Optimal. Leaf size=184 \[ \frac {3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{7/2}}-\frac {3 b^3 \sqrt {a+b x^2} (A b-2 a B)}{256 a^3 x^2}+\frac {b^2 \sqrt {a+b x^2} (A b-2 a B)}{128 a^2 x^4}+\frac {\left (a+b x^2\right )^{3/2} (A b-2 a B)}{16 a x^8}+\frac {b \sqrt {a+b x^2} (A b-2 a B)}{32 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}} \]
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Rubi [A] time = 0.15, antiderivative size = 184, 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 {3 b^3 \sqrt {a+b x^2} (A b-2 a B)}{256 a^3 x^2}+\frac {b^2 \sqrt {a+b x^2} (A b-2 a B)}{128 a^2 x^4}+\frac {3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{7/2}}+\frac {b \sqrt {a+b x^2} (A b-2 a B)}{32 a x^6}+\frac {\left (a+b x^2\right )^{3/2} (A b-2 a B)}{16 a x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{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 )^{3/2} \left (A+B x^2\right )}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2} (A+B x)}{x^6} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}+\frac {\left (-\frac {5 A b}{2}+5 a B\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^5} \, dx,x,x^2\right )}{10 a}\\ &=\frac {(A b-2 a B) \left (a+b x^2\right )^{3/2}}{16 a x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}-\frac {(3 b (A b-2 a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^4} \, dx,x,x^2\right )}{32 a}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x^2}}{32 a x^6}+\frac {(A b-2 a B) \left (a+b x^2\right )^{3/2}}{16 a x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}-\frac {\left (b^2 (A b-2 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^2\right )}{64 a}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x^2}}{32 a x^6}+\frac {b^2 (A b-2 a B) \sqrt {a+b x^2}}{128 a^2 x^4}+\frac {(A b-2 a B) \left (a+b x^2\right )^{3/2}}{16 a x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}+\frac {\left (3 b^3 (A b-2 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{256 a^2}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x^2}}{32 a x^6}+\frac {b^2 (A b-2 a B) \sqrt {a+b x^2}}{128 a^2 x^4}-\frac {3 b^3 (A b-2 a B) \sqrt {a+b x^2}}{256 a^3 x^2}+\frac {(A b-2 a B) \left (a+b x^2\right )^{3/2}}{16 a x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}-\frac {\left (3 b^4 (A b-2 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{512 a^3}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x^2}}{32 a x^6}+\frac {b^2 (A b-2 a B) \sqrt {a+b x^2}}{128 a^2 x^4}-\frac {3 b^3 (A b-2 a B) \sqrt {a+b x^2}}{256 a^3 x^2}+\frac {(A b-2 a B) \left (a+b x^2\right )^{3/2}}{16 a x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}-\frac {\left (3 b^3 (A b-2 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^3}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x^2}}{32 a x^6}+\frac {b^2 (A b-2 a B) \sqrt {a+b x^2}}{128 a^2 x^4}-\frac {3 b^3 (A b-2 a B) \sqrt {a+b x^2}}{256 a^3 x^2}+\frac {(A b-2 a B) \left (a+b x^2\right )^{3/2}}{16 a x^8}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}+\frac {3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 61, normalized size = 0.33 \begin {gather*} -\frac {\left (a+b x^2\right )^{5/2} \left (a^5 A+b^4 x^{10} (2 a B-A b) \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {b x^2}{a}+1\right )\right )}{10 a^6 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 152, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-128 a^4 A-160 a^4 B x^2-176 a^3 A b x^2-240 a^3 b B x^4-8 a^2 A b^2 x^4-20 a^2 b^2 B x^6+10 a A b^3 x^6+30 a b^3 B x^8-15 A b^4 x^8\right )}{1280 a^3 x^{10}}-\frac {3 \left (2 a b^4 B-A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 317, normalized size = 1.72 \begin {gather*} \left [-\frac {15 \, {\left (2 \, B a b^{4} - 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 (2 \, B a^{2} b^{3} - A a b^{4}\right )} x^{8} - 10 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{6} - 128 \, A a^{5} - 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2}\right )} x^{4} - 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{2560 \, a^{4} x^{10}}, \frac {15 \, {\left (2 \, B a b^{4} - A b^{5}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} x^{8} - 10 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{6} - 128 \, A a^{5} - 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2}\right )} x^{4} - 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{1280 \, a^{4} x^{10}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 212, normalized size = 1.15 \begin {gather*} \frac {\frac {15 \, {\left (2 \, B a b^{5} - A b^{6}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {30 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a b^{5} - 140 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 140 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{4} b^{5} - 30 \, \sqrt {b x^{2} + a} B a^{5} b^{5} - 15 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A b^{6} + 70 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a b^{6} - 128 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2} b^{6} - 70 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{3} b^{6} + 15 \, \sqrt {b x^{2} + a} A a^{4} b^{6}}{a^{3} b^{5} x^{10}}}{1280 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 317, normalized size = 1.72 \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 {7}{2}}}-\frac {3 B \,b^{4} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {5}{2}}}-\frac {3 \sqrt {b \,x^{2}+a}\, A \,b^{5}}{256 a^{4}}+\frac {3 \sqrt {b \,x^{2}+a}\, B \,b^{4}}{128 a^{3}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{5}}{256 a^{5}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,b^{4}}{128 a^{4}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,b^{4}}{256 a^{5} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B \,b^{3}}{128 a^{4} x^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,b^{3}}{128 a^{4} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B \,b^{2}}{64 a^{3} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,b^{2}}{32 a^{3} x^{6}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B b}{16 a^{2} x^{6}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A b}{16 a^{2} x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B}{8 a \,x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A}{10 a \,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 294, normalized size = 1.60 \begin {gather*} -\frac {3 \, B b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} + \frac {3 \, A b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, a^{\frac {7}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{4}}{128 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} B b^{4}}{128 \, a^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{5}}{256 \, a^{5}} - \frac {3 \, \sqrt {b x^{2} + a} A b^{5}}{256 \, a^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4}}{256 \, a^{5} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{64 \, a^{3} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{128 \, a^{4} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{32 \, a^{3} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{8 \, a x^{8}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{16 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{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 = 4.89, size = 205, normalized size = 1.11 \begin {gather*} \frac {3\,A\,a\,\sqrt {b\,x^2+a}}{256\,x^{10}}-\frac {11\,B\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^8}-\frac {7\,A\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^{10}}+\frac {3\,B\,a\,\sqrt {b\,x^2+a}}{128\,x^8}-\frac {A\,{\left (b\,x^2+a\right )}^{5/2}}{10\,a\,x^{10}}+\frac {7\,A\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a^2\,x^{10}}-\frac {3\,A\,{\left (b\,x^2+a\right )}^{9/2}}{256\,a^3\,x^{10}}-\frac {11\,B\,{\left (b\,x^2+a\right )}^{5/2}}{128\,a\,x^8}+\frac {3\,B\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a^2\,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^{7/2}}+\frac {B\,b^4\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,3{}\mathrm {i}}{128\,a^{5/2}} \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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