Optimal. Leaf size=48 \[ -\frac {A \left (a+b x^2\right )^3}{8 a x^8}+\frac {(A b-4 a B) \left (a+b x^2\right )^3}{24 a^2 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {457, 79, 37}
\begin {gather*} \frac {\left (a+b x^2\right )^3 (A b-4 a B)}{24 a^2 x^6}-\frac {A \left (a+b x^2\right )^3}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 79
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^9} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^3}{8 a x^8}+\frac {(-A b+4 a B) \text {Subst}\left (\int \frac {(a+b x)^2}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=-\frac {A \left (a+b x^2\right )^3}{8 a x^8}+\frac {(A b-4 a B) \left (a+b x^2\right )^3}{24 a^2 x^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 55, normalized size = 1.15 \begin {gather*} -\frac {6 b^2 x^4 \left (A+2 B x^2\right )+4 a b x^2 \left (2 A+3 B x^2\right )+a^2 \left (3 A+4 B x^2\right )}{24 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 48, normalized size = 1.00
method | result | size |
default | \(-\frac {b \left (A b +2 B a \right )}{4 x^{4}}-\frac {a \left (2 A b +B a \right )}{6 x^{6}}-\frac {b^{2} B}{2 x^{2}}-\frac {a^{2} A}{8 x^{8}}\) | \(48\) |
norman | \(\frac {-\frac {b^{2} B \,x^{6}}{2}+\left (-\frac {1}{4} b^{2} A -\frac {1}{2} a b B \right ) x^{4}+\left (-\frac {1}{3} a b A -\frac {1}{6} a^{2} B \right ) x^{2}-\frac {a^{2} A}{8}}{x^{8}}\) | \(53\) |
risch | \(\frac {-\frac {b^{2} B \,x^{6}}{2}+\left (-\frac {1}{4} b^{2} A -\frac {1}{2} a b B \right ) x^{4}+\left (-\frac {1}{3} a b A -\frac {1}{6} a^{2} B \right ) x^{2}-\frac {a^{2} A}{8}}{x^{8}}\) | \(53\) |
gosper | \(-\frac {12 b^{2} B \,x^{6}+6 A \,b^{2} x^{4}+12 B a b \,x^{4}+8 a A b \,x^{2}+4 B \,a^{2} x^{2}+3 a^{2} A}{24 x^{8}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 53, normalized size = 1.10 \begin {gather*} -\frac {12 \, B b^{2} x^{6} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.92, size = 53, normalized size = 1.10 \begin {gather*} -\frac {12 \, B b^{2} x^{6} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.86, size = 58, normalized size = 1.21 \begin {gather*} \frac {- 3 A a^{2} - 12 B b^{2} x^{6} + x^{4} \left (- 6 A b^{2} - 12 B a b\right ) + x^{2} \left (- 8 A a b - 4 B a^{2}\right )}{24 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.13, size = 55, normalized size = 1.15 \begin {gather*} -\frac {12 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 4 \, B a^{2} x^{2} + 8 \, A a b x^{2} + 3 \, A a^{2}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 53, normalized size = 1.10 \begin {gather*} -\frac {x^2\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+x^4\,\left (\frac {A\,b^2}{4}+\frac {B\,a\,b}{2}\right )+\frac {A\,a^2}{8}+\frac {B\,b^2\,x^6}{2}}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________