Optimal. Leaf size=142 \[ \frac {(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{9/2}}+\frac {d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac {2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac {d^4 x^5}{5 b^2} \]
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Rubi [A] time = 0.12, antiderivative size = 142, 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 = {390, 385, 205} \begin {gather*} \frac {d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{9/2}}+\frac {2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac {x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac {d^4 x^5}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rule 390
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^2} \, dx &=\int \left (\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{b^4}+\frac {2 d^3 (2 b c-a d) x^2}{b^3}+\frac {d^4 x^4}{b^2}+\frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^2}{b^4 \left (a+b x^2\right )^2}\right ) \, dx\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {\int \frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^2}{\left (a+b x^2\right )^2} \, dx}{b^4}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac {\left ((b c-a d)^3 (b c+7 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a b^4}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^3 (b c+7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 142, normalized size = 1.00 \begin {gather*} \frac {(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{9/2}}+\frac {d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac {2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac {d^4 x^5}{5 b^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 (c+d x^2\right )^4}{\left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.86, size = 612, normalized size = 4.31 \begin {gather*} \left [\frac {12 \, a^{2} b^{4} d^{4} x^{7} + 4 \, {\left (20 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 20 \, {\left (18 \, a^{2} b^{4} c^{2} d^{2} - 20 \, a^{3} b^{3} c d^{3} + 7 \, a^{4} b^{2} d^{4}\right )} x^{3} + 15 \, {\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 20 \, a^{4} b^{2} c d^{3} + 7 \, a^{5} b d^{4}\right )} x}{60 \, {\left (a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}, \frac {6 \, a^{2} b^{4} d^{4} x^{7} + 2 \, {\left (20 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 10 \, {\left (18 \, a^{2} b^{4} c^{2} d^{2} - 20 \, a^{3} b^{3} c d^{3} + 7 \, a^{4} b^{2} d^{4}\right )} x^{3} + 15 \, {\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 20 \, a^{4} b^{2} c d^{3} + 7 \, a^{5} b d^{4}\right )} x}{30 \, {\left (a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 220, normalized size = 1.55 \begin {gather*} \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{4}} + \frac {b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{2 \, {\left (b x^{2} + a\right )} a b^{4}} + \frac {3 \, b^{8} d^{4} x^{5} + 20 \, b^{8} c d^{3} x^{3} - 10 \, a b^{7} d^{4} x^{3} + 90 \, b^{8} c^{2} d^{2} x - 120 \, a b^{7} c d^{3} x + 45 \, a^{2} b^{6} d^{4} x}{15 \, b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 296, normalized size = 2.08 \begin {gather*} \frac {d^{4} x^{5}}{5 b^{2}}-\frac {2 a \,d^{4} x^{3}}{3 b^{3}}+\frac {4 c \,d^{3} x^{3}}{3 b^{2}}+\frac {a^{3} d^{4} x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} d^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {2 a^{2} c \,d^{3} x}{\left (b \,x^{2}+a \right ) b^{3}}+\frac {10 a^{2} c \,d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {3 a \,c^{2} d^{2} x}{\left (b \,x^{2}+a \right ) b^{2}}-\frac {9 a \,c^{2} d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {c^{4} x}{2 \left (b \,x^{2}+a \right ) a}+\frac {c^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}-\frac {2 c^{3} d x}{\left (b \,x^{2}+a \right ) b}+\frac {2 c^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {3 a^{2} d^{4} x}{b^{4}}-\frac {8 a c \,d^{3} x}{b^{3}}+\frac {6 c^{2} d^{2} x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 213, normalized size = 1.50 \begin {gather*} \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x}{2 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac {3 \, b^{2} d^{4} x^{5} + 10 \, {\left (2 \, b^{2} c d^{3} - a b d^{4}\right )} x^{3} + 15 \, {\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x}{15 \, b^{4}} + \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 261, normalized size = 1.84 \begin {gather*} x\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^4}{b^3}-\frac {4\,c\,d^3}{b^2}\right )}{b}-\frac {a^2\,d^4}{b^4}+\frac {6\,c^2\,d^2}{b^2}\right )-x^3\,\left (\frac {2\,a\,d^4}{3\,b^3}-\frac {4\,c\,d^3}{3\,b^2}\right )+\frac {d^4\,x^5}{5\,b^2}+\frac {x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,\left (b^5\,x^2+a\,b^4\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3\,\left (7\,a\,d+b\,c\right )}{\sqrt {a}\,\left (-7\,a^4\,d^4+20\,a^3\,b\,c\,d^3-18\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (7\,a\,d+b\,c\right )}{2\,a^{3/2}\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.47, size = 403, normalized size = 2.84 \begin {gather*} x^{3} \left (- \frac {2 a d^{4}}{3 b^{3}} + \frac {4 c d^{3}}{3 b^{2}}\right ) + x \left (\frac {3 a^{2} d^{4}}{b^{4}} - \frac {8 a c d^{3}}{b^{3}} + \frac {6 c^{2} d^{2}}{b^{2}}\right ) + \frac {x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{2 a^{2} b^{4} + 2 a b^{5} x^{2}} + \frac {\sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right ) \log {\left (- \frac {a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right ) \log {\left (\frac {a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} + \frac {d^{4} x^{5}}{5 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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