Optimal. Leaf size=149 \[ \frac {5 A c^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}+\frac {5 A c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]
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Rubi [A] time = 0.10, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {5 A c^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}+\frac {5 A c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7} \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^9} \, dx &=-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {\int \frac {(-8 a B+A c x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx}{8 a}\\ &=-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {(A 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}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {(A c) \operatorname {Subst}\left (\int \frac {(a+c x)^{5/2}}{x^4} \, dx,x,x^2\right )}{16 a}\\ &=\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {\left (5 A c^2\right ) \operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )}{96 a}\\ &=\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {\left (5 A c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{128 a}\\ &=\frac {5 A c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {\left (5 A c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{256 a}\\ &=\frac {5 A c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {\left (5 A 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 A c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac {B \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac {5 A 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 = 53, normalized size = 0.36 \begin {gather*} -\frac {\left (a+c x^2\right )^{7/2} \left (a^4 B+A c^4 x^7 \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {c x^2}{a}+1\right )\right )}{7 a^5 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.01, size = 139, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-336 a^3 A-384 a^3 B x-952 a^2 A c x^2-1152 a^2 B c x^3-826 a A c^2 x^4-1152 a B c^2 x^5-105 A c^3 x^6-384 B c^3 x^7\right )}{2688 a x^8}-\frac {5 A 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 = 0.53, size = 267, normalized size = 1.79 \begin {gather*} \left [\frac {105 \, A \sqrt {a} c^{4} x^{8} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{5376 \, a^{2} x^{8}}, -\frac {105 \, A \sqrt {-a} c^{4} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{2688 \, a^{2} x^{8}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 491, normalized size = 3.30 \begin {gather*} -\frac {5 \, A c^{4} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a} + \frac {105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{15} A c^{4} + 2688 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{14} B a c^{\frac {7}{2}} + 2779 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{13} A a c^{4} - 2688 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{12} B a^{2} c^{\frac {7}{2}} + 6265 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} A a^{2} c^{4} + 13440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{10} B a^{3} c^{\frac {7}{2}} + 12355 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} A a^{3} c^{4} - 13440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} B a^{4} c^{\frac {7}{2}} + 12355 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A a^{4} c^{4} + 8064 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} B a^{5} c^{\frac {7}{2}} + 6265 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a^{5} c^{4} - 8064 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{6} c^{\frac {7}{2}} + 2779 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{6} c^{4} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{7} c^{\frac {7}{2}} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{7} c^{4} - 384 \, B a^{8} c^{\frac {7}{2}}}{1344 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{8} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 185, normalized size = 1.24 \begin {gather*} \frac {5 A \,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}\, A \,c^{4}}{128 a^{2}}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{4}}{384 a^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,c^{4}}{128 a^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A \,c^{3}}{128 a^{4} x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A \,c^{2}}{192 a^{3} x^{4}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A c}{48 a^{2} x^{6}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{7 a \,x^{7}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{8 a \,x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 173, normalized size = 1.16 \begin {gather*} \frac {5 \, A 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}} A c^{4}}{128 \, a^{4}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c^{4}}{384 \, a^{3}} - \frac {5 \, \sqrt {c x^{2} + a} A c^{4}}{128 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c^{2}}{192 \, a^{3} x^{4}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{48 \, a^{2} x^{6}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{7 \, a x^{7}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{8 \, a x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 168, normalized size = 1.13 \begin {gather*} \frac {55\,A\,a\,{\left (c\,x^2+a\right )}^{3/2}}{384\,x^8}-\frac {73\,A\,{\left (c\,x^2+a\right )}^{5/2}}{384\,x^8}-\frac {5\,A\,a^2\,\sqrt {c\,x^2+a}}{128\,x^8}-\frac {5\,A\,{\left (c\,x^2+a\right )}^{7/2}}{128\,a\,x^8}-\frac {B\,a^2\,\sqrt {c\,x^2+a}}{7\,x^7}-\frac {3\,B\,c^2\,\sqrt {c\,x^2+a}}{7\,x^3}-\frac {B\,c^3\,\sqrt {c\,x^2+a}}{7\,a\,x}-\frac {3\,B\,a\,c\,\sqrt {c\,x^2+a}}{7\,x^5}-\frac {A\,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 = 25.77, size = 609, normalized size = 4.09 \begin {gather*} - \frac {A a^{3}}{8 \sqrt {c} x^{9} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {23 A a^{2} \sqrt {c}}{48 x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {127 A a c^{\frac {3}{2}}}{192 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {133 A c^{\frac {5}{2}}}{384 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 A c^{\frac {7}{2}}}{128 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {5 A c^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{128 a^{\frac {3}{2}}} - \frac {15 B a^{7} c^{\frac {9}{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 {33 B a^{6} c^{\frac {11}{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 {17 B a^{5} c^{\frac {13}{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 {3 B a^{4} c^{\frac {15}{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 {12 B a^{3} c^{\frac {17}{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 {8 B a^{2} c^{\frac {19}{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 {2 B a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {7 B c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 x^{2}} - \frac {B c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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