Optimal. Leaf size=161 \[ \frac {\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}}+\frac {x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac {a (6 b c-7 a d)}{3 c^4 x} \]
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Rubi [A] time = 0.19, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {462, 456, 453, 205} \begin {gather*} \frac {x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}+\frac {\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac {a (6 b c-7 a d)}{3 c^4 x} \end {gather*}
Antiderivative was successfully verified.
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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 )^3} \, dx &=-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\int \frac {a (6 b c-7 a d)+3 b^2 c x^2}{x^2 \left (c+d x^2\right )^3} \, dx}{3 c}\\ &=-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}-\frac {\int \frac {-\frac {4 a (6 b c-7 a d)}{c}-3 \left (3 b^2-\frac {6 a b d}{c}+\frac {7 a^2 d^2}{c^2}\right ) x^2}{x^2 \left (c+d x^2\right )^2} \, dx}{12 c}\\ &=-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac {(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac {\int \frac {\frac {8 a (6 b c-7 a d)}{c^2}+\frac {(3 b c-7 a d)^2 x^2}{c^3}}{x^2 \left (c+d x^2\right )} \, dx}{24 c}\\ &=-\frac {a (6 b c-7 a d)}{3 c^4 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac {(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^4}\\ &=-\frac {a (6 b c-7 a d)}{3 c^4 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac {(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 148, normalized size = 0.92 \begin {gather*} \frac {\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}}+\frac {x \left (11 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{8 c^4 \left (c+d x^2\right )}-\frac {a^2}{3 c^3 x^3}+\frac {a (3 a d-2 b c)}{c^4 x}+\frac {x (b c-a d)^2}{4 c^3 \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^4 \left (c+d x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.93, size = 536, normalized size = 3.33 \begin {gather*} \left [-\frac {16 \, a^{2} c^{4} d - 6 \, {\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 10 \, {\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 16 \, {\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} + 3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} 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 )}{48 \, {\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}, -\frac {8 \, a^{2} c^{4} d - 3 \, {\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 5 \, {\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 8 \, {\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} - 3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right )}{24 \, {\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 151, normalized size = 0.94 \begin {gather*} \frac {{\left (3 \, b^{2} c^{2} - 30 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{4}} + \frac {3 \, b^{2} c^{2} d x^{3} - 14 \, a b c d^{2} x^{3} + 11 \, a^{2} d^{3} x^{3} + 5 \, b^{2} c^{3} x - 18 \, a b c^{2} d x + 13 \, a^{2} c d^{2} x}{8 \, {\left (d x^{2} + c\right )}^{2} c^{4}} - \frac {6 \, a b c x^{2} - 9 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 227, normalized size = 1.41 \begin {gather*} \frac {11 a^{2} d^{3} x^{3}}{8 \left (d \,x^{2}+c \right )^{2} c^{4}}-\frac {7 a b \,d^{2} x^{3}}{4 \left (d \,x^{2}+c \right )^{2} c^{3}}+\frac {3 b^{2} d \,x^{3}}{8 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {13 a^{2} d^{2} x}{8 \left (d \,x^{2}+c \right )^{2} c^{3}}-\frac {9 a b d x}{4 \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {5 b^{2} x}{8 \left (d \,x^{2}+c \right )^{2} c}+\frac {35 a^{2} d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c^{4}}-\frac {15 a b d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{4 \sqrt {c d}\, c^{3}}+\frac {3 b^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c^{2}}+\frac {3 a^{2} d}{c^{4} x}-\frac {2 a b}{c^{3} x}-\frac {a^{2}}{3 c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.30, size = 167, normalized size = 1.04 \begin {gather*} \frac {3 \, {\left (3 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + 5 \, {\left (3 \, b^{2} c^{3} - 30 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} - 8 \, {\left (6 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}}{24 \, {\left (c^{4} d^{2} x^{7} + 2 \, c^{5} d x^{5} + c^{6} x^{3}\right )}} + \frac {{\left (3 \, b^{2} c^{2} - 30 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 156, normalized size = 0.97 \begin {gather*} \frac {\frac {5\,x^4\,\left (35\,a^2\,d^2-30\,a\,b\,c\,d+3\,b^2\,c^2\right )}{24\,c^3}-\frac {a^2}{3\,c}+\frac {a\,x^2\,\left (7\,a\,d-6\,b\,c\right )}{3\,c^2}+\frac {d\,x^6\,\left (35\,a^2\,d^2-30\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,c^4}}{c^2\,x^3+2\,c\,d\,x^5+d^2\,x^7}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (35\,a^2\,d^2-30\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,c^{9/2}\,\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 240, normalized size = 1.49 \begin {gather*} - \frac {\sqrt {- \frac {1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log {\left (- c^{5} \sqrt {- \frac {1}{c^{9} d}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log {\left (c^{5} \sqrt {- \frac {1}{c^{9} d}} + x \right )}}{16} + \frac {- 8 a^{2} c^{3} + x^{6} \left (105 a^{2} d^{3} - 90 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x^{4} \left (175 a^{2} c d^{2} - 150 a b c^{2} d + 15 b^{2} c^{3}\right ) + x^{2} \left (56 a^{2} c^{2} d - 48 a b c^{3}\right )}{24 c^{6} x^{3} + 48 c^{5} d x^{5} + 24 c^{4} d^{2} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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