Optimal. Leaf size=77 \[ \frac {a^{3/2} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}-\frac {a x (b c-a d)}{b^3}+\frac {x^3 (b c-a d)}{3 b^2}+\frac {d x^5}{5 b} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {459, 302, 205} \begin {gather*} \frac {a^{3/2} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {x^3 (b c-a d)}{3 b^2}-\frac {a x (b c-a d)}{b^3}+\frac {d x^5}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rule 459
Rubi steps
\begin {align*} \int \frac {x^4 \left (c+d x^2\right )}{a+b x^2} \, dx &=\frac {d x^5}{5 b}-\frac {(-5 b c+5 a d) \int \frac {x^4}{a+b x^2} \, dx}{5 b}\\ &=\frac {d x^5}{5 b}-\frac {(-5 b c+5 a d) \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{5 b}\\ &=-\frac {a (b c-a d) x}{b^3}+\frac {(b c-a d) x^3}{3 b^2}+\frac {d x^5}{5 b}+\frac {\left (a^2 (b c-a d)\right ) \int \frac {1}{a+b x^2} \, dx}{b^3}\\ &=-\frac {a (b c-a d) x}{b^3}+\frac {(b c-a d) x^3}{3 b^2}+\frac {d x^5}{5 b}+\frac {a^{3/2} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 1.00 \begin {gather*} -\frac {a^{3/2} (a d-b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}+\frac {a x (a d-b c)}{b^3}+\frac {x^3 (b c-a d)}{3 b^2}+\frac {d x^5}{5 b} \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^4 \left (c+d x^2\right )}{a+b x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.94, size = 178, normalized size = 2.31 \begin {gather*} \left [\frac {6 \, b^{2} d x^{5} + 10 \, {\left (b^{2} c - a b d\right )} x^{3} - 15 \, {\left (a b c - a^{2} d\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 30 \, {\left (a b c - a^{2} d\right )} x}{30 \, b^{3}}, \frac {3 \, b^{2} d x^{5} + 5 \, {\left (b^{2} c - a b d\right )} x^{3} + 15 \, {\left (a b c - a^{2} d\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 15 \, {\left (a b c - a^{2} d\right )} x}{15 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 84, normalized size = 1.09 \begin {gather*} \frac {{\left (a^{2} b c - a^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{4} d x^{5} + 5 \, b^{4} c x^{3} - 5 \, a b^{3} d x^{3} - 15 \, a b^{3} c x + 15 \, a^{2} b^{2} d x}{15 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 92, normalized size = 1.19 \begin {gather*} \frac {d \,x^{5}}{5 b}-\frac {a d \,x^{3}}{3 b^{2}}+\frac {c \,x^{3}}{3 b}-\frac {a^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {a^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {a^{2} d x}{b^{3}}-\frac {a c x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.53, size = 77, normalized size = 1.00 \begin {gather*} \frac {{\left (a^{2} b c - a^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{2} d x^{5} + 5 \, {\left (b^{2} c - a b d\right )} x^{3} - 15 \, {\left (a b c - a^{2} d\right )} x}{15 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 96, normalized size = 1.25 \begin {gather*} x^3\,\left (\frac {c}{3\,b}-\frac {a\,d}{3\,b^2}\right )+\frac {d\,x^5}{5\,b}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (a\,d-b\,c\right )}{a^3\,d-a^2\,b\,c}\right )\,\left (a\,d-b\,c\right )}{b^{7/2}}-\frac {a\,x\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.36, size = 153, normalized size = 1.99 \begin {gather*} x^{3} \left (- \frac {a d}{3 b^{2}} + \frac {c}{3 b}\right ) + x \left (\frac {a^{2} d}{b^{3}} - \frac {a c}{b^{2}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{7}}} \left (a d - b c\right ) \log {\left (- \frac {b^{3} \sqrt {- \frac {a^{3}}{b^{7}}} \left (a d - b c\right )}{a^{2} d - a b c} + x \right )}}{2} - \frac {\sqrt {- \frac {a^{3}}{b^{7}}} \left (a d - b c\right ) \log {\left (\frac {b^{3} \sqrt {- \frac {a^{3}}{b^{7}}} \left (a d - b c\right )}{a^{2} d - a b c} + x \right )}}{2} + \frac {d x^{5}}{5 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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