Optimal. Leaf size=108 \[ -\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}+b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {b^2 B \sqrt {a+b x^2}}{x}-\frac {B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b B \left (a+b x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {451, 277, 217, 206} \[ -\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}-\frac {b^2 B \sqrt {a+b x^2}}{x}+b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b B \left (a+b x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 277
Rule 451
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^8} \, dx &=-\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}+B \int \frac {\left (a+b x^2\right )^{5/2}}{x^6} \, dx\\ &=-\frac {B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}+(b B) \int \frac {\left (a+b x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {b B \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}+\left (b^2 B\right ) \int \frac {\sqrt {a+b x^2}}{x^2} \, dx\\ &=-\frac {b^2 B \sqrt {a+b x^2}}{x}-\frac {b B \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}+\left (b^3 B\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=-\frac {b^2 B \sqrt {a+b x^2}}{x}-\frac {b B \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}+\left (b^3 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=-\frac {b^2 B \sqrt {a+b x^2}}{x}-\frac {b B \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7}+b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.09, size = 78, normalized size = 0.72 \[ -\frac {a^2 B \sqrt {a+b x^2} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};-\frac {b x^2}{a}\right )}{5 x^5 \sqrt {\frac {b x^2}{a}+1}}-\frac {A \left (a+b x^2\right )^{7/2}}{7 a x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 234, normalized size = 2.17 \[ \left [\frac {105 \, B a b^{\frac {5}{2}} x^{7} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left ({\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{6} + {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, a x^{7}}, -\frac {105 \, B a \sqrt {-b} b^{2} x^{7} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left ({\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{6} + {\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 3 \, {\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, a x^{7}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 320, normalized size = 2.96 \[ -\frac {1}{2} \, B b^{\frac {5}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {5}{2}} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {7}{2}} - 1260 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {5}{2}} + 2555 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {5}{2}} + 525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {7}{2}} - 3080 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {5}{2}} + 2121 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {5}{2}} + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {7}{2}} - 812 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {5}{2}} + 161 \, B a^{7} b^{\frac {5}{2}} + 15 \, A a^{6} b^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 155, normalized size = 1.44 \[ B \,b^{\frac {5}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {\sqrt {b \,x^{2}+a}\, B \,b^{3} x}{a}+\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,b^{3} x}{3 a^{2}}+\frac {8 \left (b \,x^{2}+a \right )^{\frac {5}{2}} B \,b^{3} x}{15 a^{3}}-\frac {8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} B \,b^{2}}{15 a^{3} x}-\frac {2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} B b}{15 a^{2} x^{3}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B}{5 a \,x^{5}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A}{7 a \,x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 128, normalized size = 1.19 \[ \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3} x}{3 \, a^{2}} + \frac {\sqrt {b x^{2} + a} B b^{3} x}{a} + B b^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{15 \, a^{2} x} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b}{15 \, a^{2} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{7 \, a x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{5/2}}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 14.27, size = 592, normalized size = 5.48 \[ - \frac {15 A a^{7} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {33 A a^{6} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {17 A a^{5} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {3 A a^{4} b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {12 A a^{3} b^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {8 A a^{2} b^{\frac {19}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {2 A a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {7 A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 x^{2}} - \frac {A b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a} - \frac {B \sqrt {a} b^{2}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {11 B a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 x^{2}} - \frac {8 B b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15} + B b^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {B b^{3} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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