Optimal. Leaf size=147 \[ \frac {B c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}+\frac {B c^2 \sqrt {a+c x^2}}{16 a x^2}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {B c \left (a+c x^2\right )^{3/2}}{24 a x^4} \]
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Rubi [A] time = 0.11, antiderivative size = 147, 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 {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}+\frac {B c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}+\frac {B c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {B c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6} \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 )^{3/2}}{x^8} \, dx &=-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {\int \frac {(-7 a B+2 A c x) \left (a+c x^2\right )^{3/2}}{x^7} \, dx}{7 a}\\ &=-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {\int \frac {(-12 a A c-7 a B c x) \left (a+c x^2\right )^{3/2}}{x^6} \, dx}{42 a^2}\\ &=-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac {(B c) \int \frac {\left (a+c x^2\right )^{3/2}}{x^5} \, dx}{6 a}\\ &=-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac {(B c) \operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )}{12 a}\\ &=\frac {B c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac {\left (B c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{16 a}\\ &=\frac {B c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {B c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac {\left (B c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac {B c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {B c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac {\left (B c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{16 a}\\ &=\frac {B c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {B c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{7 a x^7}-\frac {B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}+\frac {B c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 64, normalized size = 0.44 \begin {gather*} \frac {\left (a+c x^2\right )^{5/2} \left (a^2 A \left (2 c x^2-5 a\right )+7 B c^3 x^7 \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {c x^2}{a}+1\right )\right )}{35 a^4 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.84, size = 130, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-240 a^3 A-280 a^3 B x-384 a^2 A c x^2-490 a^2 B c x^3-48 a A c^2 x^4-105 a B c^2 x^5+96 A c^3 x^6\right )}{1680 a^2 x^7}-\frac {B c^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 238, normalized size = 1.62 \begin {gather*} \left [\frac {105 \, B \sqrt {a} c^{3} x^{7} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (96 \, A c^{3} x^{6} - 105 \, B a c^{2} x^{5} - 48 \, A a c^{2} x^{4} - 490 \, B a^{2} c x^{3} - 384 \, A a^{2} c x^{2} - 280 \, B a^{3} x - 240 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{3360 \, a^{2} x^{7}}, -\frac {105 \, B \sqrt {-a} c^{3} x^{7} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (96 \, A c^{3} x^{6} - 105 \, B a c^{2} x^{5} - 48 \, A a c^{2} x^{4} - 490 \, B a^{2} c x^{3} - 384 \, A a^{2} c x^{2} - 280 \, B a^{3} x - 240 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{1680 \, a^{2} x^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 379, normalized size = 2.58 \begin {gather*} -\frac {B c^{3} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{13} B c^{3} + 1540 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} B a c^{3} + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{10} A a c^{\frac {7}{2}} + 1085 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} B a^{2} c^{3} + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} A a^{2} c^{\frac {7}{2}} + 6720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} A a^{3} c^{\frac {7}{2}} - 1085 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B a^{4} c^{3} + 1344 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{4} c^{\frac {7}{2}} - 1540 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{5} c^{3} + 672 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{5} c^{\frac {7}{2}} - 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{6} c^{3} - 96 \, A a^{6} c^{\frac {7}{2}}}{840 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{7} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 165, normalized size = 1.12 \begin {gather*} \frac {B \,c^{3} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+a}\, B \,c^{3}}{16 a^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,c^{3}}{48 a^{3}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,c^{2}}{48 a^{3} x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B c}{24 a^{2} x^{4}}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} A c}{35 a^{2} x^{5}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B}{6 a \,x^{6}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A}{7 a \,x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 153, normalized size = 1.04 \begin {gather*} \frac {B c^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{3}}{48 \, a^{3}} - \frac {\sqrt {c x^{2} + a} B c^{3}}{16 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c^{2}}{48 \, a^{3} x^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c}{24 \, a^{2} x^{4}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} A c}{35 \, a^{2} x^{5}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B}{6 \, a x^{6}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A}{7 \, a x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.77, size = 150, normalized size = 1.02 \begin {gather*} \frac {B\,a\,\sqrt {c\,x^2+a}}{16\,x^6}-\frac {A\,a\,\sqrt {c\,x^2+a}}{7\,x^7}-\frac {B\,{\left (c\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {8\,A\,c\,\sqrt {c\,x^2+a}}{35\,x^5}-\frac {B\,{\left (c\,x^2+a\right )}^{5/2}}{16\,a\,x^6}-\frac {A\,c^2\,\sqrt {c\,x^2+a}}{35\,a\,x^3}+\frac {2\,A\,c^3\,\sqrt {c\,x^2+a}}{35\,a^2\,x}-\frac {B\,c^3\,\mathrm {atan}\left (\frac {\sqrt {c\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{16\,a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.99, size = 575, normalized size = 3.91 \begin {gather*} - \frac {15 A a^{6} 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 A a^{5} 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 A a^{4} 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 A a^{3} 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 A a^{2} 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 A a 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 {A c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a x^{2}} + \frac {2 A c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{2}} - \frac {B a^{2}}{6 \sqrt {c} x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {11 B a \sqrt {c}}{24 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {17 B c^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {B c^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{16 a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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