Optimal. Leaf size=78 \[ \frac {x}{b d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} (b c-a d)}-\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{3/2} (b c-a d)} \]
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Rubi [A]
time = 0.06, antiderivative size = 78, 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 = {490, 536, 211}
\begin {gather*} \frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} (b c-a d)}-\frac {c^{3/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{3/2} (b c-a d)}+\frac {x}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 490
Rule 536
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {x}{b d}-\frac {\int \frac {a c+(b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{b d}\\ &=\frac {x}{b d}+\frac {a^2 \int \frac {1}{a+b x^2} \, dx}{b (b c-a d)}-\frac {c^2 \int \frac {1}{c+d x^2} \, dx}{d (b c-a d)}\\ &=\frac {x}{b d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} (b c-a d)}-\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{3/2} (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 74, normalized size = 0.95 \begin {gather*} \frac {-\frac {a x}{b}+\frac {c x}{d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {c^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{3/2}}}{b c-a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 73, normalized size = 0.94
method | result | size |
default | \(\frac {x}{b d}-\frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b \left (a d -b c \right ) \sqrt {a b}}+\frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{d \left (a d -b c \right ) \sqrt {c d}}\) | \(73\) |
risch | \(\frac {x}{b d}+\frac {\sqrt {-c d}\, c \ln \left (\left (-\left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d -\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}-a^{4} \sqrt {-c d}\, d^{5}-b^{4} c^{4} \sqrt {-c d}\, d \right ) x +a^{4} c \,d^{5}-a \,b^{3} c^{4} d^{2}\right )}{2 d^{2} \left (a d -b c \right )}-\frac {\sqrt {-c d}\, c \ln \left (\left (\left (-c d \right )^{\frac {3}{2}} a \,b^{3} c^{2} d +\left (-c d \right )^{\frac {3}{2}} b^{4} c^{3}+a^{4} \sqrt {-c d}\, d^{5}+b^{4} c^{4} \sqrt {-c d}\, d \right ) x +a^{4} c \,d^{5}-a \,b^{3} c^{4} d^{2}\right )}{2 d^{2} \left (a d -b c \right )}+\frac {\sqrt {-a b}\, a \ln \left (\left (-\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}-\left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}-a^{4} \sqrt {-a b}\, d^{4} b -b^{5} c^{4} \sqrt {-a b}\right ) x +a^{4} b^{2} c \,d^{3}-a \,b^{5} c^{4}\right )}{2 b^{2} \left (a d -b c \right )}-\frac {\sqrt {-a b}\, a \ln \left (\left (\left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}+\left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}+a^{4} \sqrt {-a b}\, d^{4} b +b^{5} c^{4} \sqrt {-a b}\right ) x +a^{4} b^{2} c \,d^{3}-a \,b^{5} c^{4}\right )}{2 b^{2} \left (a d -b c \right )}\) | \(426\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 72, normalized size = 0.92 \begin {gather*} \frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {a b}} - \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d - a d^{2}\right )} \sqrt {c d}} + \frac {x}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 391, normalized size = 5.01 \begin {gather*} \left [-\frac {a d \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + b c \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 (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, \frac {2 \, a d \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - b c \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 (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, -\frac {2 \, b c \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + a d \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 c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, \frac {a d \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - b c \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + {\left (b c - a d\right )} x}{b^{2} c d - a b d^{2}}\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. 921 vs.
\(2 (65) = 130\).
time = 153.14, size = 921, normalized size = 11.81 \begin {gather*} - \frac {\sqrt {- \frac {a^{3}}{b^{3}}} \log {\left (x + \frac {- \frac {a^{4} d^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c} - \frac {a^{3} b^{3} d^{6} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{4} c d^{5} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a b^{5} c^{2} d^{4} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{3} d^{3} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {a^{3}}{b^{3}}} \log {\left (x + \frac {\frac {a^{4} d^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c} + \frac {a^{3} b^{3} d^{6} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{4} c d^{5} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a b^{5} c^{2} d^{4} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{3} d^{3} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {c^{3}}{d^{3}}} \log {\left (x + \frac {- \frac {a^{4} d^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c} - \frac {a^{3} b^{3} d^{6} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{4} c d^{5} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a b^{5} c^{2} d^{4} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{3} d^{3} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {c^{3}}{d^{3}}} \log {\left (x + \frac {\frac {a^{4} d^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c} + \frac {a^{3} b^{3} d^{6} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{4} c d^{5} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a b^{5} c^{2} d^{4} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{3} d^{3} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} + \frac {x}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 72, normalized size = 0.92 \begin {gather*} \frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {a b}} - \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d - a d^{2}\right )} \sqrt {c d}} + \frac {x}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 343, normalized size = 4.40 \begin {gather*} \frac {\ln \left (a^5\,b^4\,d^3-a^2\,b^7\,c^3+d^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+b^6\,c^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,b^4\,c-2\,a\,b^3\,d}-\frac {\ln \left (a^2\,b^7\,c^3-a^5\,b^4\,d^3+d^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+b^6\,c^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,\left (b^4\,c-a\,b^3\,d\right )}+\frac {x}{b\,d}-\frac {\ln \left (a^3\,c^2\,d^7-b^3\,c^5\,d^4+b^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+a^3\,d^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,\left (a\,d^4-b\,c\,d^3\right )}+\frac {\ln \left (b^3\,c^5\,d^4-a^3\,c^2\,d^7+b^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+a^3\,d^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,a\,d^4-2\,b\,c\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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