3.106 \(\int \frac{A+B x^2}{x^6 (a+b x^2)^3} \, dx\)

Optimal. Leaf size=142 \[ -\frac{b^2 x (15 A b-11 a B)}{8 a^5 \left (a+b x^2\right )}-\frac{b^2 x (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}-\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{A}{5 a^3 x^5} \]

[Out]

-A/(5*a^3*x^5) + (3*A*b - a*B)/(3*a^4*x^3) - (3*b*(2*A*b - a*B))/(a^5*x) - (b^2*(A*b - a*B)*x)/(4*a^4*(a + b*x
^2)^2) - (b^2*(15*A*b - 11*a*B)*x)/(8*a^5*(a + b*x^2)) - (7*b^(3/2)*(9*A*b - 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(8*a^(11/2))

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Rubi [A]  time = 0.329677, antiderivative size = 142, 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, Rules used = {456, 1805, 1802, 205} \[ -\frac{b^2 x (15 A b-11 a B)}{8 a^5 \left (a+b x^2\right )}-\frac{b^2 x (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}-\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{A}{5 a^3 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^6*(a + b*x^2)^3),x]

[Out]

-A/(5*a^3*x^5) + (3*A*b - a*B)/(3*a^4*x^3) - (3*b*(2*A*b - a*B))/(a^5*x) - (b^2*(A*b - a*B)*x)/(4*a^4*(a + b*x
^2)^2) - (b^2*(15*A*b - 11*a*B)*x)/(8*a^5*(a + b*x^2)) - (7*b^(3/2)*(9*A*b - 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]
])/(8*a^(11/2))

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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^6 \left (a+b x^2\right )^3} \, dx &=-\frac{b^2 (A b-a B) x}{4 a^4 \left (a+b x^2\right )^2}-\frac{1}{4} b^2 \int \frac{-\frac{4 A}{a b^2}+\frac{4 (A b-a B) x^2}{a^2 b^2}-\frac{4 (A b-a B) x^4}{a^3 b}+\frac{3 (A b-a B) x^6}{a^4}}{x^6 \left (a+b x^2\right )^2} \, dx\\ &=-\frac{b^2 (A b-a B) x}{4 a^4 \left (a+b x^2\right )^2}-\frac{b^2 (15 A b-11 a B) x}{8 a^5 \left (a+b x^2\right )}+\frac{b^2 \int \frac{\frac{8 A}{a b^2}-\frac{8 (2 A b-a B) x^2}{a^2 b^2}+\frac{8 (3 A b-2 a B) x^4}{a^3 b}-\frac{(15 A b-11 a B) x^6}{a^4}}{x^6 \left (a+b x^2\right )} \, dx}{8 a}\\ &=-\frac{b^2 (A b-a B) x}{4 a^4 \left (a+b x^2\right )^2}-\frac{b^2 (15 A b-11 a B) x}{8 a^5 \left (a+b x^2\right )}+\frac{b^2 \int \left (\frac{8 A}{a^2 b^2 x^6}+\frac{8 (-3 A b+a B)}{a^3 b^2 x^4}-\frac{24 (-2 A b+a B)}{a^4 b x^2}+\frac{7 (-9 A b+5 a B)}{a^4 \left (a+b x^2\right )}\right ) \, dx}{8 a}\\ &=-\frac{A}{5 a^3 x^5}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 (A b-a B) x}{4 a^4 \left (a+b x^2\right )^2}-\frac{b^2 (15 A b-11 a B) x}{8 a^5 \left (a+b x^2\right )}-\frac{\left (7 b^2 (9 A b-5 a B)\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^5}\\ &=-\frac{A}{5 a^3 x^5}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 (A b-a B) x}{4 a^4 \left (a+b x^2\right )^2}-\frac{b^2 (15 A b-11 a B) x}{8 a^5 \left (a+b x^2\right )}-\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.100048, size = 139, normalized size = 0.98 \[ \frac{7 a^2 b^2 x^4 \left (125 B x^2-72 A\right )+8 a^3 b x^2 \left (9 A+35 B x^2\right )-8 a^4 \left (3 A+5 B x^2\right )+525 a b^3 x^6 \left (B x^2-3 A\right )-945 A b^4 x^8}{120 a^5 x^5 \left (a+b x^2\right )^2}+\frac{7 b^{3/2} (5 a B-9 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^3),x]

[Out]

(-945*A*b^4*x^8 + 525*a*b^3*x^6*(-3*A + B*x^2) - 8*a^4*(3*A + 5*B*x^2) + 8*a^3*b*x^2*(9*A + 35*B*x^2) + 7*a^2*
b^2*x^4*(-72*A + 125*B*x^2))/(120*a^5*x^5*(a + b*x^2)^2) + (7*b^(3/2)*(-9*A*b + 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt
[a]])/(8*a^(11/2))

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Maple [A]  time = 0.015, size = 177, normalized size = 1.3 \begin{align*} -{\frac{A}{5\,{a}^{3}{x}^{5}}}+{\frac{Ab}{{a}^{4}{x}^{3}}}-{\frac{B}{3\,{a}^{3}{x}^{3}}}-6\,{\frac{A{b}^{2}}{{a}^{5}x}}+3\,{\frac{Bb}{{a}^{4}x}}-{\frac{15\,{b}^{4}A{x}^{3}}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,{b}^{3}B{x}^{3}}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{17\,{b}^{3}Ax}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,{b}^{2}Bx}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{b}^{3}A}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{b}^{2}B}{8\,{a}^{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((B*x^2+A)/x^6/(b*x^2+a)^3,x)

