Optimal. Leaf size=104 \[ \frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}-\frac {8 A \sqrt {a+b x^2}}{3 a^3 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {837, 821, 272,
65, 214} \begin {gather*} -\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {8 A \sqrt {a+b x^2}}{3 a^3 x}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}+\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 837
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \left (a+b x^2\right )^{5/2}} \, dx &=\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-4 a A b-3 a b B x}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{3 a^2 b}\\ &=\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}+\frac {\int \frac {8 a^2 A b^2+3 a^2 b^2 B x}{x^2 \sqrt {a+b x^2}} \, dx}{3 a^4 b^2}\\ &=\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}-\frac {8 A \sqrt {a+b x^2}}{3 a^3 x}+\frac {B \int \frac {1}{x \sqrt {a+b x^2}} \, dx}{a^2}\\ &=\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}-\frac {8 A \sqrt {a+b x^2}}{3 a^3 x}+\frac {B \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}-\frac {8 A \sqrt {a+b x^2}}{3 a^3 x}+\frac {B \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a^2 b}\\ &=\frac {A+B x}{3 a x \left (a+b x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+b x^2}}-\frac {8 A \sqrt {a+b x^2}}{3 a^3 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 94, normalized size = 0.90 \begin {gather*} \frac {-8 A b^2 x^4+3 a b x^2 (-4 A+B x)+a^2 (-3 A+4 B x)}{3 a^3 x \left (a+b x^2\right )^{3/2}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 122, normalized size = 1.17
method | result | size |
default | \(B \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )+A \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )\) | \(122\) |
risch | \(-\frac {A \sqrt {b \,x^{2}+a}}{a^{3} x}+\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{2} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} b \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {5 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, A}{6 a^{3} \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {7 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {5 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, A}{6 a^{3} \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {7 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) B}{a^{\frac {5}{2}}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, A}{12 a^{2} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\, B}{12 a^{2} b \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}\) | \(563\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 100, normalized size = 0.96 \begin {gather*} -\frac {8 \, A b x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {4 \, A b x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a^{2}} + \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.19, size = 264, normalized size = 2.54 \begin {gather*} \left [\frac {3 \, {\left (B b^{2} x^{5} + 2 \, B a b x^{3} + B a^{2} x\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (8 \, A b^{2} x^{4} - 3 \, B a b x^{3} + 12 \, A a b x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}, \frac {3 \, {\left (B b^{2} x^{5} + 2 \, B a b x^{3} + B a^{2} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, A b^{2} x^{4} - 3 \, B a b x^{3} + 12 \, A a b x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}\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. 910 vs.
\(2 (88) = 176\).
time = 8.49, size = 910, normalized size = 8.75 \begin {gather*} A \left (- \frac {3 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {12 a b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {8 b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}}\right ) + B \left (\frac {8 a^{7} \sqrt {1 + \frac {b x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {3 a^{7} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {14 a^{6} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {9 a^{6} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {18 a^{6} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {6 a^{5} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {9 a^{5} b^{2} x^{4} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {18 a^{5} b^{2} x^{4} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} + \frac {3 a^{4} b^{3} x^{6} \log {\left (\frac {b x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}} - \frac {6 a^{4} b^{3} x^{6} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} b x^{2} + 18 a^{\frac {15}{2}} b^{2} x^{4} + 6 a^{\frac {13}{2}} b^{3} x^{6}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 119, normalized size = 1.14 \begin {gather*} -\frac {{\left ({\left (\frac {5 \, A b^{2} x}{a^{3}} - \frac {3 \, B b}{a^{2}}\right )} x + \frac {6 \, A b}{a^{2}}\right )} x - \frac {4 \, B}{a}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.58, size = 96, normalized size = 0.92 \begin {gather*} \frac {\frac {B}{3\,a}+\frac {B\,\left (b\,x^2+a\right )}{a^2}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A\,a^2-8\,A\,{\left (b\,x^2+a\right )}^2+4\,A\,a\,\left (b\,x^2+a\right )}{3\,a^3\,x\,{\left (b\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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