Optimal. Leaf size=109 \[ \frac {(A b-3 a B) x^2}{2 b^4}+\frac {B x^4}{4 b^3}+\frac {a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}-\frac {a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}-\frac {3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5} \]
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Rubi [A]
time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78}
\begin {gather*} \frac {a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}-\frac {a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}-\frac {3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}+\frac {x^2 (A b-3 a B)}{2 b^4}+\frac {B x^4}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A b-3 a B}{b^4}+\frac {B x}{b^3}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)^3}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)^2}+\frac {3 a (-A b+2 a B)}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {(A b-3 a B) x^2}{2 b^4}+\frac {B x^4}{4 b^3}+\frac {a^3 (A b-a B)}{4 b^5 \left (a+b x^2\right )^2}-\frac {a^2 (3 A b-4 a B)}{2 b^5 \left (a+b x^2\right )}-\frac {3 a (A b-2 a B) \log \left (a+b x^2\right )}{2 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 94, normalized size = 0.86 \begin {gather*} \frac {2 b (A b-3 a B) x^2+b^2 B x^4+\frac {a^3 (A b-a B)}{\left (a+b x^2\right )^2}+\frac {2 a^2 (-3 A b+4 a B)}{a+b x^2}+6 a (-A b+2 a B) \log \left (a+b x^2\right )}{4 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 102, normalized size = 0.94
method | result | size |
norman | \(\frac {-\frac {a^{2} \left (9 a b A -18 a^{2} B \right )}{4 b^{5}}+\frac {B \,x^{8}}{4 b}+\frac {\left (A b -2 B a \right ) x^{6}}{2 b^{2}}-\frac {a \left (3 a b A -6 a^{2} B \right ) x^{2}}{b^{4}}}{\left (b \,x^{2}+a \right )^{2}}-\frac {3 a \left (A b -2 B a \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\) | \(101\) |
default | \(\frac {\left (b B \,x^{2}+A b -3 B a \right )^{2}}{4 b^{5} B}-\frac {a \left (\frac {\left (3 A b -6 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2} \left (A b -B a \right )}{2 b \left (b \,x^{2}+a \right )^{2}}+\frac {a \left (3 A b -4 B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{4}}\) | \(102\) |
risch | \(\frac {B \,x^{4}}{4 b^{3}}+\frac {A \,x^{2}}{2 b^{3}}-\frac {3 B a \,x^{2}}{2 b^{4}}+\frac {A^{2}}{4 b^{3} B}-\frac {3 A a}{2 b^{4}}+\frac {9 B \,a^{2}}{4 b^{5}}+\frac {\left (-\frac {3}{2} A \,a^{2} b +2 B \,a^{3}\right ) x^{2}-\frac {a^{3} \left (5 A b -7 B a \right )}{4 b}}{b^{4} \left (b \,x^{2}+a \right )^{2}}-\frac {3 a \ln \left (b \,x^{2}+a \right ) A}{2 b^{4}}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right ) B}{b^{5}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 116, normalized size = 1.06 \begin {gather*} \frac {7 \, B a^{4} - 5 \, A a^{3} b + 2 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}}{4 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac {B b x^{4} - 2 \, {\left (3 \, B a - A b\right )} x^{2}}{4 \, b^{4}} + \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.37, size = 179, normalized size = 1.64 \begin {gather*} \frac {B b^{4} x^{8} - 2 \, {\left (2 \, B a b^{3} - A b^{4}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (2 \, B a^{4} - A a^{3} b + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{4} + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.81, size = 119, normalized size = 1.09 \begin {gather*} \frac {B x^{4}}{4 b^{3}} + \frac {3 a \left (- A b + 2 B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{5}} + x^{2} \left (\frac {A}{2 b^{3}} - \frac {3 B a}{2 b^{4}}\right ) + \frac {- 5 A a^{3} b + 7 B a^{4} + x^{2} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{4 a^{2} b^{5} + 8 a b^{6} x^{2} + 4 b^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 132, normalized size = 1.21 \begin {gather*} \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac {B b^{3} x^{4} - 6 \, B a b^{2} x^{2} + 2 \, A b^{3} x^{2}}{4 \, b^{6}} - \frac {18 \, B a^{2} b^{2} x^{4} - 9 \, A a b^{3} x^{4} + 28 \, B a^{3} b x^{2} - 12 \, A a^{2} b^{2} x^{2} + 11 \, B a^{4} - 4 \, A a^{3} b}{4 \, {\left (b x^{2} + a\right )}^{2} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 118, normalized size = 1.08 \begin {gather*} \frac {\frac {7\,B\,a^4-5\,A\,a^3\,b}{4\,b}+x^2\,\left (2\,B\,a^3-\frac {3\,A\,a^2\,b}{2}\right )}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+x^2\,\left (\frac {A}{2\,b^3}-\frac {3\,B\,a}{2\,b^4}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (6\,B\,a^2-3\,A\,a\,b\right )}{2\,b^5}+\frac {B\,x^4}{4\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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