Optimal. Leaf size=120 \[ -\frac {b^2 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{5/2}}+\frac {b \sqrt {a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac {\sqrt {a+b x^2} (A b-2 a B)}{8 a x^4}-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]
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Rubi [A] time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {446, 78, 47, 51, 63, 208} \[ -\frac {b^2 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{5/2}}+\frac {b \sqrt {a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac {\sqrt {a+b x^2} (A b-2 a B)}{8 a x^4}-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
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^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6}+\frac {\left (-\frac {3 A b}{2}+3 a B\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,x^2\right )}{6 a}\\ &=\frac {(A b-2 a B) \sqrt {a+b x^2}}{8 a x^4}-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6}-\frac {(b (A b-2 a B)) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac {(A b-2 a B) \sqrt {a+b x^2}}{8 a x^4}+\frac {b (A b-2 a B) \sqrt {a+b x^2}}{16 a^2 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6}+\frac {\left (b^2 (A b-2 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{32 a^2}\\ &=\frac {(A b-2 a B) \sqrt {a+b x^2}}{8 a x^4}+\frac {b (A b-2 a B) \sqrt {a+b x^2}}{16 a^2 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6}+\frac {(b (A b-2 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{16 a^2}\\ &=\frac {(A b-2 a B) \sqrt {a+b x^2}}{8 a x^4}+\frac {b (A b-2 a B) \sqrt {a+b x^2}}{16 a^2 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{6 a x^6}-\frac {b^2 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.51 \[ -\frac {\left (a+b x^2\right )^{3/2} \left (a^3 A+b^2 x^6 (2 a B-A b) \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b x^2}{a}+1\right )\right )}{6 a^4 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 221, normalized size = 1.84 \[ \left [-\frac {3 \, {\left (2 \, B a b^{2} - A b^{3}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a^{3} x^{6}}, -\frac {3 \, {\left (2 \, B a b^{2} - A b^{3}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a^{3} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 140, normalized size = 1.17 \[ -\frac {\frac {3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{3} - 6 \, \sqrt {b x^{2} + a} B a^{3} b^{3} - 3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4} + 8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{4} + 3 \, \sqrt {b x^{2} + a} A a^{2} b^{4}}{a^{2} b^{3} x^{6}}}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 197, normalized size = 1.64 \[ -\frac {A \,b^{3} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {5}{2}}}+\frac {B \,b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {3}{2}}}+\frac {\sqrt {b \,x^{2}+a}\, A \,b^{3}}{16 a^{3}}-\frac {\sqrt {b \,x^{2}+a}\, B \,b^{2}}{8 a^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{2}}{16 a^{3} x^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B b}{8 a^{2} x^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A b}{8 a^{2} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B}{4 a \,x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A}{6 a \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 174, normalized size = 1.45 \[ \frac {B b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {5}{2}}} - \frac {\sqrt {b x^{2} + a} B b^{2}}{8 \, a^{2}} + \frac {\sqrt {b x^{2} + a} A b^{3}}{16 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{16 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{4 \, a x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{8 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{6 \, a x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 134, normalized size = 1.12 \[ \frac {B\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{3/2}}-\frac {B\,\sqrt {b\,x^2+a}}{8\,x^4}-\frac {A\,\sqrt {b\,x^2+a}}{16\,x^6}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}}{6\,a\,x^6}+\frac {A\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a^2\,x^6}-\frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{8\,a\,x^4}+\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{16\,a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 144.35, size = 226, normalized size = 1.88 \[ - \frac {A a}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {3}{2}}}{48 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {5}{2}}}{16 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {5}{2}}} - \frac {B a}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {3}{2}}}{8 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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