3.70 \(\int \frac{A+B x^2}{x^8 (a+b x^2)} \, dx\)

Optimal. Leaf size=99 \[ \frac{b^2 (A b-a B)}{a^4 x}+\frac{b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}-\frac{b (A b-a B)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7} \]

[Out]

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

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Rubi [A]  time = 0.0681236, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {453, 325, 205} \[ \frac{b^2 (A b-a B)}{a^4 x}+\frac{b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}-\frac{b (A b-a B)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^8*(a + b*x^2)),x]

[Out]

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

Rule 453

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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{A+B x^2}{x^8 \left (a+b x^2\right )} \, dx &=-\frac{A}{7 a x^7}-\frac{(7 A b-7 a B) \int \frac{1}{x^6 \left (a+b x^2\right )} \, dx}{7 a}\\ &=-\frac{A}{7 a x^7}+\frac{A b-a B}{5 a^2 x^5}+\frac{(b (A b-a B)) \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac{A}{7 a x^7}+\frac{A b-a B}{5 a^2 x^5}-\frac{b (A b-a B)}{3 a^3 x^3}-\frac{\left (b^2 (A b-a B)\right ) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{a^3}\\ &=-\frac{A}{7 a x^7}+\frac{A b-a B}{5 a^2 x^5}-\frac{b (A b-a B)}{3 a^3 x^3}+\frac{b^2 (A b-a B)}{a^4 x}+\frac{\left (b^3 (A b-a B)\right ) \int \frac{1}{a+b x^2} \, dx}{a^4}\\ &=-\frac{A}{7 a x^7}+\frac{A b-a B}{5 a^2 x^5}-\frac{b (A b-a B)}{3 a^3 x^3}+\frac{b^2 (A b-a B)}{a^4 x}+\frac{b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0695324, size = 101, normalized size = 1.02 \[ -\frac{b^2 (a B-A b)}{a^4 x}-\frac{b^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{9/2}}+\frac{b (a B-A b)}{3 a^3 x^3}+\frac{A b-a B}{5 a^2 x^5}-\frac{A}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^8*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.005, size = 120, normalized size = 1.2 \begin{align*} -{\frac{A}{7\,a{x}^{7}}}+{\frac{Ab}{5\,{a}^{2}{x}^{5}}}-{\frac{B}{5\,a{x}^{5}}}-{\frac{{b}^{2}A}{3\,{a}^{3}{x}^{3}}}+{\frac{bB}{3\,{a}^{2}{x}^{3}}}+{\frac{{b}^{3}A}{{a}^{4}x}}-{\frac{{b}^{2}B}{{a}^{3}x}}+{\frac{A{b}^{4}}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}B}{{a}^{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((B*x^2+A)/x^8/(b*x^2+a),x)

[Out]

-1/7*A/a/x^7+1/5/a^2/x^5*A*b-1/5/a/x^5*B-1/3/a^3*b^2/x^3*A+1/3/a^2*b/x^3*B+1/a^4*b^3/x*A-1/a^3*b^2/x*B+b^4/a^4
/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*A-b^3/a^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((B*x^2+A)/x^8/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27974, size = 506, normalized size = 5.11 \begin{align*} \left [-\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 210 \,{\left (B a b^{2} - A b^{3}\right )} x^{6} - 70 \,{\left (B a^{2} b - A a b^{2}\right )} x^{4} + 30 \, A a^{3} + 42 \,{\left (B a^{3} - A a^{2} b\right )} x^{2}}{210 \, a^{4} x^{7}}, -\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 105 \,{\left (B a b^{2} - A b^{3}\right )} x^{6} - 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 21 \,{\left (B a^{3} - A a^{2} b\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/210*(105*(B*a*b^2 - A*b^3)*x^7*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 210*(B*a*b^2 -
A*b^3)*x^6 - 70*(B*a^2*b - A*a*b^2)*x^4 + 30*A*a^3 + 42*(B*a^3 - A*a^2*b)*x^2)/(a^4*x^7), -1/105*(105*(B*a*b^
2 - A*b^3)*x^7*sqrt(b/a)*arctan(x*sqrt(b/a)) + 105*(B*a*b^2 - A*b^3)*x^6 - 35*(B*a^2*b - A*a*b^2)*x^4 + 15*A*a
^3 + 21*(B*a^3 - A*a^2*b)*x^2)/(a^4*x^7)]

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Sympy [B]  time = 0.840096, size = 187, normalized size = 1.89 \begin{align*} \frac{\sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right ) \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right ) \log{\left (\frac{a^{5} \sqrt{- \frac{b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} - \frac{15 A a^{3} + x^{6} \left (- 105 A b^{3} + 105 B a b^{2}\right ) + x^{4} \left (35 A a b^{2} - 35 B a^{2} b\right ) + x^{2} \left (- 21 A a^{2} b + 21 B a^{3}\right )}{105 a^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**8/(b*x**2+a),x)

[Out]

sqrt(-b**5/a**9)*(-A*b + B*a)*log(-a**5*sqrt(-b**5/a**9)*(-A*b + B*a)/(-A*b**4 + B*a*b**3) + x)/2 - sqrt(-b**5
/a**9)*(-A*b + B*a)*log(a**5*sqrt(-b**5/a**9)*(-A*b + B*a)/(-A*b**4 + B*a*b**3) + x)/2 - (15*A*a**3 + x**6*(-1
05*A*b**3 + 105*B*a*b**2) + x**4*(35*A*a*b**2 - 35*B*a**2*b) + x**2*(-21*A*a**2*b + 21*B*a**3))/(105*a**4*x**7
)

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Giac [A]  time = 1.23837, size = 143, normalized size = 1.44 \begin{align*} -\frac{{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} - \frac{105 \, B a b^{2} x^{6} - 105 \, A b^{3} x^{6} - 35 \, B a^{2} b x^{4} + 35 \, A a b^{2} x^{4} + 21 \, B a^{3} x^{2} - 21 \, A a^{2} b x^{2} + 15 \, A a^{3}}{105 \, a^{4} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a),x, algorithm="giac")

[Out]

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