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

Optimal. Leaf size=94 \[ \frac{3 (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{7/2}}-\frac{x (5 A b-9 a B)}{8 b^3 \left (a+b x^2\right )}+\frac{a x (A b-a B)}{4 b^3 \left (a+b x^2\right )^2}+\frac{B x}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.208867, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{7/2}}-\frac{x (5 A b-9 a B)}{8 b^3 \left (a+b x^2\right )}+\frac{a x (A b-a B)}{4 b^3 \left (a+b x^2\right )^2}+\frac{B x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.7878, size = 88, normalized size = 0.94 \[ \frac{B x}{b^{3}} + \frac{a x \left (A b - B a\right )}{4 b^{3} \left (a + b x^{2}\right )^{2}} - \frac{x \left (5 A b - 9 B a\right )}{8 b^{3} \left (a + b x^{2}\right )} + \frac{3 \left (A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{a} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.121026, size = 91, normalized size = 0.97 \[ \frac{x \left (15 a^2 B+a \left (25 b B x^2-3 A b\right )+b^2 x^2 \left (8 B x^2-5 A\right )\right )}{8 b^3 \left (a+b x^2\right )^2}+\frac{3 (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(15*a^2*B + b^2*x^2*(-5*A + 8*B*x^2) + a*(-3*A*b + 25*b*B*x^2)))/(8*b^3*(a +
b*x^2)^2) + (3*(A*b - 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(7/2))

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Maple [A]  time = 0.014, size = 122, normalized size = 1.3 \[{\frac{Bx}{{b}^{3}}}-{\frac{5\,A{x}^{3}}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,B{x}^{3}a}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,aAx}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,Bx{a}^{2}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,A}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,Ba}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b^3-5/8/b/(b*x^2+a)^2*A*x^3+9/8/b^2/(b*x^2+a)^2*B*x^3*a-3/8/b^2/(b*x^2+a)^2*
A*x*a+7/8/b^3/(b*x^2+a)^2*B*x*a^2+3/8/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*A-
15/8/b^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*B*a

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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)*x^4/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228148, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (5 \, B a b^{2} - A b^{3}\right )} x^{4} + 5 \, B a^{3} - A a^{2} b + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (8 \, B b^{2} x^{5} + 5 \,{\left (5 \, B a b - A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - A a b\right )} x\right )} \sqrt{-a b}}{16 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{-a b}}, -\frac{3 \,{\left ({\left (5 \, B a b^{2} - A b^{3}\right )} x^{4} + 5 \, B a^{3} - A a^{2} b + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (8 \, B b^{2} x^{5} + 5 \,{\left (5 \, B a b - A b^{2}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} - A a b\right )} x\right )} \sqrt{a b}}{8 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(3*((5*B*a*b^2 - A*b^3)*x^4 + 5*B*a^3 - A*a^2*b + 2*(5*B*a^2*b - A*a*b^2)
*x^2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) - 2*(8*B*b^2*x^5 + 5*(
5*B*a*b - A*b^2)*x^3 + 3*(5*B*a^2 - A*a*b)*x)*sqrt(-a*b))/((b^5*x^4 + 2*a*b^4*x^
2 + a^2*b^3)*sqrt(-a*b)), -1/8*(3*((5*B*a*b^2 - A*b^3)*x^4 + 5*B*a^3 - A*a^2*b +
2*(5*B*a^2*b - A*a*b^2)*x^2)*arctan(sqrt(a*b)*x/a) - (8*B*b^2*x^5 + 5*(5*B*a*b
- A*b^2)*x^3 + 3*(5*B*a^2 - A*a*b)*x)*sqrt(a*b))/((b^5*x^4 + 2*a*b^4*x^2 + a^2*b
^3)*sqrt(a*b))]

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Sympy [A]  time = 4.97609, size = 194, normalized size = 2.06 \[ \frac{B x}{b^{3}} + \frac{3 \sqrt{- \frac{1}{a b^{7}}} \left (- A b + 5 B a\right ) \log{\left (- \frac{3 a b^{3} \sqrt{- \frac{1}{a b^{7}}} \left (- A b + 5 B a\right )}{- 3 A b + 15 B a} + x \right )}}{16} - \frac{3 \sqrt{- \frac{1}{a b^{7}}} \left (- A b + 5 B a\right ) \log{\left (\frac{3 a b^{3} \sqrt{- \frac{1}{a b^{7}}} \left (- A b + 5 B a\right )}{- 3 A b + 15 B a} + x \right )}}{16} + \frac{x^{3} \left (- 5 A b^{2} + 9 B a b\right ) + x \left (- 3 A a b + 7 B a^{2}\right )}{8 a^{2} b^{3} + 16 a b^{4} x^{2} + 8 b^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b**3 + 3*sqrt(-1/(a*b**7))*(-A*b + 5*B*a)*log(-3*a*b**3*sqrt(-1/(a*b**7))*(-
A*b + 5*B*a)/(-3*A*b + 15*B*a) + x)/16 - 3*sqrt(-1/(a*b**7))*(-A*b + 5*B*a)*log(
3*a*b**3*sqrt(-1/(a*b**7))*(-A*b + 5*B*a)/(-3*A*b + 15*B*a) + x)/16 + (x**3*(-5*
A*b**2 + 9*B*a*b) + x*(-3*A*a*b + 7*B*a**2))/(8*a**2*b**3 + 16*a*b**4*x**2 + 8*b
**5*x**4)

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GIAC/XCAS [A]  time = 0.227916, size = 108, normalized size = 1.15 \[ \frac{B x}{b^{3}} - \frac{3 \,{\left (5 \, B a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{3}} + \frac{9 \, B a b x^{3} - 5 \, A b^{2} x^{3} + 7 \, B a^{2} x - 3 \, A a b x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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