Optimal. Leaf size=87 \[ \frac {(b c-2 a d) x}{b^3}+\frac {d x^3}{3 b^2}+\frac {a (b c-a d) x}{2 b^3 \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 87, 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 = {466, 1167, 211}
\begin {gather*} -\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (3 b c-5 a d)}{2 b^{7/2}}+\frac {a x (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac {x (b c-2 a d)}{b^3}+\frac {d x^3}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 466
Rule 1167
Rubi steps
\begin {align*} \int \frac {x^4 \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {a (b c-a d) x}{2 b^3 \left (a+b x^2\right )}-\frac {\int \frac {a (b c-a d)-2 b (b c-a d) x^2-2 b^2 d x^4}{a+b x^2} \, dx}{2 b^3}\\ &=\frac {a (b c-a d) x}{2 b^3 \left (a+b x^2\right )}-\frac {\int \left (-2 (b c-2 a d)-2 b d x^2+\frac {3 a b c-5 a^2 d}{a+b x^2}\right ) \, dx}{2 b^3}\\ &=\frac {(b c-2 a d) x}{b^3}+\frac {d x^3}{3 b^2}+\frac {a (b c-a d) x}{2 b^3 \left (a+b x^2\right )}-\frac {(a (3 b c-5 a d)) \int \frac {1}{a+b x^2} \, dx}{2 b^3}\\ &=\frac {(b c-2 a d) x}{b^3}+\frac {d x^3}{3 b^2}+\frac {a (b c-a d) x}{2 b^3 \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 89, normalized size = 1.02 \begin {gather*} \frac {(b c-2 a d) x}{b^3}+\frac {d x^3}{3 b^2}+\frac {\left (a b c-a^2 d\right ) x}{2 b^3 \left (a+b x^2\right )}+\frac {\sqrt {a} (-3 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 76, normalized size = 0.87
method | result | size |
default | \(-\frac {-\frac {1}{3} b d \,x^{3}+2 a d x -b c x}{b^{3}}+\frac {a \left (\frac {\left (-\frac {a d}{2}+\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (5 a d -3 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{3}}\) | \(76\) |
risch | \(\frac {d \,x^{3}}{3 b^{2}}-\frac {2 a d x}{b^{3}}+\frac {c x}{b^{2}}+\frac {\left (-\frac {1}{2} a^{2} d +\frac {1}{2} a b c \right ) x}{b^{3} \left (b \,x^{2}+a \right )}+\frac {5 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a d}{4 b^{4}}-\frac {3 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) c}{4 b^{3}}-\frac {5 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a d}{4 b^{4}}+\frac {3 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) c}{4 b^{3}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 84, normalized size = 0.97 \begin {gather*} \frac {{\left (a b c - a^{2} d\right )} x}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} - \frac {{\left (3 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} + \frac {b d x^{3} + 3 \, {\left (b c - 2 \, a d\right )} x}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.56, size = 240, normalized size = 2.76 \begin {gather*} \left [\frac {4 \, b^{2} d x^{5} + 4 \, {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{3} - 3 \, {\left (3 \, a b c - 5 \, a^{2} d + {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, {\left (3 \, a b c - 5 \, a^{2} d\right )} x}{12 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {2 \, b^{2} d x^{5} + 2 \, {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{3} - 3 \, {\left (3 \, a b c - 5 \, a^{2} d + {\left (3 \, b^{2} c - 5 \, a b d\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3 \, {\left (3 \, a b c - 5 \, a^{2} d\right )} x}{6 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 129, normalized size = 1.48 \begin {gather*} x \left (- \frac {2 a d}{b^{3}} + \frac {c}{b^{2}}\right ) + \frac {x \left (- a^{2} d + a b c\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {\sqrt {- \frac {a}{b^{7}}} \cdot \left (5 a d - 3 b c\right ) \log {\left (- b^{3} \sqrt {- \frac {a}{b^{7}}} + x \right )}}{4} + \frac {\sqrt {- \frac {a}{b^{7}}} \cdot \left (5 a d - 3 b c\right ) \log {\left (b^{3} \sqrt {- \frac {a}{b^{7}}} + x \right )}}{4} + \frac {d x^{3}}{3 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 88, normalized size = 1.01 \begin {gather*} -\frac {{\left (3 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} + \frac {a b c x - a^{2} d x}{2 \, {\left (b x^{2} + a\right )} b^{3}} + \frac {b^{4} d x^{3} + 3 \, b^{4} c x - 6 \, a b^{3} d x}{3 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 104, normalized size = 1.20 \begin {gather*} x\,\left (\frac {c}{b^2}-\frac {2\,a\,d}{b^3}\right )+\frac {d\,x^3}{3\,b^2}-\frac {x\,\left (\frac {a^2\,d}{2}-\frac {a\,b\,c}{2}\right )}{b^4\,x^2+a\,b^3}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (5\,a\,d-3\,b\,c\right )}{5\,a^2\,d-3\,a\,b\,c}\right )\,\left (5\,a\,d-3\,b\,c\right )}{2\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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