Optimal. Leaf size=108 \[ \frac {b x}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\sqrt {b} (b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} (b c-a d)^2}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {425, 536, 211}
\begin {gather*} \frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-3 a d)}{2 a^{3/2} (b c-a d)^2}+\frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^2}+\frac {b x}{2 a \left (a+b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 425
Rule 536
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {b x}{2 a (b c-a d) \left (a+b x^2\right )}-\frac {\int \frac {-b c+2 a d-b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=\frac {b x}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {d^2 \int \frac {1}{c+d x^2} \, dx}{(b c-a d)^2}+\frac {(b (b c-3 a d)) \int \frac {1}{a+b x^2} \, dx}{2 a (b c-a d)^2}\\ &=\frac {b x}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\sqrt {b} (b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} (b c-a d)^2}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 109, normalized size = 1.01 \begin {gather*} -\frac {b x}{2 a (-b c+a d) \left (a+b x^2\right )}-\frac {\sqrt {b} (-b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} (-b c+a d)^2}+\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 95, normalized size = 0.88
method | result | size |
default | \(-\frac {b \left (\frac {\left (a d -b c \right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (3 a d -b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a d -b c \right )^{2}}+\frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}}\) | \(95\) |
risch | \(-\frac {b x}{2 a \left (a d -b c \right ) \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {-a b}\, \ln \left (\left (-9 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{3}-3 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{2}+5 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d -\left (-a b \right )^{\frac {3}{2}} b^{3} c^{3}-13 \sqrt {-a b}\, a^{4} b \,d^{3}+6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{2}-\sqrt {-a b}\, a^{2} b^{3} c^{2} d \right ) x -4 a^{5} b \,d^{3}+9 b^{2} c \,d^{2} a^{4}-6 a^{3} b^{3} c^{2} d +a^{2} b^{4} c^{3}\right ) d}{4 a \left (a d -b c \right )^{2}}-\frac {\sqrt {-a b}\, \ln \left (\left (-9 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{3}-3 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{2}+5 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d -\left (-a b \right )^{\frac {3}{2}} b^{3} c^{3}-13 \sqrt {-a b}\, a^{4} b \,d^{3}+6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{2}-\sqrt {-a b}\, a^{2} b^{3} c^{2} d \right ) x -4 a^{5} b \,d^{3}+9 b^{2} c \,d^{2} a^{4}-6 a^{3} b^{3} c^{2} d +a^{2} b^{4} c^{3}\right ) b c}{4 a^{2} \left (a d -b c \right )^{2}}-\frac {3 \sqrt {-a b}\, \ln \left (\left (9 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{3}+3 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{2}-5 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d +\left (-a b \right )^{\frac {3}{2}} b^{3} c^{3}+13 \sqrt {-a b}\, a^{4} b \,d^{3}-6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{2}+\sqrt {-a b}\, a^{2} b^{3} c^{2} d \right ) x -4 a^{5} b \,d^{3}+9 b^{2} c \,d^{2} a^{4}-6 a^{3} b^{3} c^{2} d +a^{2} b^{4} c^{3}\right ) d}{4 a \left (a d -b c \right )^{2}}+\frac {\sqrt {-a b}\, \ln \left (\left (9 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{3}+3 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{2}-5 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d +\left (-a b \right )^{\frac {3}{2}} b^{3} c^{3}+13 \sqrt {-a b}\, a^{4} b \,d^{3}-6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{2}+\sqrt {-a b}\, a^{2} b^{3} c^{2} d \right ) x -4 a^{5} b \,d^{3}+9 b^{2} c \,d^{2} a^{4}-6 a^{3} b^{3} c^{2} d +a^{2} b^{4} c^{3}\right ) b c}{4 a^{2} \left (a d -b c \right )^{2}}+\frac {\sqrt {-c d}\, d \ln \left (\left (-4 \left (-c d \right )^{\frac {3}{2}} a^{3} d^{2}-4 \left (-c d \right )^{\frac {3}{2}} a^{2} b c d -13 \sqrt {-c d}\, a^{2} b \,c^{2} d^{2}+6 \sqrt {-c d}\, a \,b^{2} c^{3} d -\sqrt {-c d}\, b^{3} c^{4}\right ) x -4 a^{3} c^{2} d^{3}+9 a^{2} b \,c^{3} d^{2}-6 a \,b^{2} c^{4} d +b^{3} c^{5}\right )}{2 c \left (a d -b c \right )^{2}}-\frac {\sqrt {-c d}\, d \ln \left (\left (4 \left (-c d \right )^{\frac {3}{2}} a^{3} d^{2}+4 \left (-c d \right )^{\frac {3}{2}} a^{2} b c d +13 \sqrt {-c d}\, a^{2} b \,c^{2} d^{2}-6 \sqrt {-c d}\, a \,b^{2} c^{3} d +\sqrt {-c d}\, b^{3} c^{4}\right ) x -4 a^{3} c^{2} d^{3}+9 a^{2} b \,c^{3} d^{2}-6 a \,b^{2} c^{4} d +b^{3} c^{5}\right )}{2 c \left (a d -b c \right )^{2}}\) | \(1035\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 132, normalized size = 1.22 \begin {gather*} \frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {b x}{2 \, {\left (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )}} + \frac {{\left (b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 699, normalized size = 6.47 \begin {gather*} \left [-\frac {{\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) - 2 \, {\left (b^{2} c - a b d\right )} x}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, \frac {4 \, {\left (a b d x^{2} + a^{2} d\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) - {\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 2 \, {\left (b^{2} c - a b d\right )} x}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, \frac {{\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) + {\left (b^{2} c - a b d\right )} x}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}, \frac {{\left (a b c - 3 \, a^{2} d + {\left (b^{2} c - 3 \, a b d\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, {\left (a b d x^{2} + a^{2} d\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + {\left (b^{2} c - a b d\right )} x}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A]
time = 0.45, size = 121, normalized size = 1.12 \begin {gather*} \frac {d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {{\left (b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {a b}} + \frac {b x}{2 \, {\left (a b c - a^{2} d\right )} {\left (b x^{2} + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.54, size = 2500, normalized size = 23.15 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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