Optimal. Leaf size=143 \[ -\frac {\left (a+b x^2\right )^{5/2} (4 a B+3 A b)}{8 a x^2}+\frac {5 b \left (a+b x^2\right )^{3/2} (4 a B+3 A b)}{24 a}+\frac {5}{8} b \sqrt {a+b x^2} (4 a B+3 A b)-\frac {5}{8} \sqrt {a} b (4 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4} \]
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Rubi [A] time = 0.10, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {446, 78, 47, 50, 63, 208} \begin {gather*} -\frac {\left (a+b x^2\right )^{5/2} (4 a B+3 A b)}{8 a x^2}+\frac {5 b \left (a+b x^2\right )^{3/2} (4 a B+3 A b)}{24 a}+\frac {5}{8} b \sqrt {a+b x^2} (4 a B+3 A b)-\frac {5}{8} \sqrt {a} b (4 a B+3 A b) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2} (A+B x)}{x^3} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4}+\frac {\left (\frac {3 A b}{2}+2 a B\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^2} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {(3 A b+4 a B) \left (a+b x^2\right )^{5/2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4}+\frac {(5 b (3 A b+4 a B)) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )}{16 a}\\ &=\frac {5 b (3 A b+4 a B) \left (a+b x^2\right )^{3/2}}{24 a}-\frac {(3 A b+4 a B) \left (a+b x^2\right )^{5/2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4}+\frac {1}{16} (5 b (3 A b+4 a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac {5}{8} b (3 A b+4 a B) \sqrt {a+b x^2}+\frac {5 b (3 A b+4 a B) \left (a+b x^2\right )^{3/2}}{24 a}-\frac {(3 A b+4 a B) \left (a+b x^2\right )^{5/2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4}+\frac {1}{16} (5 a b (3 A b+4 a B)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {5}{8} b (3 A b+4 a B) \sqrt {a+b x^2}+\frac {5 b (3 A b+4 a B) \left (a+b x^2\right )^{3/2}}{24 a}-\frac {(3 A b+4 a B) \left (a+b x^2\right )^{5/2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4}+\frac {1}{8} (5 a (3 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 )\\ &=\frac {5}{8} b (3 A b+4 a B) \sqrt {a+b x^2}+\frac {5 b (3 A b+4 a B) \left (a+b x^2\right )^{3/2}}{24 a}-\frac {(3 A b+4 a B) \left (a+b x^2\right )^{5/2}}{8 a x^2}-\frac {A \left (a+b x^2\right )^{7/2}}{4 a x^4}-\frac {5}{8} \sqrt {a} b (3 A b+4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 60, normalized size = 0.42 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (b x^4 (4 a B+3 A b) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {b x^2}{a}+1\right )-7 a^2 A\right )}{28 a^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 112, normalized size = 0.78 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-6 a^2 A-12 a^2 B x^2-27 a A b x^2+56 a b B x^4+24 A b^2 x^4+8 b^2 B x^6\right )}{24 x^4}-\frac {5}{8} \left (4 a^{3/2} b B+3 \sqrt {a} A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 221, normalized size = 1.55 \begin {gather*} \left [\frac {15 \, {\left (4 \, B a b + 3 \, 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 (8 \, B b^{2} x^{6} + 8 \, {\left (7 \, B a b + 3 \, A b^{2}\right )} x^{4} - 6 \, A a^{2} - 3 \, {\left (4 \, B a^{2} + 9 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, x^{4}}, \frac {15 \, {\left (4 \, B a b + 3 \, A b^{2}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (8 \, B b^{2} x^{6} + 8 \, {\left (7 \, B a b + 3 \, A b^{2}\right )} x^{4} - 6 \, A a^{2} - 3 \, {\left (4 \, B a^{2} + 9 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{24 \, x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 171, normalized size = 1.20 \begin {gather*} \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2} + 48 \, \sqrt {b x^{2} + a} B a b^{2} + 24 \, \sqrt {b x^{2} + a} A b^{3} + \frac {15 \, {\left (4 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3 \, {\left (4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{2} - 4 \, \sqrt {b x^{2} + a} B a^{3} b^{2} + 9 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{3} - 7 \, \sqrt {b x^{2} + a} A a^{2} b^{3}\right )}}{b^{2} x^{4}}}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 213, normalized size = 1.49 \begin {gather*} -\frac {15 A \sqrt {a}\, b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8}-\frac {5 B \,a^{\frac {3}{2}} b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}+\frac {15 \sqrt {b \,x^{2}+a}\, A \,b^{2}}{8}+\frac {5 \sqrt {b \,x^{2}+a}\, B a b}{2}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{2}}{8 a}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B b}{6}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,b^{2}}{8 a^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B b}{2 a}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} A b}{8 a^{2} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B}{2 a \,x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 190, normalized size = 1.33 \begin {gather*} -\frac {5}{2} \, B a^{\frac {3}{2}} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {15}{8} \, A \sqrt {a} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {5}{6} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{2 \, a} + \frac {5}{2} \, \sqrt {b x^{2} + a} B a b + \frac {15}{8} \, \sqrt {b x^{2} + a} A b^{2} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{8 \, a^{2}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{2 \, a x^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 144, normalized size = 1.01 \begin {gather*} A\,b^2\,\sqrt {b\,x^2+a}+\frac {B\,b\,{\left (b\,x^2+a\right )}^{3/2}}{3}+2\,B\,a\,b\,\sqrt {b\,x^2+a}+\frac {A\,\sqrt {a}\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,15{}\mathrm {i}}{8}-\frac {9\,A\,a\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4}+\frac {7\,A\,a^2\,\sqrt {b\,x^2+a}}{8\,x^4}-\frac {B\,a^2\,\sqrt {b\,x^2+a}}{2\,x^2}+\frac {B\,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 = 165.19, size = 279, normalized size = 1.95 \begin {gather*} - \frac {15 A \sqrt {a} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8} - \frac {A a^{3}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A a^{2} \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x} + \frac {7 A a b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {5}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 B a^{\frac {3}{2}} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {B a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {2 B a^{2} \sqrt {b}}{x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {2 B a b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + B 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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