Optimal. Leaf size=98 \[ -\frac {a^2}{6 c x^6}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4}-\frac {d \log (x) (b c-a d)^2}{c^4}-\frac {(b c-a d)^2}{2 c^3 x^2}-\frac {a (2 b c-a d)}{4 c^2 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 88} \[ -\frac {a^2}{6 c x^6}-\frac {a (2 b c-a d)}{4 c^2 x^4}-\frac {(b c-a d)^2}{2 c^3 x^2}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4}-\frac {d \log (x) (b c-a d)^2}{c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^4 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{c x^4}-\frac {a (-2 b c+a d)}{c^2 x^3}+\frac {(b c-a d)^2}{c^3 x^2}-\frac {d (b c-a d)^2}{c^4 x}+\frac {d^2 (b c-a d)^2}{c^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{6 c x^6}-\frac {a (2 b c-a d)}{4 c^2 x^4}-\frac {(b c-a d)^2}{2 c^3 x^2}-\frac {d (b c-a d)^2 \log (x)}{c^4}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 108, normalized size = 1.10 \[ -\frac {c \left (a^2 \left (2 c^2-3 c d x^2+6 d^2 x^4\right )+6 a b c x^2 \left (c-2 d x^2\right )+6 b^2 c^2 x^4\right )+12 d x^6 \log (x) (b c-a d)^2-6 d x^6 (b c-a d)^2 \log \left (c+d x^2\right )}{12 c^4 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 136, normalized size = 1.39 \[ \frac {6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (d x^{2} + c\right ) - 12 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \relax (x) - 2 \, a^{2} c^{3} - 6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (2 \, a b c^{3} - a^{2} c^{2} d\right )} x^{2}}{12 \, c^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.32, size = 184, normalized size = 1.88 \[ -\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} + \frac {{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{4} d} + \frac {11 \, b^{2} c^{2} d x^{6} - 22 \, a b c d^{2} x^{6} + 11 \, a^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} x^{4} + 12 \, a b c^{2} d x^{4} - 6 \, a^{2} c d^{2} x^{4} - 6 \, a b c^{3} x^{2} + 3 \, a^{2} c^{2} d x^{2} - 2 \, a^{2} c^{3}}{12 \, c^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 160, normalized size = 1.63 \[ -\frac {a^{2} d^{3} \ln \relax (x )}{c^{4}}+\frac {a^{2} d^{3} \ln \left (d \,x^{2}+c \right )}{2 c^{4}}+\frac {2 a b \,d^{2} \ln \relax (x )}{c^{3}}-\frac {a b \,d^{2} \ln \left (d \,x^{2}+c \right )}{c^{3}}-\frac {b^{2} d \ln \relax (x )}{c^{2}}+\frac {b^{2} d \ln \left (d \,x^{2}+c \right )}{2 c^{2}}-\frac {a^{2} d^{2}}{2 c^{3} x^{2}}+\frac {a b d}{c^{2} x^{2}}-\frac {b^{2}}{2 c \,x^{2}}+\frac {a^{2} d}{4 c^{2} x^{4}}-\frac {a b}{2 c \,x^{4}}-\frac {a^{2}}{6 c \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 134, normalized size = 1.37 \[ \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{4}} - \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} - \frac {6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + 3 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{12 \, c^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 129, normalized size = 1.32 \[ \frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{2\,c^4}-\frac {\frac {a^2}{6\,c}+\frac {x^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3}-\frac {a\,x^2\,\left (a\,d-2\,b\,c\right )}{4\,c^2}}{x^6}-\frac {\ln \relax (x)\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.51, size = 105, normalized size = 1.07 \[ \frac {- 2 a^{2} c^{2} + x^{4} \left (- 6 a^{2} d^{2} + 12 a b c d - 6 b^{2} c^{2}\right ) + x^{2} \left (3 a^{2} c d - 6 a b c^{2}\right )}{12 c^{3} x^{6}} - \frac {d \left (a d - b c\right )^{2} \log {\relax (x )}}{c^{4}} + \frac {d \left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________