Optimal. Leaf size=88 \[ \frac {b (A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {\sqrt {a+b x^2} (A b-4 a B)}{8 a x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{4 a x^4} \]
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Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 47, 63, 208} \[ \frac {b (A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {\sqrt {a+b x^2} (A b-4 a B)}{8 a x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (A+B x)}{x^3} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{4 a x^4}+\frac {\left (-\frac {A b}{2}+2 a B\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac {(A b-4 a B) \sqrt {a+b x^2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{4 a x^4}-\frac {(b (A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac {(A b-4 a B) \sqrt {a+b x^2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{4 a x^4}-\frac {(A b-4 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{8 a}\\ &=\frac {(A b-4 a B) \sqrt {a+b x^2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{4 a x^4}+\frac {b (A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 93, normalized size = 1.06 \[ \frac {-\left (a+b x^2\right ) \left (2 a \left (A+2 B x^2\right )+A b x^2\right )-b x^4 \sqrt {\frac {b x^2}{a}+1} (4 a B-A b) \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{8 a x^4 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 170, normalized size = 1.93 \[ \left [-\frac {{\left (4 \, B a b - A b^{2}\right )} \sqrt {a} x^{4} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, A a^{2} + {\left (4 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{16 \, a^{2} x^{4}}, \frac {{\left (4 \, B a b - A b^{2}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, A a^{2} + {\left (4 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{8 \, a^{2} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 120, normalized size = 1.36 \[ \frac {\frac {{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b^{2} - 4 \, \sqrt {b x^{2} + a} B a^{2} b^{2} + {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3} + \sqrt {b x^{2} + a} A a b^{3}}{a b^{2} x^{4}}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 153, normalized size = 1.74 \[ \frac {A \,b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {3}{2}}}-\frac {B b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 \sqrt {a}}-\frac {\sqrt {b \,x^{2}+a}\, A \,b^{2}}{8 a^{2}}+\frac {\sqrt {b \,x^{2}+a}\, B b}{2 a}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A b}{8 a^{2} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B}{2 a \,x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A}{4 a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 130, normalized size = 1.48 \[ -\frac {B b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \frac {A b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} + \frac {\sqrt {b x^{2} + a} B b}{2 \, a} - \frac {\sqrt {b x^{2} + a} A b^{2}}{8 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{2 \, a x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{4 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.70, size = 93, normalized size = 1.06 \[ \frac {A\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{3/2}}-\frac {B\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {A\,\sqrt {b\,x^2+a}}{8\,x^4}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}}{8\,a\,x^4}-\frac {B\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 143.85, size = 144, normalized size = 1.64 \[ - \frac {A a}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {3}{2}}}{8 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {3}{2}}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {B b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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