3.58 \(\int \frac{x^4 (A+B x^2)}{a+b x^2} \, dx\)

Optimal. Leaf size=77 \[ \frac{a^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^3 (A b-a B)}{3 b^2}-\frac{a x (A b-a B)}{b^3}+\frac{B x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0501443, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {459, 302, 205} \[ \frac{a^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^3 (A b-a B)}{3 b^2}-\frac{a x (A b-a B)}{b^3}+\frac{B x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x^2))/(a + b*x^2),x]

[Out]

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^4 \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac{B x^5}{5 b}-\frac{(-5 A b+5 a B) \int \frac{x^4}{a+b x^2} \, dx}{5 b}\\ &=\frac{B x^5}{5 b}-\frac{(-5 A b+5 a B) \int \left (-\frac{a}{b^2}+\frac{x^2}{b}+\frac{a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{5 b}\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^3}{3 b^2}+\frac{B x^5}{5 b}+\frac{\left (a^2 (A b-a B)\right ) \int \frac{1}{a+b x^2} \, dx}{b^3}\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^3}{3 b^2}+\frac{B x^5}{5 b}+\frac{a^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0484471, size = 77, normalized size = 1. \[ -\frac{a^{3/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^3 (A b-a B)}{3 b^2}+\frac{a x (a B-A b)}{b^3}+\frac{B x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 92, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{5\,b}}+{\frac{A{x}^{3}}{3\,b}}-{\frac{B{x}^{3}a}{3\,{b}^{2}}}-{\frac{aAx}{{b}^{2}}}+{\frac{{a}^{2}Bx}{{b}^{3}}}+{\frac{{a}^{2}A}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{B{a}^{3}}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x^2+A)/(b*x^2+a),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26106, size = 381, normalized size = 4.95 \begin{align*} \left [\frac{6 \, B b^{2} x^{5} - 10 \,{\left (B a b - A b^{2}\right )} x^{3} - 15 \,{\left (B a^{2} - A a b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 30 \,{\left (B a^{2} - A a b\right )} x}{30 \, b^{3}}, \frac{3 \, B b^{2} x^{5} - 5 \,{\left (B a b - A b^{2}\right )} x^{3} - 15 \,{\left (B a^{2} - A a b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 15 \,{\left (B a^{2} - A a b\right )} x}{15 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(6*B*b^2*x^5 - 10*(B*a*b - A*b^2)*x^3 - 15*(B*a^2 - A*a*b)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)
/(b*x^2 + a)) + 30*(B*a^2 - A*a*b)*x)/b^3, 1/15*(3*B*b^2*x^5 - 5*(B*a*b - A*b^2)*x^3 - 15*(B*a^2 - A*a*b)*sqrt
(a/b)*arctan(b*x*sqrt(a/b)/a) + 15*(B*a^2 - A*a*b)*x)/b^3]

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Sympy [B]  time = 0.478674, size = 150, normalized size = 1.95 \begin{align*} \frac{B x^{5}}{5 b} + \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right ) \log{\left (- \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right )}{- A a b + B a^{2}} + x \right )}}{2} - \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right ) \log{\left (\frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (- A b + B a\right )}{- A a b + B a^{2}} + x \right )}}{2} - \frac{x^{3} \left (- A b + B a\right )}{3 b^{2}} + \frac{x \left (- A a b + B a^{2}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x**2+A)/(b*x**2+a),x)

[Out]

B*x**5/(5*b) + sqrt(-a**3/b**7)*(-A*b + B*a)*log(-b**3*sqrt(-a**3/b**7)*(-A*b + B*a)/(-A*a*b + B*a**2) + x)/2
- sqrt(-a**3/b**7)*(-A*b + B*a)*log(b**3*sqrt(-a**3/b**7)*(-A*b + B*a)/(-A*a*b + B*a**2) + x)/2 - x**3*(-A*b +
B*a)/(3*b**2) + x*(-A*a*b + B*a**2)/b**3

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Giac [A]  time = 1.13787, size = 115, normalized size = 1.49 \begin{align*} -\frac{{\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, B b^{4} x^{5} - 5 \, B a b^{3} x^{3} + 5 \, A b^{4} x^{3} + 15 \, B a^{2} b^{2} x - 15 \, A a b^{3} x}{15 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*a^3 - A*a^2*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*B*b^4*x^5 - 5*B*a*b^3*x^3 + 5*A*b^4*x^3 + 1
5*B*a^2*b^2*x - 15*A*a*b^3*x)/b^5