3.1.97 \(\int \frac {A+B x^2}{x^7 (a+b x^2)^3} \, dx\)

Optimal. Leaf size=149 \[ \frac {b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}-\frac {2 b^2 \log (x) (5 A b-3 a B)}{a^6}-\frac {b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac {3 b (2 A b-a B)}{2 a^5 x^2}-\frac {b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac {3 A b-a B}{4 a^4 x^4}-\frac {A}{6 a^3 x^6} \]

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Rubi [A]  time = 0.17, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \begin {gather*} -\frac {b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac {b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac {b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}-\frac {2 b^2 \log (x) (5 A b-3 a B)}{a^6}-\frac {3 b (2 A b-a B)}{2 a^5 x^2}+\frac {3 A b-a B}{4 a^4 x^4}-\frac {A}{6 a^3 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^7*(a + b*x^2)^3),x]

[Out]

-A/(6*a^3*x^6) + (3*A*b - a*B)/(4*a^4*x^4) - (3*b*(2*A*b - a*B))/(2*a^5*x^2) - (b^2*(A*b - a*B))/(4*a^4*(a + b
*x^2)^2) - (b^2*(4*A*b - 3*a*B))/(2*a^5*(a + b*x^2)) - (2*b^2*(5*A*b - 3*a*B)*Log[x])/a^6 + (b^2*(5*A*b - 3*a*
B)*Log[a + b*x^2])/a^6

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

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

Rubi steps

\begin {align*} \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^4 (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{a^3 x^4}+\frac {-3 A b+a B}{a^4 x^3}-\frac {3 b (-2 A b+a B)}{a^5 x^2}+\frac {2 b^2 (-5 A b+3 a B)}{a^6 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)^3}-\frac {b^3 (-4 A b+3 a B)}{a^5 (a+b x)^2}-\frac {2 b^3 (-5 A b+3 a B)}{a^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{6 a^3 x^6}+\frac {3 A b-a B}{4 a^4 x^4}-\frac {3 b (2 A b-a B)}{2 a^5 x^2}-\frac {b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}-\frac {b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac {2 b^2 (5 A b-3 a B) \log (x)}{a^6}+\frac {b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 135, normalized size = 0.91 \begin {gather*} \frac {-\frac {2 a^3 A}{x^6}+\frac {3 a^2 b^2 (a B-A b)}{\left (a+b x^2\right )^2}-\frac {3 a^2 (a B-3 A b)}{x^4}+\frac {6 a b^2 (3 a B-4 A b)}{a+b x^2}+12 b^2 (5 A b-3 a B) \log \left (a+b x^2\right )+24 b^2 \log (x) (3 a B-5 A b)+\frac {18 a b (a B-2 A b)}{x^2}}{12 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^3),x]

[Out]

((-2*a^3*A)/x^6 - (3*a^2*(-3*A*b + a*B))/x^4 + (18*a*b*(-2*A*b + a*B))/x^2 + (3*a^2*b^2*(-(A*b) + a*B))/(a + b
*x^2)^2 + (6*a*b^2*(-4*A*b + 3*a*B))/(a + b*x^2) + 24*b^2*(-5*A*b + 3*a*B)*Log[x] + 12*b^2*(5*A*b - 3*a*B)*Log
[a + b*x^2])/(12*a^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x^2)/(x^7*(a + b*x^2)^3),x]

[Out]

IntegrateAlgebraic[(A + B*x^2)/(x^7*(a + b*x^2)^3), x]

