Optimal. Leaf size=189 \[ -\frac {b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{5/2}}+\frac {b^3 \sqrt {a+b x^2} (3 A b-10 a B)}{256 a^2 x^2}+\frac {b^2 \sqrt {a+b x^2} (3 A b-10 a B)}{128 a x^4}+\frac {\left (a+b x^2\right )^{5/2} (3 A b-10 a B)}{80 a x^8}+\frac {b \left (a+b x^2\right )^{3/2} (3 A b-10 a B)}{96 a x^6}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}} \]
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Rubi [A] time = 0.15, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \begin {gather*} \frac {b^3 \sqrt {a+b x^2} (3 A b-10 a B)}{256 a^2 x^2}-\frac {b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{5/2}}+\frac {b^2 \sqrt {a+b x^2} (3 A b-10 a B)}{128 a x^4}+\frac {b \left (a+b x^2\right )^{3/2} (3 A b-10 a B)}{96 a x^6}+\frac {\left (a+b x^2\right )^{5/2} (3 A b-10 a B)}{80 a x^8}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}} \end {gather*}
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 {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2} (A+B x)}{x^6} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}+\frac {\left (-\frac {3 A b}{2}+5 a B\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^5} \, dx,x,x^2\right )}{10 a}\\ &=\frac {(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac {(b (3 A b-10 a B)) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^4} \, dx,x,x^2\right )}{32 a}\\ &=\frac {b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac {(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac {\left (b^2 (3 A b-10 a B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,x^2\right )}{64 a}\\ &=\frac {b^2 (3 A b-10 a B) \sqrt {a+b x^2}}{128 a x^4}+\frac {b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac {(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac {\left (b^3 (3 A b-10 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{256 a}\\ &=\frac {b^2 (3 A b-10 a B) \sqrt {a+b x^2}}{128 a x^4}+\frac {b^3 (3 A b-10 a B) \sqrt {a+b x^2}}{256 a^2 x^2}+\frac {b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac {(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}+\frac {\left (b^4 (3 A b-10 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{512 a^2}\\ &=\frac {b^2 (3 A b-10 a B) \sqrt {a+b x^2}}{128 a x^4}+\frac {b^3 (3 A b-10 a B) \sqrt {a+b x^2}}{256 a^2 x^2}+\frac {b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac {(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}+\frac {\left (b^3 (3 A b-10 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{256 a^2}\\ &=\frac {b^2 (3 A b-10 a B) \sqrt {a+b x^2}}{128 a x^4}+\frac {b^3 (3 A b-10 a B) \sqrt {a+b x^2}}{256 a^2 x^2}+\frac {b (3 A b-10 a B) \left (a+b x^2\right )^{3/2}}{96 a x^6}+\frac {(3 A b-10 a B) \left (a+b x^2\right )^{5/2}}{80 a x^8}-\frac {A \left (a+b x^2\right )^{7/2}}{10 a x^{10}}-\frac {b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 62, normalized size = 0.33 \begin {gather*} -\frac {\left (a+b x^2\right )^{7/2} \left (7 a^5 A+b^4 x^{10} (10 a B-3 A b) \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {b x^2}{a}+1\right )\right )}{70 a^6 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 152, normalized size = 0.80 \begin {gather*} \frac {\left (10 a b^4 B-3 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{5/2}}+\frac {\sqrt {a+b x^2} \left (-384 a^4 A-480 a^4 B x^2-1008 a^3 A b x^2-1360 a^3 b B x^4-744 a^2 A b^2 x^4-1180 a^2 b^2 B x^6-30 a A b^3 x^6-150 a b^3 B x^8+45 A b^4 x^8\right )}{3840 a^2 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 319, normalized size = 1.69 \begin {gather*} \left [-\frac {15 \, {\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} \sqrt {a} x^{10} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (15 \, {\left (10 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{8} + 10 \, {\left (118 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \, {\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2}\right )} x^{4} + 48 \, {\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{7680 \, a^{3} x^{10}}, -\frac {15 \, {\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, {\left (10 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{8} + 10 \, {\left (118 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{6} + 384 \, A a^{5} + 8 \, {\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2}\right )} x^{4} + 48 \, {\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3840 \, a^{3} x^{10}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 230, normalized size = 1.22 \begin {gather*} -\frac {\frac {15 \, {\left (10 \, B a b^{5} - 3 \, A b^{6}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {150 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a b^{5} + 580 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} b^{5} - 1280 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3} b^{5} + 700 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{4} b^{5} - 150 \, \sqrt {b x^{2} + a} B a^{5} b^{5} - 45 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A b^{6} + 210 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a b^{6} + 384 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2} b^{6} - 210 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{3} b^{6} + 45 \, \sqrt {b x^{2} + a} A a^{4} b^{6}}{a^{2} b^{5} x^{10}}}{3840 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 353, normalized size = 1.87 \begin {gather*} -\frac {3 A \,b^{5} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{256 a^{\frac {5}{2}}}+\frac {5 B \,b^{4} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {3}{2}}}+\frac {3 \sqrt {b \,x^{2}+a}\, A \,b^{5}}{256 a^{3}}-\frac {5 \sqrt {b \,x^{2}+a}\, B \,b^{4}}{128 a^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{5}}{256 a^{4}}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,b^{4}}{384 a^{3}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,b^{5}}{1280 a^{5}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B \,b^{4}}{128 a^{4}}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} A \,b^{4}}{1280 a^{5} x^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B \,b^{3}}{128 a^{4} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A \,b^{3}}{640 a^{4} x^{4}}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B \,b^{2}}{192 a^{3} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A \,b^{2}}{160 a^{3} x^{6}}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B b}{48 a^{2} x^{6}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} A b}{80 a^{2} x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B}{8 a \,x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A}{10 a \,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.27, size = 330, normalized size = 1.75 \begin {gather*} \frac {5 \, B b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {3}{2}}} - \frac {3 \, A b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, a^{\frac {5}{2}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{4}}{128 \, a^{4}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{4}}{384 \, a^{3}} - \frac {5 \, \sqrt {b x^{2} + a} B b^{4}}{128 \, a^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{5}}{1280 \, a^{5}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{5}}{256 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} A b^{5}}{256 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B b^{3}}{128 \, a^{4} x^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{4}}{1280 \, a^{5} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B b^{2}}{192 \, a^{3} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{3}}{640 \, a^{4} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B b}{48 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{2}}{160 \, a^{3} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{8 \, a x^{8}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{80 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{10 \, a x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.06, size = 205, normalized size = 1.08 \begin {gather*} \frac {7\,A\,a\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^{10}}-\frac {73\,B\,{\left (b\,x^2+a\right )}^{5/2}}{384\,x^8}-\frac {A\,{\left (b\,x^2+a\right )}^{5/2}}{10\,x^{10}}+\frac {55\,B\,a\,{\left (b\,x^2+a\right )}^{3/2}}{384\,x^8}-\frac {3\,A\,a^2\,\sqrt {b\,x^2+a}}{256\,x^{10}}-\frac {7\,A\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a\,x^{10}}+\frac {3\,A\,{\left (b\,x^2+a\right )}^{9/2}}{256\,a^2\,x^{10}}-\frac {5\,B\,a^2\,\sqrt {b\,x^2+a}}{128\,x^8}-\frac {5\,B\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a\,x^8}+\frac {A\,b^5\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,3{}\mathrm {i}}{256\,a^{5/2}}-\frac {B\,b^4\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{128\,a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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