Optimal. Leaf size=48 \[ -\frac {A b^2}{5 x^5}-\frac {b (b B+2 A c)}{3 x^3}-\frac {c (2 b B+A c)}{x}+B c^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1598, 459}
\begin {gather*} -\frac {A b^2}{5 x^5}-\frac {b (2 A c+b B)}{3 x^3}-\frac {c (A c+2 b B)}{x}+B c^2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 459
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^{10}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x^6} \, dx\\ &=\int \left (B c^2+\frac {A b^2}{x^6}+\frac {b (b B+2 A c)}{x^4}+\frac {c (2 b B+A c)}{x^2}\right ) \, dx\\ &=-\frac {A b^2}{5 x^5}-\frac {b (b B+2 A c)}{3 x^3}-\frac {c (2 b B+A c)}{x}+B c^2 x\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 48, normalized size = 1.00 \begin {gather*} -\frac {A b^2}{5 x^5}-\frac {b (b B+2 A c)}{3 x^3}-\frac {c (2 b B+A c)}{x}+B c^2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.36, size = 45, normalized size = 0.94
method | result | size |
default | \(-\frac {A \,b^{2}}{5 x^{5}}-\frac {b \left (2 A c +B b \right )}{3 x^{3}}-\frac {c \left (A c +2 B b \right )}{x}+B \,c^{2} x\) | \(45\) |
risch | \(B \,c^{2} x +\frac {\left (-A \,c^{2}-2 b B c \right ) x^{4}+\left (-\frac {2}{3} A b c -\frac {1}{3} b^{2} B \right ) x^{2}-\frac {b^{2} A}{5}}{x^{5}}\) | \(51\) |
norman | \(\frac {\left (-\frac {2}{3} A b c -\frac {1}{3} b^{2} B \right ) x^{6}+\left (-A \,c^{2}-2 b B c \right ) x^{8}+B \,c^{2} x^{10}-\frac {A \,b^{2} x^{4}}{5}}{x^{9}}\) | \(55\) |
gosper | \(-\frac {-15 B \,c^{2} x^{6}+15 A \,c^{2} x^{4}+30 x^{4} b B c +10 A b c \,x^{2}+5 b^{2} B \,x^{2}+3 b^{2} A}{15 x^{5}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 51, normalized size = 1.06 \begin {gather*} B c^{2} x - \frac {15 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 3 \, A b^{2} + 5 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.32, size = 53, normalized size = 1.10 \begin {gather*} \frac {15 \, B c^{2} x^{6} - 15 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} - 3 \, A b^{2} - 5 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.31, size = 54, normalized size = 1.12 \begin {gather*} B c^{2} x + \frac {- 3 A b^{2} + x^{4} \left (- 15 A c^{2} - 30 B b c\right ) + x^{2} \left (- 10 A b c - 5 B b^{2}\right )}{15 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.12, size = 53, normalized size = 1.10 \begin {gather*} B c^{2} x - \frac {30 \, B b c x^{4} + 15 \, A c^{2} x^{4} + 5 \, B b^{2} x^{2} + 10 \, A b c x^{2} + 3 \, A b^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 50, normalized size = 1.04 \begin {gather*} B\,c^2\,x-\frac {x^2\,\left (\frac {B\,b^2}{3}+\frac {2\,A\,c\,b}{3}\right )+x^4\,\left (A\,c^2+2\,B\,b\,c\right )+\frac {A\,b^2}{5}}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________