Optimal. Leaf size=83 \[ -\frac {\sqrt {c} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{7/2}}+\frac {x (b c-a d)^2}{d^3}-\frac {b x^3 (b c-2 a d)}{3 d^2}+\frac {b^2 x^5}{5 d} \]
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Rubi [A] time = 0.06, antiderivative size = 83, 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} \begin {gather*} -\frac {b x^3 (b c-2 a d)}{3 d^2}+\frac {x (b c-a d)^2}{d^3}-\frac {\sqrt {c} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{7/2}}+\frac {b^2 x^5}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 461
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (\frac {(b c-a d)^2}{d^3}-\frac {b (b c-2 a d) x^2}{d^2}+\frac {b^2 x^4}{d}+\frac {-b^2 c^3+2 a b c^2 d-a^2 c d^2}{d^3 \left (c+d x^2\right )}\right ) \, dx\\ &=\frac {(b c-a d)^2 x}{d^3}-\frac {b (b c-2 a d) x^3}{3 d^2}+\frac {b^2 x^5}{5 d}-\frac {\left (c (b c-a d)^2\right ) \int \frac {1}{c+d x^2} \, dx}{d^3}\\ &=\frac {(b c-a d)^2 x}{d^3}-\frac {b (b c-2 a d) x^3}{3 d^2}+\frac {b^2 x^5}{5 d}-\frac {\sqrt {c} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{7/2}}+\frac {x (a d-b c)^2}{d^3}-\frac {b x^3 (b c-2 a d)}{3 d^2}+\frac {b^2 x^5}{5 d} \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 {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.57, size = 228, normalized size = 2.75 \begin {gather*} \left [\frac {6 \, b^{2} d^{2} x^{5} - 10 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} 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 ) + 30 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{30 \, d^{3}}, \frac {3 \, b^{2} d^{2} x^{5} - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} - 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 113, normalized size = 1.36 \begin {gather*} -\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{2} d^{4} x^{5} - 5 \, b^{2} c d^{3} x^{3} + 10 \, a b d^{4} x^{3} + 15 \, b^{2} c^{2} d^{2} x - 30 \, a b c d^{3} x + 15 \, a^{2} d^{4} x}{15 \, d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 135, normalized size = 1.63 \begin {gather*} \frac {b^{2} x^{5}}{5 d}+\frac {2 a b \,x^{3}}{3 d}-\frac {b^{2} c \,x^{3}}{3 d^{2}}-\frac {a^{2} c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d}+\frac {2 a b \,c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{2}}-\frac {b^{2} c^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{3}}+\frac {a^{2} x}{d}-\frac {2 a b c x}{d^{2}}+\frac {b^{2} c^{2} x}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 104, normalized size = 1.25 \begin {gather*} -\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{2} d^{2} x^{5} - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 128, normalized size = 1.54 \begin {gather*} x\,\left (\frac {a^2}{d}+\frac {c\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )}{d}\right )-x^3\,\left (\frac {b^2\,c}{3\,d^2}-\frac {2\,a\,b}{3\,d}\right )+\frac {b^2\,x^5}{5\,d}-\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^2}{a^2\,c\,d^2-2\,a\,b\,c^2\,d+b^2\,c^3}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.49, size = 194, normalized size = 2.34 \begin {gather*} \frac {b^{2} x^{5}}{5 d} + x^{3} \left (\frac {2 a b}{3 d} - \frac {b^{2} c}{3 d^{2}}\right ) + x \left (\frac {a^{2}}{d} - \frac {2 a b c}{d^{2}} + \frac {b^{2} c^{2}}{d^{3}}\right ) + \frac {\sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2} \log {\left (- \frac {d^{3} \sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac {\sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2} \log {\left (\frac {d^{3} \sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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