Optimal. Leaf size=118 \[ -\frac {3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{7/2}}+\frac {3 b (5 A b-4 a B)}{8 a^3 \sqrt {a+b x^2}}+\frac {5 A b-4 a B}{8 a^2 x^2 \sqrt {a+b x^2}}-\frac {A}{4 a x^4 \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ \frac {3 \sqrt {a+b x^2} (5 A b-4 a B)}{8 a^3 x^2}-\frac {5 A b-4 a B}{4 a^2 x^2 \sqrt {a+b x^2}}-\frac {3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {A}{4 a x^4 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {A}{4 a x^4 \sqrt {a+b x^2}}+\frac {\left (-\frac {5 A b}{2}+2 a B\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {A}{4 a x^4 \sqrt {a+b x^2}}-\frac {5 A b-4 a B}{4 a^2 x^2 \sqrt {a+b x^2}}-\frac {(3 (5 A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{8 a^2}\\ &=-\frac {A}{4 a x^4 \sqrt {a+b x^2}}-\frac {5 A b-4 a B}{4 a^2 x^2 \sqrt {a+b x^2}}+\frac {3 (5 A b-4 a B) \sqrt {a+b x^2}}{8 a^3 x^2}+\frac {(3 b (5 A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{16 a^3}\\ &=-\frac {A}{4 a x^4 \sqrt {a+b x^2}}-\frac {5 A b-4 a B}{4 a^2 x^2 \sqrt {a+b x^2}}+\frac {3 (5 A b-4 a B) \sqrt {a+b x^2}}{8 a^3 x^2}+\frac {(3 (5 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^3}\\ &=-\frac {A}{4 a x^4 \sqrt {a+b x^2}}-\frac {5 A b-4 a B}{4 a^2 x^2 \sqrt {a+b x^2}}+\frac {3 (5 A b-4 a B) \sqrt {a+b x^2}}{8 a^3 x^2}-\frac {3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 60, normalized size = 0.51 \[ \frac {b x^4 (5 A b-4 a B) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x^2}{a}+1\right )-a^2 A}{4 a^3 x^4 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 287, normalized size = 2.43 \[ \left [-\frac {3 \, {\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4}\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} + 2 \, A a^{3} + {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{16 \, {\left (a^{4} b x^{6} + a^{5} x^{4}\right )}}, -\frac {3 \, {\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} + 2 \, A a^{3} + {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{8 \, {\left (a^{4} b x^{6} + a^{5} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 137, normalized size = 1.16 \[ -\frac {3 \, {\left (4 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{3}} - \frac {B a b - A b^{2}}{\sqrt {b x^{2} + a} a^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b - 4 \, \sqrt {b x^{2} + a} B a^{2} b - 7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} + 9 \, \sqrt {b x^{2} + a} A a b^{2}}{8 \, a^{3} b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 153, normalized size = 1.30 \[ -\frac {15 A \,b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {7}{2}}}+\frac {3 B b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}}}+\frac {15 A \,b^{2}}{8 \sqrt {b \,x^{2}+a}\, a^{3}}-\frac {3 B b}{2 \sqrt {b \,x^{2}+a}\, a^{2}}+\frac {5 A b}{8 \sqrt {b \,x^{2}+a}\, a^{2} x^{2}}-\frac {B}{2 \sqrt {b \,x^{2}+a}\, a \,x^{2}}-\frac {A}{4 \sqrt {b \,x^{2}+a}\, a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 130, normalized size = 1.10 \[ \frac {3 \, B b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {15 \, A b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {7}{2}}} - \frac {3 \, B b}{2 \, \sqrt {b x^{2} + a} a^{2}} + \frac {15 \, A b^{2}}{8 \, \sqrt {b x^{2} + a} a^{3}} - \frac {B}{2 \, \sqrt {b x^{2} + a} a x^{2}} + \frac {5 \, A b}{8 \, \sqrt {b x^{2} + a} a^{2} x^{2}} - \frac {A}{4 \, \sqrt {b x^{2} + a} a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.91, size = 134, normalized size = 1.14 \[ \frac {15\,A\,b^2}{8\,a^3\,\sqrt {b\,x^2+a}}-\frac {3\,B\,b}{2\,a^2\,\sqrt {b\,x^2+a}}-\frac {15\,A\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{7/2}}-\frac {A}{4\,a\,x^4\,\sqrt {b\,x^2+a}}-\frac {B}{2\,a\,x^2\,\sqrt {b\,x^2+a}}+\frac {3\,B\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}}+\frac {5\,A\,b}{8\,a^2\,x^2\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 84.40, size = 180, normalized size = 1.53 \[ A \left (- \frac {1}{4 a \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {5 \sqrt {b}}{8 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {15 b^{\frac {3}{2}}}{8 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {15 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {7}{2}}}\right ) + B \left (- \frac {1}{2 a \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{2 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {5}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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