\(\int \frac {(a+b x^2)^2 (A+B x^2)}{x^7} \, dx\) [20]

Optimal result

Integrand size = 20, antiderivative size = 51 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=-\frac {a^2 A}{6 x^6}-\frac {a (2 A b+a B)}{4 x^4}-\frac {b (A b+2 a B)}{2 x^2}+b^2 B \log (x) \]

[Out]

-1/6*a^2*A/x^6-1/4*a*(2*A*b+B*a)/x^4-1/2*b*(A*b+2*B*a)/x^2+b^2*B*ln(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=-\frac {a^2 A}{6 x^6}-\frac {a (a B+2 A b)}{4 x^4}-\frac {b (2 a B+A b)}{2 x^2}+b^2 B \log (x) \]

[In]

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

[Out]

-1/6*(a^2*A)/x^6 - (a*(2*A*b + a*B))/(4*x^4) - (b*(A*b + 2*a*B))/(2*x^2) + b^2*B*Log[x]

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 457

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=-\frac {6 A b^2 x^4+6 a b x^2 \left (A+2 B x^2\right )+a^2 \left (2 A+3 B x^2\right )}{12 x^6}+b^2 B \log (x) \]

[In]

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

[Out]

-1/12*(6*A*b^2*x^4 + 6*a*b*x^2*(A + 2*B*x^2) + a^2*(2*A + 3*B*x^2))/x^6 + b^2*B*Log[x]

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {a^{2} A}{6 x^{6}}-\frac {a \left (2 A b +B a \right )}{4 x^{4}}-\frac {b \left (A b +2 B a \right )}{2 x^{2}}+b^{2} B \ln \left (x \right )\)	\(46\)
norman	\(\frac {\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{4}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} B \right ) x^{2}-\frac {a^{2} A}{6}}{x^{6}}+b^{2} B \ln \left (x \right )\)	\(52\)
risch	\(\frac {\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{4}+\left (-\frac {1}{2} a b A -\frac {1}{4} a^{2} B \right ) x^{2}-\frac {a^{2} A}{6}}{x^{6}}+b^{2} B \ln \left (x \right )\)	\(52\)
parallelrisch	\(-\frac {-12 B \,b^{2} \ln \left (x \right ) x^{6}+6 A \,b^{2} x^{4}+12 B a b \,x^{4}+6 a A b \,x^{2}+3 a^{2} B \,x^{2}+2 a^{2} A}{12 x^{6}}\)	\(58\)

[In]

int((b*x^2+a)^2*(B*x^2+A)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^2*A/x^6-1/4*a*(2*A*b+B*a)/x^4-1/2*b*(A*b+2*B*a)/x^2+b^2*B*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=\frac {12 \, B b^{2} x^{6} \log \left (x\right ) - 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 2 \, A a^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \]

[In]

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

[Out]

1/12*(12*B*b^2*x^6*log(x) - 6*(2*B*a*b + A*b^2)*x^4 - 2*A*a^2 - 3*(B*a^2 + 2*A*a*b)*x^2)/x^6

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=B b^{2} \log {\left (x \right )} + \frac {- 2 A a^{2} + x^{4} \left (- 6 A b^{2} - 12 B a b\right ) + x^{2} \left (- 6 A a b - 3 B a^{2}\right )}{12 x^{6}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=\frac {1}{2} \, B b^{2} \log \left (x^{2}\right ) - \frac {6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 2 \, A a^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \]

[In]

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

[Out]

1/2*B*b^2*log(x^2) - 1/12*(6*(2*B*a*b + A*b^2)*x^4 + 2*A*a^2 + 3*(B*a^2 + 2*A*a*b)*x^2)/x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=\frac {1}{2} \, B b^{2} \log \left (x^{2}\right ) - \frac {11 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + 2 \, A a^{2}}{12 \, x^{6}} \]

[In]

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

[Out]

1/2*B*b^2*log(x^2) - 1/12*(11*B*b^2*x^6 + 12*B*a*b*x^4 + 6*A*b^2*x^4 + 3*B*a^2*x^2 + 6*A*a*b*x^2 + 2*A*a^2)/x^
6

Mupad [B] (verification not implemented)

Time = 4.89 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx=B\,b^2\,\ln \left (x\right )-\frac {x^2\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^4\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+\frac {A\,a^2}{6}}{x^6} \]

[In]

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

[Out]

B*b^2*log(x) - (x^2*((B*a^2)/4 + (A*a*b)/2) + x^4*((A*b^2)/2 + B*a*b) + (A*a^2)/6)/x^6