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fricas [A]  time = 0.44, size = 267, normalized size = 1.79 \begin {gather*} \frac {12 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + 18 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6} - 2 \, A a^{5} + 4 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{4} - {\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x^{2} - 12 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \relax (x)}{12 \, {\left (a^{6} b^{2} x^{10} + 2 \, a^{7} b x^{8} + a^{8} x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/12*(12*(3*B*a^2*b^3 - 5*A*a*b^4)*x^8 + 18*(3*B*a^3*b^2 - 5*A*a^2*b^3)*x^6 - 2*A*a^5 + 4*(3*B*a^4*b - 5*A*a^3
*b^2)*x^4 - (3*B*a^5 - 5*A*a^4*b)*x^2 - 12*((3*B*a*b^4 - 5*A*b^5)*x^10 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^8 + (3*
B*a^3*b^2 - 5*A*a^2*b^3)*x^6)*log(b*x^2 + a) + 24*((3*B*a*b^4 - 5*A*b^5)*x^10 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^
8 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*x^6)*log(x))/(a^6*b^2*x^10 + 2*a^7*b*x^8 + a^8*x^6)

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giac [A]  time = 0.40, size = 201, normalized size = 1.35 \begin {gather*} \frac {{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x^{2}\right )}{a^{6}} - \frac {{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6} b} + \frac {18 \, B a b^{4} x^{4} - 30 \, A b^{5} x^{4} + 42 \, B a^{2} b^{3} x^{2} - 68 \, A a b^{4} x^{2} + 25 \, B a^{3} b^{2} - 39 \, A a^{2} b^{3}}{4 \, {\left (b x^{2} + a\right )}^{2} a^{6}} - \frac {66 \, B a b^{2} x^{6} - 110 \, A b^{3} x^{6} - 18 \, B a^{2} b x^{4} + 36 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 9 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{6} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(b*x^2+a)^3,x, algorithm="giac")

[Out]

(3*B*a*b^2 - 5*A*b^3)*log(x^2)/a^6 - (3*B*a*b^3 - 5*A*b^4)*log(abs(b*x^2 + a))/(a^6*b) + 1/4*(18*B*a*b^4*x^4 -
30*A*b^5*x^4 + 42*B*a^2*b^3*x^2 - 68*A*a*b^4*x^2 + 25*B*a^3*b^2 - 39*A*a^2*b^3)/((b*x^2 + a)^2*a^6) - 1/12*(6
6*B*a*b^2*x^6 - 110*A*b^3*x^6 - 18*B*a^2*b*x^4 + 36*A*a*b^2*x^4 + 3*B*a^3*x^2 - 9*A*a^2*b*x^2 + 2*A*a^3)/(a^6*
x^6)

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maple [A]  time = 0.02, size = 180, normalized size = 1.21 \begin {gather*} -\frac {A \,b^{3}}{4 \left (b \,x^{2}+a \right )^{2} a^{4}}+\frac {B \,b^{2}}{4 \left (b \,x^{2}+a \right )^{2} a^{3}}-\frac {2 A \,b^{3}}{\left (b \,x^{2}+a \right ) a^{5}}-\frac {10 A \,b^{3} \ln \relax (x )}{a^{6}}+\frac {5 A \,b^{3} \ln \left (b \,x^{2}+a \right )}{a^{6}}+\frac {3 B \,b^{2}}{2 \left (b \,x^{2}+a \right ) a^{4}}+\frac {6 B \,b^{2} \ln \relax (x )}{a^{5}}-\frac {3 B \,b^{2} \ln \left (b \,x^{2}+a \right )}{a^{5}}-\frac {3 A \,b^{2}}{a^{5} x^{2}}+\frac {3 B b}{2 a^{4} x^{2}}+\frac {3 A b}{4 a^{4} x^{4}}-\frac {B}{4 a^{3} x^{4}}-\frac {A}{6 a^{3} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^7/(b*x^2+a)^3,x)

[Out]

-2/a^5*b^3/(b*x^2+a)*A+3/2/a^4*b^2/(b*x^2+a)*B-1/4/a^4*b^3/(b*x^2+a)^2*A+1/4/a^3*b^2/(b*x^2+a)^2*B+5/a^6*b^3*l
n(b*x^2+a)*A-3/a^5*b^2*ln(b*x^2+a)*B-1/6*A/a^3/x^6+3/4/a^4/x^4*A*b-1/4/a^3/x^4*B-3*b^2/a^5/x^2*A+3/2*b/a^4/x^2
*B-10*b^3/a^6*ln(x)*A+6*b^2/a^5*ln(x)*B

