Optimal. Leaf size=109 \[ \frac {a x}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 109, 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 = {481, 536, 211}
\begin {gather*} -\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b c-a d)}{2 b^{3/2} (b c-a d)^2}+\frac {c^{3/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 481
Rule 536
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {a x}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {\int \frac {a c+(-2 b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 b (b c-a d)}\\ &=\frac {a x}{2 b (b c-a d) \left (a+b x^2\right )}+\frac {c^2 \int \frac {1}{c+d x^2} \, dx}{(b c-a d)^2}-\frac {(a (3 b c-a d)) \int \frac {1}{a+b x^2} \, dx}{2 b (b c-a d)^2}\\ &=\frac {a x}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 95, normalized size = 0.87 \begin {gather*} \frac {\frac {a (b c-a d) x}{b \left (a+b x^2\right )}+\frac {\sqrt {a} (-3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+\frac {2 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d}}}{2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 94, normalized size = 0.86
method | result | size |
default | \(-\frac {a \left (\frac {\left (a d -b c \right ) x}{2 b \left (b \,x^{2}+a \right )}-\frac {\left (a d -3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\right )}{\left (a d -b c \right )^{2}}+\frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}}\) | \(94\) |
risch | \(-\frac {a x}{2 \left (a d -b c \right ) b \left (b \,x^{2}+a \right )}+\frac {\sqrt {-c d}\, c \ln \left (\left (-4 \left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d -4 \left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}-a^{4} \sqrt {-c d}\, d^{5}+6 \sqrt {-c d}\, a^{3} b c \,d^{4}-9 \sqrt {-c d}\, a^{2} b^{2} c^{2} d^{3}-4 b^{4} c^{4} \sqrt {-c d}\, d \right ) x +a^{4} c \,d^{5}-6 b \,c^{2} d^{4} a^{3}+9 b^{2} c^{3} d^{3} a^{2}-4 a \,b^{3} c^{4} d^{2}\right )}{2 d \left (a d -b c \right )^{2}}-\frac {\sqrt {-c d}\, c \ln \left (\left (4 \left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d +4 \left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}+a^{4} \sqrt {-c d}\, d^{5}-6 \sqrt {-c d}\, a^{3} b c \,d^{4}+9 \sqrt {-c d}\, a^{2} b^{2} c^{2} d^{3}+4 b^{4} c^{4} \sqrt {-c d}\, d \right ) x +a^{4} c \,d^{5}-6 b \,c^{2} d^{4} a^{3}+9 b^{2} c^{3} d^{3} a^{2}-4 a \,b^{3} c^{4} d^{2}\right )}{2 d \left (a d -b c \right )^{2}}+\frac {\sqrt {-a b}\, \ln \left (\left (-\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}+5 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}-3 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d^{2}-9 \left (-a b \right )^{\frac {3}{2}} b^{3} c^{3} d -a^{4} \sqrt {-a b}\, d^{4} b +6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}-9 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}-4 b^{5} c^{4} \sqrt {-a b}\right ) x -a^{4} b^{2} c \,d^{3}+6 a^{3} b^{3} c^{2} d^{2}-9 a^{2} b^{4} c^{3} d +4 a \,b^{5} c^{4}\right ) a d}{4 b^{2} \left (a d -b c \right )^{2}}-\frac {3 \sqrt {-a b}\, \ln \left (\left (-\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}+5 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}-3 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d^{2}-9 \left (-a b \right )^{\frac {3}{2}} b^{3} c^{3} d -a^{4} \sqrt {-a b}\, d^{4} b +6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}-9 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}-4 b^{5} c^{4} \sqrt {-a b}\right ) x -a^{4} b^{2} c \,d^{3}+6 a^{3} b^{3} c^{2} d^{2}-9 a^{2} b^{4} c^{3} d +4 a \,b^{5} c^{4}\right ) c}{4 b \left (a d -b c \right )^{2}}-\frac {\sqrt {-a b}\, \ln \left (\left (\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}-5 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}+3 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d^{2}+9 \left (-a b \right )^{\frac {3}{2}} b^{3} c^{3} d +a^{4} \sqrt {-a b}\, d^{4} b -6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}+9 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}+4 b^{5} c^{4} \sqrt {-a b}\right ) x -a^{4} b^{2} c \,d^{3}+6 a^{3} b^{3} c^{2} d^{2}-9 a^{2} b^{4} c^{3} d +4 a \,b^{5} c^{4}\right ) a d}{4 b^{2} \left (a d -b c \right )^{2}}+\frac {3 \sqrt {-a b}\, \ln \left (\left (\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}-5 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}+3 \left (-a b \right )^{\frac {3}{2}} a \,b^{2} c^{2} d^{2}+9 \left (-a b \right )^{\frac {3}{2}} b^{3} c^{3} d +a^{4} \sqrt {-a b}\, d^{4} b -6 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}+9 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}+4 b^{5} c^{4} \sqrt {-a b}\right ) x -a^{4} b^{2} c \,d^{3}+6 a^{3} b^{3} c^{2} d^{2}-9 a^{2} b^{4} c^{3} d +4 a \,b^{5} c^{4}\right ) c}{4 b \left (a d -b c \right )^{2}}\) | \(1177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 133, normalized size = 1.22 \begin {gather*} \frac {c^{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 {a x}{2 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )}} - \frac {{\left (3 \, a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b 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 = 0.91, size = 726, normalized size = 6.66 \begin {gather*} \left [-\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 2 \, {\left (a b c - a^{2} d\right )} x}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - {\left (a b c - a^{2} d\right )} x}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {4 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - {\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 2 \, {\left (a b c - a^{2} d\right )} x}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {{\left (3 \, a b c - a^{2} d + {\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - {\left (a b c - a^{2} d\right )} x}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} 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 = 0.48, size = 122, normalized size = 1.12 \begin {gather*} \frac {c^{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 (3 \, a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {a b}} + \frac {a x}{2 \, {\left (b^{2} c - a b 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 = 0.55, size = 2500, normalized size = 22.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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