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

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

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Rubi [A]  time = 0.33, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {456, 1805, 1802, 205} \begin {gather*} -\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} \end {gather*}

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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.10, size = 139, normalized size = 0.98 \begin {gather*} \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}}+\frac {-8 a^4 \left (3 A+5 B x^2\right )+8 a^3 b x^2 \left (9 A+35 B x^2\right )+7 a^2 b^2 x^4 \left (125 B x^2-72 A\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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 426, normalized size = 3.00 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.36, size = 135, normalized size = 0.95 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.03, size = 177, normalized size = 1.25 \begin {gather*} -\frac {15 A \,b^{4} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{5}}+\frac {11 B \,b^{3} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{4}}-\frac {17 A \,b^{3} x}{8 \left (b \,x^{2}+a \right )^{2} a^{4}}+\frac {13 B \,b^{2} x}{8 \left (b \,x^{2}+a \right )^{2} a^{3}}-\frac {63 A \,b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{5}}+\frac {35 B \,b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{4}}-\frac {6 A \,b^{2}}{a^{5} x}+\frac {3 B b}{a^{4} x}+\frac {A b}{a^{4} x^{3}}-\frac {B}{3 a^{3} x^{3}}-\frac {A}{5 a^{3} x^{5}} \end {gather*}

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]

-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(1/(a*b)^(1/2)*b*x)*A+35/8/a^4*b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*
b*x)*B-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

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maxima [A]  time = 2.36, size = 154, normalized size = 1.08 \begin {gather*} \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}}{120 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} + \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}} \end {gather*}

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]

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)/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5) + 7/8*(5*B*a*b^2 - 9*A*b^3)*arct
an(b*x/sqrt(a*b))/(sqrt(a*b)*a^5)

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mupad [B]  time = 0.20, size = 135, normalized size = 0.95 \begin {gather*} -\frac {\frac {A}{5\,a}-\frac {x^2\,\left (9\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {35\,b^2\,x^6\,\left (9\,A\,b-5\,B\,a\right )}{24\,a^4}+\frac {7\,b^3\,x^8\,\left (9\,A\,b-5\,B\,a\right )}{8\,a^5}+\frac {7\,b\,x^4\,\left (9\,A\,b-5\,B\,a\right )}{15\,a^3}}{a^2\,x^5+2\,a\,b\,x^7+b^2\,x^9}-\frac {7\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{8\,a^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (A/(5*a) - (x^2*(9*A*b - 5*B*a))/(15*a^2) + (35*b^2*x^6*(9*A*b - 5*B*a))/(24*a^4) + (7*b^3*x^8*(9*A*b - 5*B*
a))/(8*a^5) + (7*b*x^4*(9*A*b - 5*B*a))/(15*a^3))/(a^2*x^5 + b^2*x^9 + 2*a*b*x^7) - (7*b^(3/2)*atan((b^(1/2)*x
)/a^(1/2))*(9*A*b - 5*B*a))/(8*a^(11/2))

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sympy [A]  time = 0.87, size = 260, normalized size = 1.83 \begin {gather*} - \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 {gather*}

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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