Optimal. Leaf size=160 \[ -\frac {1}{4 a^2 c x^4}+\frac {2 b c+a d}{2 a^3 c^2 x^2}+\frac {b^3}{2 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {\left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) \log (x)}{a^4 c^3}-\frac {b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}-\frac {d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 160, 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 = {457, 90}
\begin {gather*} -\frac {b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}+\frac {b^3}{2 a^3 \left (a+b x^2\right ) (b c-a d)}+\frac {a d+2 b c}{2 a^3 c^2 x^2}-\frac {1}{4 a^2 c x^4}+\frac {\log (x) \left (a^2 d^2+2 a b c d+3 b^2 c^2\right )}{a^4 c^3}-\frac {d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^2 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 c x^3}+\frac {-2 b c-a d}{a^3 c^2 x^2}+\frac {3 b^2 c^2+2 a b c d+a^2 d^2}{a^4 c^3 x}+\frac {b^4}{a^3 (-b c+a d) (a+b x)^2}+\frac {b^4 (-3 b c+4 a d)}{a^4 (-b c+a d)^2 (a+b x)}-\frac {d^5}{c^3 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^2 c x^4}+\frac {2 b c+a d}{2 a^3 c^2 x^2}+\frac {b^3}{2 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {\left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) \log (x)}{a^4 c^3}-\frac {b^3 (3 b c-4 a d) \log \left (a+b x^2\right )}{2 a^4 (b c-a d)^2}-\frac {d^4 \log \left (c+d x^2\right )}{2 c^3 (b c-a d)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 155, normalized size = 0.97 \begin {gather*} \frac {1}{4} \left (-\frac {1}{a^2 c x^4}+\frac {4 b c+2 a d}{a^3 c^2 x^2}-\frac {2 b^3}{a^3 (-b c+a d) \left (a+b x^2\right )}+\frac {4 \left (3 b^2 c^2+2 a b c d+a^2 d^2\right ) \log (x)}{a^4 c^3}+\frac {2 b^3 (-3 b c+4 a d) \log \left (a+b x^2\right )}{a^4 (b c-a d)^2}-\frac {2 d^4 \log \left (c+d x^2\right )}{c^3 (b c-a d)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 153, normalized size = 0.96
method | result | size |
default | \(\frac {b^{4} \left (\frac {\left (4 a d -3 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a \left (a d -b c \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{4} \left (a d -b c \right )^{2}}-\frac {d^{4} \ln \left (d \,x^{2}+c \right )}{2 c^{3} \left (a d -b c \right )^{2}}-\frac {1}{4 a^{2} c \,x^{4}}-\frac {-a d -2 b c}{2 a^{3} c^{2} x^{2}}+\frac {\left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{3}}\) | \(153\) |
norman | \(\frac {-\frac {1}{4 a c}+\frac {\left (2 a d +3 b c \right ) x^{2}}{4 a^{2} c^{2}}+\frac {\left (-a^{2} b \,d^{2}-a \,b^{2} c d +3 b^{3} c^{2}\right ) b \,x^{6}}{2 c^{2} a^{4} \left (a d -b c \right )}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {\left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{3}}-\frac {d^{4} \ln \left (d \,x^{2}+c \right )}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b^{3} \left (4 a d -3 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(210\) |
risch | \(\frac {\frac {b \left (a^{2} d^{2}+a b c d -3 b^{2} c^{2}\right ) x^{4}}{2 a^{3} c^{2} \left (a d -b c \right )}+\frac {\left (2 a d +3 b c \right ) x^{2}}{4 a^{2} c^{2}}-\frac {1}{4 a c}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {\ln \left (x \right ) d^{2}}{a^{2} c^{3}}+\frac {2 \ln \left (x \right ) b d}{a^{3} c^{2}}+\frac {3 \ln \left (x \right ) b^{2}}{a^{4} c}-\frac {d^{4} \ln \left (-d \,x^{2}-c \right )}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 b^{3} \ln \left (b \,x^{2}+a \right ) d}{a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {3 b^{4} \ln \left (b \,x^{2}+a \right ) c}{2 a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 258, normalized size = 1.61 \begin {gather*} -\frac {d^{4} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac {{\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )}} - \frac {a^{2} b c^{2} - a^{3} c d - 2 \, {\left (3 \, b^{3} c^{2} - a b^{2} c d - a^{2} b d^{2}\right )} x^{4} - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}}{4 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{6} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{4}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs.
\(2 (150) = 300\).
time = 6.81, size = 356, normalized size = 2.22 \begin {gather*} -\frac {a^{3} b^{2} c^{4} - 2 \, a^{4} b c^{3} d + a^{5} c^{2} d^{2} - 2 \, {\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + a^{4} b c d^{3}\right )} x^{4} - {\left (3 \, a^{2} b^{3} c^{4} - 4 \, a^{3} b^{2} c^{3} d - a^{4} b c^{2} d^{2} + 2 \, a^{5} c d^{3}\right )} x^{2} + 2 \, {\left ({\left (3 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d\right )} x^{6} + {\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left (a^{4} b d^{4} x^{6} + a^{5} d^{4} x^{4}\right )} \log \left (d x^{2} + c\right ) - 4 \, {\left ({\left (3 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d + a^{4} b d^{4}\right )} x^{6} + {\left (3 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + a^{5} d^{4}\right )} x^{4}\right )} \log \left (x\right )}{4 \, {\left ({\left (a^{4} b^{3} c^{5} - 2 \, a^{5} b^{2} c^{4} d + a^{6} b c^{3} d^{2}\right )} x^{6} + {\left (a^{5} b^{2} c^{5} - 2 \, a^{6} b c^{4} d + a^{7} c^{3} d^{2}\right )} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.55, size = 281, normalized size = 1.76 \begin {gather*} -\frac {d^{5} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )}} - \frac {{\left (3 \, b^{5} c - 4 \, a b^{4} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )}} + \frac {3 \, b^{5} c x^{2} - 4 \, a b^{4} d x^{2} + 4 \, a b^{4} c - 5 \, a^{2} b^{3} d}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{3}} - \frac {9 \, b^{2} c^{2} x^{4} + 6 \, a b c d x^{4} + 3 \, a^{2} d^{2} x^{4} - 4 \, a b c^{2} x^{2} - 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{4 \, a^{4} c^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.59, size = 217, normalized size = 1.36 \begin {gather*} \frac {\frac {x^2\,\left (2\,a\,d+3\,b\,c\right )}{4\,a^2\,c^2}-\frac {1}{4\,a\,c}+\frac {x^4\,\left (a^2\,b\,d^2+a\,b^2\,c\,d-3\,b^3\,c^2\right )}{2\,a^3\,c^2\,\left (a\,d-b\,c\right )}}{b\,x^6+a\,x^4}-\frac {\ln \left (b\,x^2+a\right )\,\left (3\,b^4\,c-4\,a\,b^3\,d\right )}{2\,a^6\,d^2-4\,a^5\,b\,c\,d+2\,a^4\,b^2\,c^2}-\frac {d^4\,\ln \left (d\,x^2+c\right )}{2\,\left (a^2\,c^3\,d^2-2\,a\,b\,c^4\,d+b^2\,c^5\right )}+\frac {\ln \left (x\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^4\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________