Optimal. Leaf size=104 \[ \frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}-\frac {c x (b c-a d)^2}{d^4}+\frac {x^3 (b c-a d)^2}{3 d^3}-\frac {b x^5 (b c-2 a d)}{5 d^2}+\frac {b^2 x^7}{7 d} \]
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Rubi [A] time = 0.07, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {461, 205} \[ \frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}-\frac {b x^5 (b c-2 a d)}{5 d^2}+\frac {x^3 (b c-a d)^2}{3 d^3}-\frac {c x (b c-a d)^2}{d^4}+\frac {b^2 x^7}{7 d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 461
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac {c (b c-a d)^2}{d^4}+\frac {(b c-a d)^2 x^2}{d^3}-\frac {b (b c-2 a d) x^4}{d^2}+\frac {b^2 x^6}{d}+\frac {b^2 c^4-2 a b c^3 d+a^2 c^2 d^2}{d^4 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {c (b c-a d)^2 x}{d^4}+\frac {(b c-a d)^2 x^3}{3 d^3}-\frac {b (b c-2 a d) x^5}{5 d^2}+\frac {b^2 x^7}{7 d}+\frac {\left (c^2 (b c-a d)^2\right ) \int \frac {1}{c+d x^2} \, dx}{d^4}\\ &=-\frac {c (b c-a d)^2 x}{d^4}+\frac {(b c-a d)^2 x^3}{3 d^3}-\frac {b (b c-2 a d) x^5}{5 d^2}+\frac {b^2 x^7}{7 d}+\frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 104, normalized size = 1.00 \[ \frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}-\frac {c x (b c-a d)^2}{d^4}+\frac {x^3 (a d-b c)^2}{3 d^3}-\frac {b x^5 (b c-2 a d)}{5 d^2}+\frac {b^2 x^7}{7 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 302, normalized size = 2.90 \[ \left [\frac {30 \, b^{2} d^{3} x^{7} - 42 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 70 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 210 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{210 \, d^{4}}, \frac {15 \, b^{2} d^{3} x^{7} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{105 \, d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 153, normalized size = 1.47 \[ \frac {{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{4}} + \frac {15 \, b^{2} d^{6} x^{7} - 21 \, b^{2} c d^{5} x^{5} + 42 \, a b d^{6} x^{5} + 35 \, b^{2} c^{2} d^{4} x^{3} - 70 \, a b c d^{5} x^{3} + 35 \, a^{2} d^{6} x^{3} - 105 \, b^{2} c^{3} d^{3} x + 210 \, a b c^{2} d^{4} x - 105 \, a^{2} c d^{5} x}{105 \, d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 176, normalized size = 1.69 \[ \frac {b^{2} x^{7}}{7 d}+\frac {2 a b \,x^{5}}{5 d}-\frac {b^{2} c \,x^{5}}{5 d^{2}}+\frac {a^{2} x^{3}}{3 d}-\frac {2 a b c \,x^{3}}{3 d^{2}}+\frac {b^{2} c^{2} x^{3}}{3 d^{3}}+\frac {a^{2} c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{2}}-\frac {2 a b \,c^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{3}}+\frac {b^{2} c^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{4}}-\frac {a^{2} c x}{d^{2}}+\frac {2 a b \,c^{2} x}{d^{3}}-\frac {b^{2} c^{3} x}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.41, size = 139, normalized size = 1.34 \[ \frac {{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{4}} + \frac {15 \, b^{2} d^{3} x^{7} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{105 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 169, normalized size = 1.62 \[ x^3\,\left (\frac {a^2}{3\,d}+\frac {c\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )}{3\,d}\right )-x^5\,\left (\frac {b^2\,c}{5\,d^2}-\frac {2\,a\,b}{5\,d}\right )+\frac {b^2\,x^7}{7\,d}+\frac {c^{3/2}\,\mathrm {atan}\left (\frac {c^{3/2}\,\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^2}{a^2\,c^2\,d^2-2\,a\,b\,c^3\,d+b^2\,c^4}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{9/2}}-\frac {c\,x\,\left (\frac {a^2}{d}+\frac {c\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )}{d}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.53, size = 246, normalized size = 2.37 \[ \frac {b^{2} x^{7}}{7 d} + x^{5} \left (\frac {2 a b}{5 d} - \frac {b^{2} c}{5 d^{2}}\right ) + x^{3} \left (\frac {a^{2}}{3 d} - \frac {2 a b c}{3 d^{2}} + \frac {b^{2} c^{2}}{3 d^{3}}\right ) + x \left (- \frac {a^{2} c}{d^{2}} + \frac {2 a b c^{2}}{d^{3}} - \frac {b^{2} c^{3}}{d^{4}}\right ) - \frac {\sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log {\left (- \frac {d^{4} \sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log {\left (\frac {d^{4} \sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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