Optimal. Leaf size=172 \[ \frac {5 B c^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}+\frac {5 B c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6} \]
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Rubi [A] time = 0.12, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {835, 807, 266, 47, 63, 208} \begin {gather*} \frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}+\frac {5 B c^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}+\frac {5 B c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {\int \frac {(-9 a B+2 A c x) \left (a+c x^2\right )^{5/2}}{x^9} \, dx}{9 a}\\ &=-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {\int \frac {(-16 a A c-9 a B c x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx}{72 a^2}\\ &=-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac {(B c) \int \frac {\left (a+c x^2\right )^{5/2}}{x^7} \, dx}{8 a}\\ &=-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac {(B c) \operatorname {Subst}\left (\int \frac {(a+c x)^{5/2}}{x^4} \, dx,x,x^2\right )}{16 a}\\ &=\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (5 B c^2\right ) \operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )}{96 a}\\ &=\frac {5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (5 B c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{128 a}\\ &=\frac {5 B c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (5 B c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{256 a}\\ &=\frac {5 B c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (5 B c^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{128 a}\\ &=\frac {5 B c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}+\frac {5 B c^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 64, normalized size = 0.37 \begin {gather*} -\frac {\left (a+c x^2\right )^{7/2} \left (a^3 A \left (7 a-2 c x^2\right )+9 B c^4 x^9 \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {c x^2}{a}+1\right )\right )}{63 a^5 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.08, size = 154, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-896 a^4 A-1008 a^4 B x-2432 a^3 A c x^2-2856 a^3 B c x^3-1920 a^2 A c^2 x^4-2478 a^2 B c^2 x^5-128 a A c^3 x^6-315 a B c^3 x^7+256 A c^4 x^8\right )}{8064 a^2 x^9}-\frac {5 B c^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{64 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 286, normalized size = 1.66 \begin {gather*} \left [\frac {315 \, B \sqrt {a} c^{4} x^{9} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{16128 \, a^{2} x^{9}}, -\frac {315 \, B \sqrt {-a} c^{4} x^{9} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{8064 \, a^{2} x^{9}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 491, normalized size = 2.85 \begin {gather*} -\frac {5 \, B c^{4} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a} + \frac {315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{17} B c^{4} + 8022 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{15} B a c^{4} + 16128 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{14} A a c^{\frac {9}{2}} + 10458 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{13} B a^{2} c^{4} + 26880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{12} A a^{2} c^{\frac {9}{2}} + 18270 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} B a^{3} c^{4} + 80640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{10} A a^{3} c^{\frac {9}{2}} + 48384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} A a^{4} c^{\frac {9}{2}} - 18270 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} B a^{5} c^{4} + 48384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} A a^{5} c^{\frac {9}{2}} - 10458 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B a^{6} c^{4} + 6912 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{6} c^{\frac {9}{2}} - 8022 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{7} c^{4} + 2304 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{7} c^{\frac {9}{2}} - 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{8} c^{4} - 256 \, A a^{8} c^{\frac {9}{2}}}{4032 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{9} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 204, normalized size = 1.19 \begin {gather*} \frac {5 B \,c^{4} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {3}{2}}}-\frac {5 \sqrt {c \,x^{2}+a}\, B \,c^{4}}{128 a^{2}}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,c^{4}}{384 a^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,c^{4}}{128 a^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,c^{3}}{128 a^{4} x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,c^{2}}{192 a^{3} x^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B c}{48 a^{2} x^{6}}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {7}{2}} A c}{63 a^{2} x^{7}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{8 a \,x^{8}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{9 a \,x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 192, normalized size = 1.12 \begin {gather*} \frac {5 \, B c^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{128 \, a^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c^{4}}{128 \, a^{4}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{4}}{384 \, a^{3}} - \frac {5 \, \sqrt {c x^{2} + a} B c^{4}}{128 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{2}}{192 \, a^{3} x^{4}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c}{48 \, a^{2} x^{6}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{63 \, a^{2} x^{7}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{8 \, a x^{8}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{9 \, a x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.34, size = 189, normalized size = 1.10 \begin {gather*} \frac {55\,B\,a\,{\left (c\,x^2+a\right )}^{3/2}}{384\,x^8}-\frac {73\,B\,{\left (c\,x^2+a\right )}^{5/2}}{384\,x^8}-\frac {A\,a^2\,\sqrt {c\,x^2+a}}{9\,x^9}-\frac {5\,B\,a^2\,\sqrt {c\,x^2+a}}{128\,x^8}-\frac {5\,B\,{\left (c\,x^2+a\right )}^{7/2}}{128\,a\,x^8}-\frac {5\,A\,c^2\,\sqrt {c\,x^2+a}}{21\,x^5}-\frac {A\,c^3\,\sqrt {c\,x^2+a}}{63\,a\,x^3}+\frac {2\,A\,c^4\,\sqrt {c\,x^2+a}}{63\,a^2\,x}-\frac {19\,A\,a\,c\,\sqrt {c\,x^2+a}}{63\,x^7}-\frac {B\,c^4\,\mathrm {atan}\left (\frac {\sqrt {c\,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 [B] time = 25.67, size = 1202, normalized size = 6.99 \begin {gather*} - \frac {35 A a^{9} c^{\frac {19}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {110 A a^{8} c^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {114 A a^{7} c^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {40 A a^{6} c^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {30 A a^{6} c^{\frac {11}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {5 A a^{5} c^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {66 A a^{5} c^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {30 A a^{4} c^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {34 A a^{4} c^{\frac {15}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {40 A a^{3} c^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {6 A a^{3} c^{\frac {17}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {16 A a^{2} c^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {24 A a^{2} c^{\frac {19}{2}} x^{8} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {16 A a c^{\frac {21}{2}} x^{10} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {A c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a x^{2}} + \frac {2 A c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{2}} - \frac {B a^{3}}{8 \sqrt {c} x^{9} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {23 B a^{2} \sqrt {c}}{48 x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {127 B a c^{\frac {3}{2}}}{192 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {133 B c^{\frac {5}{2}}}{384 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 B c^{\frac {7}{2}}}{128 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {5 B c^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{128 a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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