Optimal. Leaf size=68 \[ -\frac {1}{b \sqrt {a+b x^2}}-\frac {1}{3 b \left (a+b x^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
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Rubi [A] time = 0.51, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6, 6715, 897, 1261, 207} \[ -\frac {1}{b \sqrt {a+b x^2}}-\frac {1}{3 b \left (a+b x^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
Antiderivative was successfully verified.
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Rule 6
Rule 207
Rule 897
Rule 1261
Rule 6715
Rubi steps
\begin {align*} \int \frac {x \left (1+a+a^2+x^2+a x^2+b x^2+2 a b x^2+b x^4+b^2 x^4\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx &=\int \frac {x \left (1+a+a^2+(1+a) x^2+b x^2+2 a b x^2+b x^4+b^2 x^4\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx\\ &=\int \frac {x \left (1+a+a^2+2 a b x^2+(1+a+b) x^2+b x^4+b^2 x^4\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx\\ &=\int \frac {x \left (1+a+a^2+(1+a+b+2 a b) x^2+b x^4+b^2 x^4\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx\\ &=\int \frac {x \left (1+a+a^2+(1+a+b+2 a b) x^2+\left (b+b^2\right ) x^4\right )}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+a+a^2+(1+a+b+2 a b) x+\left (b+b^2\right ) x^2}{(1+x) (a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\frac {\left (1+a+a^2\right ) b^2-a b (1+a+b+2 a b)+a^2 \left (b+b^2\right )}{b^2}-\frac {\left (-b (1+a+b+2 a b)+2 a \left (b+b^2\right )\right ) x^2}{b^2}+\frac {\left (b+b^2\right ) x^4}{b^2}}{x^4 \left (\frac {-a+b}{b}+\frac {x^2}{b}\right )} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^4}+\frac {1}{x^2}+\frac {b}{-a+b+x^2}\right ) \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=-\frac {1}{3 b \left (a+b x^2\right )^{3/2}}-\frac {1}{b \sqrt {a+b x^2}}+\operatorname {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b x^2}\right )\\ &=-\frac {1}{3 b \left (a+b x^2\right )^{3/2}}-\frac {1}{b \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 63, normalized size = 0.93 \[ \frac {-3 a-3 b x^2-1}{3 b \left (a+b x^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 382, normalized size = 5.62 \[ \left [\frac {3 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sqrt {a - b} \log \left (\frac {b^{2} x^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} x^{2} - 4 \, {\left (b x^{2} + 2 \, a - b\right )} \sqrt {b x^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{x^{4} + 2 \, x^{2} + 1}\right ) - 4 \, {\left (3 \, {\left (a b - b^{2}\right )} x^{2} + 3 \, a^{2} - {\left (3 \, a + 1\right )} b + a\right )} \sqrt {b x^{2} + a}}{12 \, {\left ({\left (a b^{3} - b^{4}\right )} x^{4} + a^{3} b - a^{2} b^{2} + 2 \, {\left (a^{2} b^{2} - a b^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sqrt {-a + b} \arctan \left (-\frac {{\left (b x^{2} + 2 \, a - b\right )} \sqrt {b x^{2} + a} \sqrt {-a + b}}{2 \, {\left ({\left (a b - b^{2}\right )} x^{2} + a^{2} - a b\right )}}\right ) + 2 \, {\left (3 \, {\left (a b - b^{2}\right )} x^{2} + 3 \, a^{2} - {\left (3 \, a + 1\right )} b + a\right )} \sqrt {b x^{2} + a}}{6 \, {\left ({\left (a b^{3} - b^{4}\right )} x^{4} + a^{3} b - a^{2} b^{2} + 2 \, {\left (a^{2} b^{2} - a b^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 52, normalized size = 0.76 \[ \frac {\arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a + b}}\right )}{\sqrt {-a + b}} - \frac {3 \, b x^{2} + 3 \, a + 1}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 314, normalized size = 4.62 \[ \frac {a^{2} \arctan \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {-a +b}}\right )}{\left (a -b \right )^{2} \sqrt {-a +b}}-\frac {2 a b \arctan \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {-a +b}}\right )}{\left (a -b \right )^{2} \sqrt {-a +b}}+\frac {b^{2} \arctan \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {-a +b}}\right )}{\left (a -b \right )^{2} \sqrt {-a +b}}-\frac {b \,x^{2}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a^{2}}{\left (a -b \right )^{2} \sqrt {b \,x^{2}+a}}+\frac {a^{2}}{3 \left (a -b \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a b}{\left (a -b \right )^{2} \sqrt {b \,x^{2}+a}}-\frac {2 a b}{3 \left (a -b \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {b^{2}}{\left (a -b \right )^{2} \sqrt {b \,x^{2}+a}}+\frac {b^{2}}{3 \left (a -b \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {b}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {a}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} b}-\frac {1}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 50, normalized size = 0.74 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {b\,x^2+a+\frac {1}{3}}{b\,{\left (b\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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