Optimal. Leaf size=136 \[ a^{5/2} (-B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {15}{8} a^2 A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {1}{8} a \sqrt {a+b x^2} (8 a B+15 A b x)-\frac {\left (a+b x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac {1}{12} \left (a+b x^2\right )^{3/2} (4 a B+15 A b x) \]
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Rubi [A] time = 0.13, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {813, 815, 844, 217, 206, 266, 63, 208} \begin {gather*} \frac {15}{8} a^2 A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+a^{5/2} (-B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {1}{8} a \sqrt {a+b x^2} (8 a B+15 A b x)-\frac {\left (a+b x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac {1}{12} \left (a+b x^2\right )^{3/2} (4 a B+15 A b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 813
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x^2\right )^{5/2}}{x^2} \, dx &=-\frac {(5 A-B x) \left (a+b x^2\right )^{5/2}}{5 x}-\frac {1}{2} \int \frac {(-2 a B-10 A b x) \left (a+b x^2\right )^{3/2}}{x} \, dx\\ &=\frac {1}{12} (4 a B+15 A b x) \left (a+b x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+b x^2\right )^{5/2}}{5 x}-\frac {\int \frac {\left (-8 a^2 b B-30 a A b^2 x\right ) \sqrt {a+b x^2}}{x} \, dx}{8 b}\\ &=\frac {1}{8} a (8 a B+15 A b x) \sqrt {a+b x^2}+\frac {1}{12} (4 a B+15 A b x) \left (a+b x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+b x^2\right )^{5/2}}{5 x}-\frac {\int \frac {-16 a^3 b^2 B-30 a^2 A b^3 x}{x \sqrt {a+b x^2}} \, dx}{16 b^2}\\ &=\frac {1}{8} a (8 a B+15 A b x) \sqrt {a+b x^2}+\frac {1}{12} (4 a B+15 A b x) \left (a+b x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+b x^2\right )^{5/2}}{5 x}+\frac {1}{8} \left (15 a^2 A b\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx+\left (a^3 B\right ) \int \frac {1}{x \sqrt {a+b x^2}} \, dx\\ &=\frac {1}{8} a (8 a B+15 A b x) \sqrt {a+b x^2}+\frac {1}{12} (4 a B+15 A b x) \left (a+b x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+b x^2\right )^{5/2}}{5 x}+\frac {1}{8} \left (15 a^2 A b\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )+\frac {1}{2} \left (a^3 B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {1}{8} a (8 a B+15 A b x) \sqrt {a+b x^2}+\frac {1}{12} (4 a B+15 A b x) \left (a+b x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+b x^2\right )^{5/2}}{5 x}+\frac {15}{8} a^2 A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {\left (a^3 B\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=\frac {1}{8} a (8 a B+15 A b x) \sqrt {a+b x^2}+\frac {1}{12} (4 a B+15 A b x) \left (a+b x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+b x^2\right )^{5/2}}{5 x}+\frac {15}{8} a^2 A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-a^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.23, size = 117, normalized size = 0.86 \begin {gather*} -a^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {a^3 A \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \sqrt {a+b x^2}}+\frac {1}{15} B \sqrt {a+b x^2} \left (23 a^2+11 a b x^2+3 b^2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 141, normalized size = 1.04 \begin {gather*} 2 a^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {\sqrt {a+b x^2} \left (-120 a^2 A+184 a^2 B x+135 a A b x^2+88 a b B x^3+30 A b^2 x^4+24 b^2 B x^5\right )}{120 x}-\frac {15}{8} a^2 A \sqrt {b} \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 519, normalized size = 3.82 \begin {gather*} \left [\frac {225 \, A a^{2} \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 120 \, B a^{\frac {5}{2}} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (24 \, B b^{2} x^{5} + 30 \, A b^{2} x^{4} + 88 \, B a b x^{3} + 135 \, A a b x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{240 \, x}, -\frac {225 \, A a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 60 \, B a^{\frac {5}{2}} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (24 \, B b^{2} x^{5} + 30 \, A b^{2} x^{4} + 88 \, B a b x^{3} + 135 \, A a b x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{120 \, x}, \frac {240 \, B \sqrt {-a} a^{2} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + 225 \, A a^{2} \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (24 \, B b^{2} x^{5} + 30 \, A b^{2} x^{4} + 88 \, B a b x^{3} + 135 \, A a b x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{240 \, x}, -\frac {225 \, A a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 120 \, B \sqrt {-a} a^{2} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (24 \, B b^{2} x^{5} + 30 \, A b^{2} x^{4} + 88 \, B a b x^{3} + 135 \, A a b x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{120 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 150, normalized size = 1.10 \begin {gather*} \frac {2 \, B a^{3} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {15}{8} \, A a^{2} \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {2 \, A a^{3} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{120} \, {\left (184 \, B a^{2} + {\left (135 \, A a b + 2 \, {\left (44 \, B a b + 3 \, {\left (4 \, B b^{2} x + 5 \, A b^{2}\right )} x\right )} x\right )} x\right )} \sqrt {b x^{2} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 158, normalized size = 1.16 \begin {gather*} \frac {15 A \,a^{2} \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8}-B \,a^{\frac {5}{2}} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {15 \sqrt {b \,x^{2}+a}\, A a b x}{8}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A b x}{4}+\sqrt {b \,x^{2}+a}\, B \,a^{2}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A b x}{a}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B a}{3}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B}{5}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 120, normalized size = 0.88 \begin {gather*} \frac {5}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b x + \frac {15}{8} \, \sqrt {b x^{2} + a} A a b x + \frac {15}{8} \, A a^{2} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - B a^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{5} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a + \sqrt {b x^{2} + a} B a^{2} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.16, size = 104, normalized size = 0.76 \begin {gather*} \frac {B\,{\left (b\,x^2+a\right )}^{5/2}}{5}+B\,a^2\,\sqrt {b\,x^2+a}+\frac {B\,a\,{\left (b\,x^2+a\right )}^{3/2}}{3}-\frac {A\,{\left (b\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{5/2}}+B\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.91, size = 318, normalized size = 2.34 \begin {gather*} - \frac {A a^{\frac {5}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + A a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x^{2}}{a}} - \frac {7 A a^{\frac {3}{2}} b x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 A \sqrt {a} b^{2} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 A a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8} + \frac {A b^{3} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - B a^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a^{3}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B a^{2} \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + 2 B a b \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) + B b^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + b x^{2}}}{15 b^{2}} + \frac {a x^{2} \sqrt {a + b x^{2}}}{15 b} + \frac {x^{4} \sqrt {a + b x^{2}}}{5} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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