Optimal. Leaf size=172 \[ -\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^7}{7 b^2}+\frac {f x^9}{9 b}+\frac {a^{3/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1816, 211}
\begin {gather*} \frac {x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}-\frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac {x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^{11/2}}+\frac {x^7 (b e-a f)}{7 b^2}+\frac {f x^9}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1816
Rubi steps
\begin {align*} \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx &=\int \left (-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^4}{b^3}+\frac {(b e-a f) x^6}{b^2}+\frac {f x^8}{b}+\frac {a^2 b^3 c-a^3 b^2 d+a^4 b e-a^5 f}{b^5 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^7}{7 b^2}+\frac {f x^9}{9 b}+\frac {\left (a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{b^5}\\ &=-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^7}{7 b^2}+\frac {f x^9}{9 b}+\frac {a^{3/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 162, normalized size = 0.94 \begin {gather*} \frac {x \left (315 a^4 f-105 a^3 b \left (3 e+f x^2\right )+21 a^2 b^2 \left (15 d+5 e x^2+3 f x^4\right )-3 a b^3 \left (105 c+35 d x^2+21 e x^4+15 f x^6\right )+b^4 x^2 \left (105 c+63 d x^2+45 e x^4+35 f x^6\right )\right )}{315 b^5}-\frac {a^{3/2} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 185, normalized size = 1.08
method | result | size |
default | \(\frac {\frac {1}{9} f \,x^{9} b^{4}-\frac {1}{7} a \,b^{3} f \,x^{7}+\frac {1}{7} b^{4} e \,x^{7}+\frac {1}{5} a^{2} b^{2} f \,x^{5}-\frac {1}{5} a \,b^{3} e \,x^{5}+\frac {1}{5} b^{4} d \,x^{5}-\frac {1}{3} a^{3} b f \,x^{3}+\frac {1}{3} a^{2} b^{2} e \,x^{3}-\frac {1}{3} a \,b^{3} d \,x^{3}+\frac {1}{3} b^{4} c \,x^{3}+a^{4} f x -a^{3} b e x +a^{2} b^{2} d x -a \,b^{3} c x}{b^{5}}-\frac {a^{2} \left (a^{3} f -a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{5} \sqrt {a b}}\) | \(185\) |
risch | \(\frac {f \,x^{9}}{9 b}-\frac {a f \,x^{7}}{7 b^{2}}+\frac {e \,x^{7}}{7 b}+\frac {a^{2} f \,x^{5}}{5 b^{3}}-\frac {a e \,x^{5}}{5 b^{2}}+\frac {d \,x^{5}}{5 b}-\frac {a^{3} f \,x^{3}}{3 b^{4}}+\frac {a^{2} e \,x^{3}}{3 b^{3}}-\frac {a d \,x^{3}}{3 b^{2}}+\frac {c \,x^{3}}{3 b}+\frac {a^{4} f x}{b^{5}}-\frac {a^{3} e x}{b^{4}}+\frac {a^{2} d x}{b^{3}}-\frac {a c x}{b^{2}}+\frac {\sqrt {-a b}\, a^{4} \ln \left (-\sqrt {-a b}\, x -a \right ) f}{2 b^{6}}-\frac {\sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x -a \right ) e}{2 b^{5}}+\frac {\sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) d}{2 b^{4}}-\frac {\sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x -a \right ) c}{2 b^{3}}-\frac {\sqrt {-a b}\, a^{4} \ln \left (\sqrt {-a b}\, x -a \right ) f}{2 b^{6}}+\frac {\sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x -a \right ) e}{2 b^{5}}-\frac {\sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right ) d}{2 b^{4}}+\frac {\sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x -a \right ) c}{2 b^{3}}\) | \(364\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 177, normalized size = 1.03 \begin {gather*} \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f + a^{4} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{4} f x^{9} - 45 \, {\left (a b^{3} f - b^{4} e\right )} x^{7} + 63 \, {\left (b^{4} d + a^{2} b^{2} f - a b^{3} e\right )} x^{5} + 105 \, {\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} x^{3} - 315 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} x}{315 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.87, size = 392, normalized size = 2.28 \begin {gather*} \left [\frac {70 \, b^{4} f x^{9} - 90 \, a b^{3} f x^{7} + 126 \, {\left (b^{4} d + a^{2} b^{2} f\right )} x^{5} + 210 \, {\left (b^{4} c - a b^{3} d - a^{3} b f\right )} x^{3} + 315 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f\right )} x + 6 \, {\left (15 \, b^{4} x^{7} - 21 \, a b^{3} x^{5} + 35 \, a^{2} b^{2} x^{3} - 105 \, a^{3} b x\right )} e}{630 \, b^{5}}, \frac {35 \, b^{4} f x^{9} - 45 \, a b^{3} f x^{7} + 63 \, {\left (b^{4} d + a^{2} b^{2} f\right )} x^{5} + 105 \, {\left (b^{4} c - a b^{3} d - a^{3} b f\right )} x^{3} + 315 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (a b^{3} c - a^{2} b^{2} d - a^{4} f\right )} x + 3 \, {\left (15 \, b^{4} x^{7} - 21 \, a b^{3} x^{5} + 35 \, a^{2} b^{2} x^{3} - 105 \, a^{3} b x\right )} e}{315 \, b^{5}}\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. 337 vs.
\(2 (167) = 334\).
time = 0.43, size = 337, normalized size = 1.96 \begin {gather*} x^{7} \left (- \frac {a f}{7 b^{2}} + \frac {e}{7 b}\right ) + x^{5} \left (\frac {a^{2} f}{5 b^{3}} - \frac {a e}{5 b^{2}} + \frac {d}{5 b}\right ) + x^{3} \left (- \frac {a^{3} f}{3 b^{4}} + \frac {a^{2} e}{3 b^{3}} - \frac {a d}{3 b^{2}} + \frac {c}{3 b}\right ) + x \left (\frac {a^{4} f}{b^{5}} - \frac {a^{3} e}{b^{4}} + \frac {a^{2} d}{b^{3}} - \frac {a c}{b^{2}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c} + x \right )}}{2} - \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c} + x \right )}}{2} + \frac {f x^{9}}{9 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 200, normalized size = 1.16 \begin {gather*} \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f + a^{4} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{8} f x^{9} - 45 \, a b^{7} f x^{7} + 45 \, b^{8} x^{7} e + 63 \, b^{8} d x^{5} + 63 \, a^{2} b^{6} f x^{5} - 63 \, a b^{7} x^{5} e + 105 \, b^{8} c x^{3} - 105 \, a b^{7} d x^{3} - 105 \, a^{3} b^{5} f x^{3} + 105 \, a^{2} b^{6} x^{3} e - 315 \, a b^{7} c x + 315 \, a^{2} b^{6} d x + 315 \, a^{4} b^{4} f x - 315 \, a^{3} b^{5} x e}{315 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.94, size = 243, normalized size = 1.41 \begin {gather*} x^7\,\left (\frac {e}{7\,b}-\frac {a\,f}{7\,b^2}\right )+x^5\,\left (\frac {d}{5\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{5\,b}\right )+x^3\,\left (\frac {c}{3\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{3\,b}\right )+\frac {f\,x^9}{9\,b}-\frac {a\,x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{f\,a^5-e\,a^4\,b+d\,a^3\,b^2-c\,a^2\,b^3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{b^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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