Optimal. Leaf size=152 \[ -\frac {x \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac {a^2}{c x \left (c+d x^2\right )^2}+\frac {\left (3 a d (2 b c-5 a d)+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} d^{3/2}}+\frac {x \left (3 a d (2 b c-5 a d)+b^2 c^2\right )}{8 c^3 d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 151, normalized size of antiderivative = 0.99, 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, 385, 199, 205} \begin {gather*} -\frac {x \left (5 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac {a^2}{c x \left (c+d x^2\right )^2}+\frac {\left (3 a d (2 b c-5 a d)+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} d^{3/2}}+\frac {x \left (\frac {3 a (2 b c-5 a d)}{c^2}+\frac {b^2}{d}\right )}{8 c \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 385
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^3} \, dx &=-\frac {a^2}{c x \left (c+d x^2\right )^2}+\frac {\int \frac {a (2 b c-5 a d)+b^2 c x^2}{\left (c+d x^2\right )^3} \, dx}{c}\\ &=-\frac {a^2}{c x \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x}{4 c \left (c+d x^2\right )^2}+\frac {1}{4} \left (\frac {b^2}{d}+\frac {3 a (2 b c-5 a d)}{c^2}\right ) \int \frac {1}{\left (c+d x^2\right )^2} \, dx\\ &=-\frac {a^2}{c x \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {b^2}{d}+\frac {3 a (2 b c-5 a d)}{c^2}\right ) x}{8 c \left (c+d x^2\right )}+\frac {\left (\frac {b^2}{d}+\frac {3 a (2 b c-5 a d)}{c^2}\right ) \int \frac {1}{c+d x^2} \, dx}{8 c}\\ &=-\frac {a^2}{c x \left (c+d x^2\right )^2}+\frac {\left (2 a b-\frac {b^2 c}{d}-\frac {5 a^2 d}{c}\right ) x}{4 c \left (c+d x^2\right )^2}+\frac {\left (\frac {b^2}{d}+\frac {3 a (2 b c-5 a d)}{c^2}\right ) x}{8 c \left (c+d x^2\right )}+\frac {\left (b^2 c^2+6 a b c d-15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 133, normalized size = 0.88 \begin {gather*} \frac {\left (-15 a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{7/2} d^{3/2}}+\frac {x \left (-7 a^2 d^2+6 a b c d+b^2 c^2\right )}{8 c^3 d \left (c+d x^2\right )}-\frac {a^2}{c^3 x}-\frac {x (b c-a d)^2}{4 c^2 d \left (c+d x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.85, size = 475, normalized size = 3.12 \begin {gather*} \left [-\frac {16 \, a^{2} c^{3} d^{2} - 2 \, {\left (b^{2} c^{3} d^{2} + 6 \, a b c^{2} d^{3} - 15 \, a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 10 \, a b c^{3} d^{2} + 25 \, a^{2} c^{2} d^{3}\right )} x^{2} - {\left ({\left (b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{5} + 2 \, {\left (b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{3} + {\left (b^{2} c^{4} + 6 \, a b c^{3} d - 15 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {-c d} \log \left (\frac {d x^{2} + 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right )}{16 \, {\left (c^{4} d^{4} x^{5} + 2 \, c^{5} d^{3} x^{3} + c^{6} d^{2} x\right )}}, -\frac {8 \, a^{2} c^{3} d^{2} - {\left (b^{2} c^{3} d^{2} + 6 \, a b c^{2} d^{3} - 15 \, a^{2} c d^{4}\right )} x^{4} + {\left (b^{2} c^{4} d - 10 \, a b c^{3} d^{2} + 25 \, a^{2} c^{2} d^{3}\right )} x^{2} - {\left ({\left (b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - 15 \, a^{2} d^{4}\right )} x^{5} + 2 \, {\left (b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - 15 \, a^{2} c d^{3}\right )} x^{3} + {\left (b^{2} c^{4} + 6 \, a b c^{3} d - 15 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right )}{8 \, {\left (c^{4} d^{4} x^{5} + 2 \, c^{5} d^{3} x^{3} + c^{6} d^{2} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 135, normalized size = 0.89 \begin {gather*} -\frac {a^{2}}{c^{3} x} + \frac {{\left (b^{2} c^{2} + 6 \, a b c d - 15 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{3} d} + \frac {b^{2} c^{2} d x^{3} + 6 \, a b c d^{2} x^{3} - 7 \, a^{2} d^{3} x^{3} - b^{2} c^{3} x + 10 \, a b c^{2} d x - 9 \, a^{2} c d^{2} x}{8 \, {\left (d x^{2} + c\right )}^{2} c^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 199, normalized size = 1.31 \begin {gather*} -\frac {7 a^{2} d^{2} x^{3}}{8 \left (d \,x^{2}+c \right )^{2} c^{3}}+\frac {3 a b d \,x^{3}}{4 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {b^{2} x^{3}}{8 \left (d \,x^{2}+c \right )^{2} c}-\frac {9 a^{2} d x}{8 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {5 a b x}{4 \left (d \,x^{2}+c \right )^{2} c}-\frac {b^{2} x}{8 \left (d \,x^{2}+c \right )^{2} d}-\frac {15 a^{2} d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c^{3}}+\frac {3 a b \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{4 \sqrt {c d}\, c^{2}}+\frac {b^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c d}-\frac {a^{2}}{c^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.41, size = 146, normalized size = 0.96 \begin {gather*} -\frac {8 \, a^{2} c^{2} d - {\left (b^{2} c^{2} d + 6 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{4} + {\left (b^{2} c^{3} - 10 \, a b c^{2} d + 25 \, a^{2} c d^{2}\right )} x^{2}}{8 \, {\left (c^{3} d^{3} x^{5} + 2 \, c^{4} d^{2} x^{3} + c^{5} d x\right )}} + \frac {{\left (b^{2} c^{2} + 6 \, a b c d - 15 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 135, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (-15\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{8\,c^{7/2}\,d^{3/2}}-\frac {\frac {a^2}{c}-\frac {x^4\,\left (-15\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{8\,c^3}+\frac {x^2\,\left (25\,a^2\,d^2-10\,a\,b\,c\,d+b^2\,c^2\right )}{8\,c^2\,d}}{c^2\,x+2\,c\,d\,x^3+d^2\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 224, normalized size = 1.47 \begin {gather*} \frac {\sqrt {- \frac {1}{c^{7} d^{3}}} \left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \log {\left (- c^{4} d \sqrt {- \frac {1}{c^{7} d^{3}}} + x \right )}}{16} - \frac {\sqrt {- \frac {1}{c^{7} d^{3}}} \left (15 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right ) \log {\left (c^{4} d \sqrt {- \frac {1}{c^{7} d^{3}}} + x \right )}}{16} + \frac {- 8 a^{2} c^{2} d + x^{4} \left (- 15 a^{2} d^{3} + 6 a b c d^{2} + b^{2} c^{2} d\right ) + x^{2} \left (- 25 a^{2} c d^{2} + 10 a b c^{2} d - b^{2} c^{3}\right )}{8 c^{5} d x + 16 c^{4} d^{2} x^{3} + 8 c^{3} d^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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