Optimal. Leaf size=141 \[ -\frac {5}{2} a^{3/2} A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {15}{8} a^2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\left (a+b x^2\right )^{5/2} (2 A-B x)}{4 x^2}-\frac {5 \left (a+b x^2\right )^{3/2} (3 a B-2 A b x)}{12 x}+\frac {5}{8} a b \sqrt {a+b x^2} (4 A+3 B x) \]
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Rubi [A] time = 0.12, antiderivative size = 141, 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 {5}{2} a^{3/2} A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {15}{8} a^2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\left (a+b x^2\right )^{5/2} (2 A-B x)}{4 x^2}-\frac {5 \left (a+b x^2\right )^{3/2} (3 a B-2 A b x)}{12 x}+\frac {5}{8} a b \sqrt {a+b x^2} (4 A+3 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^3} \, dx &=-\frac {(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}-\frac {5}{16} \int \frac {(-4 a B-8 A b x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac {5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac {(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac {5}{32} \int \frac {(16 a A b+24 a b B x) \sqrt {a+b x^2}}{x} \, dx\\ &=\frac {5}{8} a b (4 A+3 B x) \sqrt {a+b x^2}-\frac {5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac {(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac {5 \int \frac {32 a^2 A b^2+24 a^2 b^2 B x}{x \sqrt {a+b x^2}} \, dx}{64 b}\\ &=\frac {5}{8} a b (4 A+3 B x) \sqrt {a+b x^2}-\frac {5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac {(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac {1}{2} \left (5 a^2 A b\right ) \int \frac {1}{x \sqrt {a+b x^2}} \, dx+\frac {1}{8} \left (15 a^2 b B\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {5}{8} a b (4 A+3 B x) \sqrt {a+b x^2}-\frac {5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac {(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac {1}{4} \left (5 a^2 A b\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )+\frac {1}{8} \left (15 a^2 b B\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {5}{8} a b (4 A+3 B x) \sqrt {a+b x^2}-\frac {5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac {(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac {15}{8} a^2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {1}{2} \left (5 a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=\frac {5}{8} a b (4 A+3 B x) \sqrt {a+b x^2}-\frac {5 (3 a B-2 A b x) \left (a+b x^2\right )^{3/2}}{12 x}-\frac {(2 A-B x) \left (a+b x^2\right )^{5/2}}{4 x^2}+\frac {15}{8} a^2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {5}{2} a^{3/2} A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 92, normalized size = 0.65 \begin {gather*} \frac {A b \left (a+b x^2\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {b x^2}{a}+1\right )}{7 a^2}-\frac {a^2 B \sqrt {a+b x^2} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \sqrt {\frac {b x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.51, size = 142, normalized size = 1.01 \begin {gather*} 5 a^{3/2} A 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 (-12 a^2 A-24 a^2 B x+56 a A b x^2+27 a b B x^3+8 A b^2 x^4+6 b^2 B x^5\right )}{24 x^2}-\frac {15}{8} a^2 \sqrt {b} 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 = 0.58, size = 535, normalized size = 3.79 \begin {gather*} \left [\frac {45 \, B a^{2} \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 60 \, A a^{\frac {3}{2}} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{48 \, x^{2}}, -\frac {45 \, B a^{2} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 30 \, A a^{\frac {3}{2}} b x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{24 \, x^{2}}, \frac {120 \, A \sqrt {-a} a b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + 45 \, B a^{2} \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{48 \, x^{2}}, -\frac {45 \, B a^{2} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 60 \, A \sqrt {-a} a b x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, B b^{2} x^{5} + 8 \, A b^{2} x^{4} + 27 \, B a b x^{3} + 56 \, A a b x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt {b x^{2} + a}}{24 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 219, normalized size = 1.55 \begin {gather*} \frac {5 \, A a^{2} b \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {15}{8} \, B a^{2} \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {1}{24} \, {\left (56 \, A a b + {\left (27 \, B a b + 2 \, {\left (3 \, B b^{2} x + 4 \, A b^{2}\right )} x\right )} x\right )} \sqrt {b x^{2} + a} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a^{2} b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{3} b - 2 \, B a^{4} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 181, normalized size = 1.28 \begin {gather*} -\frac {5 A \,a^{\frac {3}{2}} b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}+\frac {15 B \,a^{2} \sqrt {b}\, \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8}+\frac {15 \sqrt {b \,x^{2}+a}\, B a b x}{8}+\frac {5 \sqrt {b \,x^{2}+a}\, A a b}{2}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B b x}{4}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A b}{6}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B b x}{a}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A b}{2 a}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B}{a x}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 143, normalized size = 1.01 \begin {gather*} \frac {5}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b x + \frac {15}{8} \, \sqrt {b x^{2} + a} B a b x + \frac {15}{8} \, B a^{2} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {5}{2} \, A a^{\frac {3}{2}} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {5}{6} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{2 \, a} + \frac {5}{2} \, \sqrt {b x^{2} + a} A a b - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{x} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 111, normalized size = 0.79 \begin {gather*} \frac {A\,b\,{\left (b\,x^2+a\right )}^{3/2}}{3}+2\,A\,a\,b\,\sqrt {b\,x^2+a}-\frac {A\,a^2\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {B\,{\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}}+\frac {A\,a^{3/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.98, size = 279, normalized size = 1.98 \begin {gather*} - \frac {5 A a^{\frac {3}{2}} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {A a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {2 A a^{2} \sqrt {b}}{x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {2 A a b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + A b^{2} \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 ) - \frac {B a^{\frac {5}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + B a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x^{2}}{a}} - \frac {7 B a^{\frac {3}{2}} b x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B \sqrt {a} b^{2} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 B a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8} + \frac {B b^{3} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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