a*b) + x)/16 - 5*sqrt(-b/a**9)*(-7*A*b + 3*B*a)*log(5*a**5*sqrt(-
b/a**9)*(-7*A*b + 3*B*a)/(-35*A*b**2 + 15*B*a*b) + x)/16 - (8*A*a**3 + x**6*(-10
5*A*b**3 + 45*B*a*b**2) + x**4*(-175*A*a*b**2 + 75*B*a**2*b) + x**2*(-56*A*a**2*
b + 24*B*a**3))/(24*a**6*x**3 + 48*a**5*b*x**5 + 24*a**4*b**2*x**7)

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GIAC/XCAS [A]  time = 0.250669, size = 146, normalized size = 1.25 \[ -\frac{5 \,{\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{4}} - \frac{7 \, B a b^{2} x^{3} - 11 \, A b^{3} x^{3} + 9 \, B a^{2} b x - 13 \, A a b^{2} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{4}} - \frac{3 \, B a x^{2} - 9 \, A b x^{2} + A a}{3 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^3*x^4),x, algorithm="giac")

[Out]

-5/8*(3*B*a*b - 7*A*b^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/8*(7*B*a*b^2*
x^3 - 11*A*b^3*x^3 + 9*B*a^2*b*x - 13*A*a*b^2*x)/((b*x^2 + a)^2*a^4) - 1/3*(3*B*
a*x^2 - 9*A*b*x^2 + A*a)/(a^4*x^3)