Optimal. Leaf size=81 \[ -\frac {1}{a c x}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (b c-a d)}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)} \]
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Rubi [A]
time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {491, 536, 211}
\begin {gather*} -\frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (b c-a d)}+\frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)}-\frac {1}{a c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 491
Rule 536
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=-\frac {1}{a c x}+\frac {\int \frac {-b c-a d-b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{a c}\\ &=-\frac {1}{a c x}-\frac {b^2 \int \frac {1}{a+b x^2} \, dx}{a (b c-a d)}+\frac {d^2 \int \frac {1}{c+d x^2} \, dx}{c (b c-a d)}\\ &=-\frac {1}{a c x}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (b c-a d)}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 76, normalized size = 0.94 \begin {gather*} \frac {-\frac {b}{a}+\frac {d}{c}-\frac {b^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2}}}{b c x-a d x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 76, normalized size = 0.94
method | result | size |
default | \(\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \left (a d -b c \right ) \sqrt {a b}}-\frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{c \left (a d -b c \right ) \sqrt {c d}}-\frac {1}{a c x}\) | \(76\) |
risch | \(-\frac {1}{a c x}+\frac {\sqrt {-c d}\, d \ln \left (c \,d^{2} x +\left (-c d \right )^{\frac {3}{2}}\right )}{2 c^{2} \left (a d -b c \right )}-\frac {\sqrt {-c d}\, d \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right )}{2 c^{2} \left (a d -b c \right )}+\frac {\sqrt {-a b}\, b \ln \left (-a \,b^{2} x +\left (-a b \right )^{\frac {3}{2}}\right )}{2 a^{2} \left (a d -b c \right )}-\frac {\sqrt {-a b}\, b \ln \left (-a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right )}{2 a^{2} \left (a d -b c \right )}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 75, normalized size = 0.93 \begin {gather*} -\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a b c - a^{2} d\right )} \sqrt {a b}} + \frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c^{2} - a c d\right )} \sqrt {c d}} - \frac {1}{a c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.02, size = 384, normalized size = 4.74 \begin {gather*} \left [-\frac {b c x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + a d x \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \, {\left (a b c^{2} - a^{2} c d\right )} x}, \frac {2 \, a d x \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) - b c x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, b c + 2 \, a d}{2 \, {\left (a b c^{2} - a^{2} c d\right )} x}, -\frac {2 \, b c x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + a d x \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) + 2 \, b c - 2 \, a d}{2 \, {\left (a b c^{2} - a^{2} c d\right )} x}, -\frac {b c x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - a d x \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + b c - a d}{{\left (a b c^{2} - a^{2} c d\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1093 vs.
\(2 (66) = 132\).
time = 138.92, size = 1093, normalized size = 13.49 \begin {gather*} - \frac {\sqrt {- \frac {b^{3}}{a^{3}}} \log {\left (x + \frac {- \frac {a^{7} c^{3} d^{4} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {2 a^{6} b c^{4} d^{3} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {2 a^{5} b^{2} c^{5} d^{2} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{5} d^{5} \sqrt {- \frac {b^{3}}{a^{3}}}}{a d - b c} + \frac {2 a^{4} b^{3} c^{6} d \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{3} b^{4} c^{7} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{5} c^{5} \sqrt {- \frac {b^{3}}{a^{3}}}}{a d - b c}}{a^{2} b^{2} d^{4} + a b^{3} c d^{3} + b^{4} c^{2} d^{2}} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {b^{3}}{a^{3}}} \log {\left (x + \frac {\frac {a^{7} c^{3} d^{4} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {2 a^{6} b c^{4} d^{3} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {2 a^{5} b^{2} c^{5} d^{2} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{5} d^{5} \sqrt {- \frac {b^{3}}{a^{3}}}}{a d - b c} - \frac {2 a^{4} b^{3} c^{6} d \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b^{4} c^{7} \left (- \frac {b^{3}}{a^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{5} c^{5} \sqrt {- \frac {b^{3}}{a^{3}}}}{a d - b c}}{a^{2} b^{2} d^{4} + a b^{3} c d^{3} + b^{4} c^{2} d^{2}} \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {d^{3}}{c^{3}}} \log {\left (x + \frac {- \frac {a^{7} c^{3} d^{4} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {2 a^{6} b c^{4} d^{3} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {2 a^{5} b^{2} c^{5} d^{2} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{5} d^{5} \sqrt {- \frac {d^{3}}{c^{3}}}}{a d - b c} + \frac {2 a^{4} b^{3} c^{6} d \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{3} b^{4} c^{7} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{5} c^{5} \sqrt {- \frac {d^{3}}{c^{3}}}}{a d - b c}}{a^{2} b^{2} d^{4} + a b^{3} c d^{3} + b^{4} c^{2} d^{2}} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {d^{3}}{c^{3}}} \log {\left (x + \frac {\frac {a^{7} c^{3} d^{4} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {2 a^{6} b c^{4} d^{3} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {2 a^{5} b^{2} c^{5} d^{2} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{5} d^{5} \sqrt {- \frac {d^{3}}{c^{3}}}}{a d - b c} - \frac {2 a^{4} b^{3} c^{6} d \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b^{4} c^{7} \left (- \frac {d^{3}}{c^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{5} c^{5} \sqrt {- \frac {d^{3}}{c^{3}}}}{a d - b c}}{a^{2} b^{2} d^{4} + a b^{3} c d^{3} + b^{4} c^{2} d^{2}} \right )}}{2 \left (a d - b c\right )} - \frac {1}{a c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 75, normalized size = 0.93 \begin {gather*} -\frac {b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a b c - a^{2} d\right )} \sqrt {a b}} + \frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c^{2} - a c d\right )} \sqrt {c d}} - \frac {1}{a c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 338, normalized size = 4.17 \begin {gather*} \frac {\ln \left (a^3\,c^5\,d^4-b^3\,c^8\,d+a^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+b^3\,c^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,b\,c^4-2\,a\,c^3\,d}-\frac {\ln \left (b^3\,c^8\,d-a^3\,c^5\,d^4+a^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+b^3\,c^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,\left (b\,c^4-a\,c^3\,d\right )}-\frac {1}{a\,c\,x}-\frac {\ln \left (a^8\,b\,d^3-a^5\,b^4\,c^3+c^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+a^6\,d^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,\left (a^4\,d-a^3\,b\,c\right )}+\frac {\ln \left (a^5\,b^4\,c^3-a^8\,b\,d^3+c^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+a^6\,d^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,a^4\,d-2\,a^3\,b\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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