3.56 \(\int \frac{x^6 (A+B x^2)}{a+b x^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{a^2 x (A b-a B)}{b^4}-\frac{a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{x^5 (A b-a B)}{5 b^2}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{B x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.0605939, antiderivative size = 98, 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} \[ \frac{a^2 x (A b-a B)}{b^4}-\frac{a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{x^5 (A b-a B)}{5 b^2}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{B x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x^2))/(a + b*x^2),x]

[Out]

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

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^6 \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac{B x^7}{7 b}-\frac{(-7 A b+7 a B) \int \frac{x^6}{a+b x^2} \, dx}{7 b}\\ &=\frac{B x^7}{7 b}-\frac{(-7 A b+7 a B) \int \left (\frac{a^2}{b^3}-\frac{a x^2}{b^2}+\frac{x^4}{b}-\frac{a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx}{7 b}\\ &=\frac{a^2 (A b-a B) x}{b^4}-\frac{a (A b-a B) x^3}{3 b^3}+\frac{(A b-a B) x^5}{5 b^2}+\frac{B x^7}{7 b}-\frac{\left (a^3 (A b-a B)\right ) \int \frac{1}{a+b x^2} \, dx}{b^4}\\ &=\frac{a^2 (A b-a B) x}{b^4}-\frac{a (A b-a B) x^3}{3 b^3}+\frac{(A b-a B) x^5}{5 b^2}+\frac{B x^7}{7 b}-\frac{a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0709316, size = 98, normalized size = 1. \[ -\frac{a^2 x (a B-A b)}{b^4}+\frac{a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{x^5 (A b-a B)}{5 b^2}+\frac{a x^3 (a B-A b)}{3 b^3}+\frac{B x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(A + B*x^2))/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 116, normalized size = 1.2 \begin{align*}{\frac{B{x}^{7}}{7\,b}}+{\frac{A{x}^{5}}{5\,b}}-{\frac{B{x}^{5}a}{5\,{b}^{2}}}-{\frac{aA{x}^{3}}{3\,{b}^{2}}}+{\frac{B{x}^{3}{a}^{2}}{3\,{b}^{3}}}+{\frac{{a}^{2}Ax}{{b}^{3}}}-{\frac{B{a}^{3}x}{{b}^{4}}}-{\frac{{a}^{3}A}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B{a}^{4}}{{b}^{4}}\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^6*(B*x^2+A)/(b*x^2+a),x)

[Out]

1/7*B*x^7/b+1/5/b*A*x^5-1/5/b^2*B*x^5*a-1/3/b^2*A*x^3*a+1/3/b^3*B*x^3*a^2+1/b^3*A*a^2*x-1/b^4*B*a^3*x-a^3/b^3/
(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*A+a^4/b^4/(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^6*(B*x^2+A)/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26916, size = 487, normalized size = 4.97 \begin{align*} \left [\frac{30 \, B b^{3} x^{7} - 42 \,{\left (B a b^{2} - A b^{3}\right )} x^{5} + 70 \,{\left (B a^{2} b - A a b^{2}\right )} x^{3} - 105 \,{\left (B a^{3} - A a^{2} 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 ) - 210 \,{\left (B a^{3} - A a^{2} b\right )} x}{210 \, b^{4}}, \frac{15 \, B b^{3} x^{7} - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{5} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{3} + 105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 105 \,{\left (B a^{3} - A a^{2} b\right )} x}{105 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/210*(30*B*b^3*x^7 - 42*(B*a*b^2 - A*b^3)*x^5 + 70*(B*a^2*b - A*a*b^2)*x^3 - 105*(B*a^3 - A*a^2*b)*sqrt(-a/b
)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 210*(B*a^3 - A*a^2*b)*x)/b^4, 1/105*(15*B*b^3*x^7 - 21*(B*
a*b^2 - A*b^3)*x^5 + 35*(B*a^2*b - A*a*b^2)*x^3 + 105*(B*a^3 - A*a^2*b)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 10
5*(B*a^3 - A*a^2*b)*x)/b^4]

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Sympy [A]  time = 0.501098, size = 173, normalized size = 1.77 \begin{align*} \frac{B x^{7}}{7 b} - \frac{\sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right ) \log{\left (- \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right ) \log{\left (\frac{b^{4} \sqrt{- \frac{a^{5}}{b^{9}}} \left (- A b + B a\right )}{- A a^{2} b + B a^{3}} + x \right )}}{2} - \frac{x^{5} \left (- A b + B a\right )}{5 b^{2}} + \frac{x^{3} \left (- A a b + B a^{2}\right )}{3 b^{3}} - \frac{x \left (- A a^{2} b + B a^{3}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(B*x**2+A)/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.12601, size = 146, normalized size = 1.49 \begin{align*} \frac{{\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, B b^{6} x^{7} - 21 \, B a b^{5} x^{5} + 21 \, A b^{6} x^{5} + 35 \, B a^{2} b^{4} x^{3} - 35 \, A a b^{5} x^{3} - 105 \, B a^{3} b^{3} x + 105 \, A a^{2} b^{4} x}{105 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a),x, algorithm="giac")

[Out]

(B*a^4 - A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*B*b^6*x^7 - 21*B*a*b^5*x^5 + 21*A*b^6*x^5
+ 35*B*a^2*b^4*x^3 - 35*A*a*b^5*x^3 - 105*B*a^3*b^3*x + 105*A*a^2*b^4*x)/b^7