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maxima [A]  time = 1.03, size = 170, normalized size = 1.14 \begin {gather*} \frac {12 \, {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{8} + 18 \, {\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 2 \, A a^{4} + 4 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{4} - {\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x^{2}}{12 \, {\left (a^{5} b^{2} x^{10} + 2 \, a^{6} b x^{8} + a^{7} x^{6}\right )}} - \frac {{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x^{2} + a\right )}{a^{6}} + \frac {{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x^{2}\right )}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/12*(12*(3*B*a*b^3 - 5*A*b^4)*x^8 + 18*(3*B*a^2*b^2 - 5*A*a*b^3)*x^6 - 2*A*a^4 + 4*(3*B*a^3*b - 5*A*a^2*b^2)*
x^4 - (3*B*a^4 - 5*A*a^3*b)*x^2)/(a^5*b^2*x^10 + 2*a^6*b*x^8 + a^7*x^6) - (3*B*a*b^2 - 5*A*b^3)*log(b*x^2 + a)
/a^6 + (3*B*a*b^2 - 5*A*b^3)*log(x^2)/a^6

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mupad [B]  time = 0.18, size = 155, normalized size = 1.04 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )\,\left (5\,A\,b^3-3\,B\,a\,b^2\right )}{a^6}-\frac {\frac {A}{6\,a}-\frac {x^2\,\left (5\,A\,b-3\,B\,a\right )}{12\,a^2}+\frac {3\,b^2\,x^6\,\left (5\,A\,b-3\,B\,a\right )}{2\,a^4}+\frac {b^3\,x^8\,\left (5\,A\,b-3\,B\,a\right )}{a^5}+\frac {b\,x^4\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^3}}{a^2\,x^6+2\,a\,b\,x^8+b^2\,x^{10}}-\frac {\ln \relax (x)\,\left (10\,A\,b^3-6\,B\,a\,b^2\right )}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^7*(a + b*x^2)^3),x)

[Out]

(log(a + b*x^2)*(5*A*b^3 - 3*B*a*b^2))/a^6 - (A/(6*a) - (x^2*(5*A*b - 3*B*a))/(12*a^2) + (3*b^2*x^6*(5*A*b - 3
*B*a))/(2*a^4) + (b^3*x^8*(5*A*b - 3*B*a))/a^5 + (b*x^4*(5*A*b - 3*B*a))/(3*a^3))/(a^2*x^6 + b^2*x^10 + 2*a*b*
x^8) - (log(x)*(10*A*b^3 - 6*B*a*b^2))/a^6

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sympy [A]  time = 1.26, size = 165, normalized size = 1.11 \begin {gather*} \frac {- 2 A a^{4} + x^{8} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{6} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{4} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x^{2} \left (5 A a^{3} b - 3 B a^{4}\right )}{12 a^{7} x^{6} + 24 a^{6} b x^{8} + 12 a^{5} b^{2} x^{10}} + \frac {2 b^{2} \left (- 5 A b + 3 B a\right ) \log {\relax (x )}}{a^{6}} - \frac {b^{2} \left (- 5 A b + 3 B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**7/(b*x**2+a)**3,x)

[Out]

(-2*A*a**4 + x**8*(-60*A*b**4 + 36*B*a*b**3) + x**6*(-90*A*a*b**3 + 54*B*a**2*b**2) + x**4*(-20*A*a**2*b**2 +
12*B*a**3*b) + x**2*(5*A*a**3*b - 3*B*a**4))/(12*a**7*x**6 + 24*a**6*b*x**8 + 12*a**5*b**2*x**10) + 2*b**2*(-5
*A*b + 3*B*a)*log(x)/a**6 - b**2*(-5*A*b + 3*B*a)*log(a/b + x**2)/a**6

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