Optimal. Leaf size=126 \[ \frac {x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}}-\frac {a (6 b c-5 a d)}{3 c^3 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {462, 456, 453, 205} \[ \frac {x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}-\frac {a (6 b c-5 a d)}{3 c^3 x}+\frac {(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 453
Rule 456
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^2} \, dx &=-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\int \frac {a (6 b c-5 a d)+3 b^2 c x^2}{x^2 \left (c+d x^2\right )^2} \, dx}{3 c}\\ &=-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{6 c^3 \left (c+d x^2\right )}-\frac {\int \frac {-\frac {2 a (6 b c-5 a d)}{c}-\left (3 b^2-\frac {6 a b d}{c}+\frac {5 a^2 d^2}{c^2}\right ) x^2}{x^2 \left (c+d x^2\right )} \, dx}{6 c}\\ &=-\frac {a (6 b c-5 a d)}{3 c^3 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{6 c^3 \left (c+d x^2\right )}+\frac {((b c-5 a d) (b c-a d)) \int \frac {1}{c+d x^2} \, dx}{2 c^3}\\ &=-\frac {a (6 b c-5 a d)}{3 c^3 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{6 c^3 \left (c+d x^2\right )}+\frac {(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 107, normalized size = 0.85 \[ \frac {\left (5 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}}-\frac {a^2}{3 c^2 x^3}+\frac {x (b c-a d)^2}{2 c^3 \left (c+d x^2\right )}+\frac {2 a (a d-b c)}{c^3 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 356, normalized size = 2.83 \[ \left [-\frac {4 \, a^{2} c^{3} d - 6 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{4} + 4 \, {\left (6 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} + {\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right )}{12 \, {\left (c^{4} d^{2} x^{5} + c^{5} d x^{3}\right )}}, -\frac {2 \, a^{2} c^{3} d - 3 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{4} + 2 \, {\left (6 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2} - 3 \, {\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} + {\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right )}{6 \, {\left (c^{4} d^{2} x^{5} + c^{5} d x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.39, size = 111, normalized size = 0.88 \[ \frac {{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c^{3}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (d x^{2} + c\right )} c^{3}} - \frac {6 \, a b c x^{2} - 6 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 161, normalized size = 1.28 \[ \frac {a^{2} d^{2} x}{2 \left (d \,x^{2}+c \right ) c^{3}}+\frac {5 a^{2} d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, c^{3}}-\frac {a b d x}{\left (d \,x^{2}+c \right ) c^{2}}-\frac {3 a b d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, c^{2}}+\frac {b^{2} x}{2 \left (d \,x^{2}+c \right ) c}+\frac {b^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, c}+\frac {2 a^{2} d}{c^{3} x}-\frac {2 a b}{c^{2} x}-\frac {a^{2}}{3 c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.49, size = 118, normalized size = 0.94 \[ \frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} - 2 \, {\left (6 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}}{6 \, {\left (c^{3} d x^{5} + c^{4} x^{3}\right )}} + \frac {{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.19, size = 147, normalized size = 1.17 \[ \frac {\frac {x^4\,\left (5\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3}-\frac {a^2}{3\,c}+\frac {a\,x^2\,\left (5\,a\,d-6\,b\,c\right )}{3\,c^2}}{d\,x^5+c\,x^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-b\,c\right )}{\sqrt {c}\,\left (5\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-b\,c\right )}{2\,c^{7/2}\,\sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.99, size = 248, normalized size = 1.97 \[ - \frac {\sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log {\left (- \frac {c^{4} \sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log {\left (\frac {c^{4} \sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac {- 2 a^{2} c^{2} + x^{4} \left (15 a^{2} d^{2} - 18 a b c d + 3 b^{2} c^{2}\right ) + x^{2} \left (10 a^{2} c d - 12 a b c^{2}\right )}{6 c^{4} x^{3} + 6 c^{3} d x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________