[Out]

-1/5*A/a^3/x^5+1/a^4/x^3*A*b-1/3/a^3/x^3*B-6*b^2/a^5/x*A+3*b/a^4/x*B-15/8/a^5*b^4/(b*x^2+a)^2*A*x^3+11/8/a^4*b
^3/(b*x^2+a)^2*B*x^3-17/8/a^4*b^3/(b*x^2+a)^2*A*x+13/8/a^3*b^2/(b*x^2+a)^2*B*x-63/8/a^5*b^3/(a*b)^(1/2)*arctan
(b*x/(a*b)^(1/2))*A+35/8/a^4*b^2/(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^6/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34927, size = 909, normalized size = 6.4 \begin{align*} \left [\frac{210 \,{\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{8} + 350 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} - 48 \, A a^{4} + 112 \,{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} - 16 \,{\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x^{2} - 105 \,{\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{9} + 2 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{7} +{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{240 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}, \frac{105 \,{\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{8} + 175 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{6} - 24 \, A a^{4} + 56 \,{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{4} - 8 \,{\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x^{2} + 105 \,{\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{9} + 2 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{7} +{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{120 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/240*(210*(5*B*a*b^3 - 9*A*b^4)*x^8 + 350*(5*B*a^2*b^2 - 9*A*a*b^3)*x^6 - 48*A*a^4 + 112*(5*B*a^3*b - 9*A*a^
2*b^2)*x^4 - 16*(5*B*a^4 - 9*A*a^3*b)*x^2 - 105*((5*B*a*b^3 - 9*A*b^4)*x^9 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*x^7 +
(5*B*a^3*b - 9*A*a^2*b^2)*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^5*b^2*x^9 + 2*a
^6*b*x^7 + a^7*x^5), 1/120*(105*(5*B*a*b^3 - 9*A*b^4)*x^8 + 175*(5*B*a^2*b^2 - 9*A*a*b^3)*x^6 - 24*A*a^4 + 56*
(5*B*a^3*b - 9*A*a^2*b^2)*x^4 - 8*(5*B*a^4 - 9*A*a^3*b)*x^2 + 105*((5*B*a*b^3 - 9*A*b^4)*x^9 + 2*(5*B*a^2*b^2
- 9*A*a*b^3)*x^7 + (5*B*a^3*b - 9*A*a^2*b^2)*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^5*b^2*x^9 + 2*a^6*b*x^7 +
a^7*x^5)]

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Sympy [A]  time = 1.64817, size = 260, normalized size = 1.83 \begin{align*} - \frac{7 \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right ) \log{\left (- \frac{7 a^{6} \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right )}{- 63 A b^{3} + 35 B a b^{2}} + x \right )}}{16} + \frac{7 \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right ) \log{\left (\frac{7 a^{6} \sqrt{- \frac{b^{3}}{a^{11}}} \left (- 9 A b + 5 B a\right )}{- 63 A b^{3} + 35 B a b^{2}} + x \right )}}{16} + \frac{- 24 A a^{4} + x^{8} \left (- 945 A b^{4} + 525 B a b^{3}\right ) + x^{6} \left (- 1575 A a b^{3} + 875 B a^{2} b^{2}\right ) + x^{4} \left (- 504 A a^{2} b^{2} + 280 B a^{3} b\right ) + x^{2} \left (72 A a^{3} b - 40 B a^{4}\right )}{120 a^{7} x^{5} + 240 a^{6} b x^{7} + 120 a^{5} b^{2} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**6/(b*x**2+a)**3,x)

[Out]

-7*sqrt(-b**3/a**11)*(-9*A*b + 5*B*a)*log(-7*a**6*sqrt(-b**3/a**11)*(-9*A*b + 5*B*a)/(-63*A*b**3 + 35*B*a*b**2
) + x)/16 + 7*sqrt(-b**3/a**11)*(-9*A*b + 5*B*a)*log(7*a**6*sqrt(-b**3/a**11)*(-9*A*b + 5*B*a)/(-63*A*b**3 + 3
5*B*a*b**2) + x)/16 + (-24*A*a**4 + x**8*(-945*A*b**4 + 525*B*a*b**3) + x**6*(-1575*A*a*b**3 + 875*B*a**2*b**2
) + x**4*(-504*A*a**2*b**2 + 280*B*a**3*b) + x**2*(72*A*a**3*b - 40*B*a**4))/(120*a**7*x**5 + 240*a**6*b*x**7
+ 120*a**5*b**2*x**9)

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Giac [A]  time = 1.15926, size = 182, normalized size = 1.28 \begin{align*} \frac{7 \,{\left (5 \, B a b^{2} - 9 \, A b^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} + \frac{11 \, B a b^{3} x^{3} - 15 \, A b^{4} x^{3} + 13 \, B a^{2} b^{2} x - 17 \, A a b^{3} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{5}} + \frac{45 \, B a b x^{4} - 90 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 15 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{5} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^3,x, algorithm="giac")

[Out]

7/8*(5*B*a*b^2 - 9*A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/8*(11*B*a*b^3*x^3 - 15*A*b^4*x^3 + 13*B*a^
2*b^2*x - 17*A*a*b^3*x)/((b*x^2 + a)^2*a^5) + 1/15*(45*B*a*b*x^4 - 90*A*b^2*x^4 - 5*B*a^2*x^2 + 15*A*a*b*x^2 -
3*A*a^2)/(a^5*x^5)