Optimal. Leaf size=124 \[ \frac {A c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {A c^2 \sqrt {a+c x^2}}{16 a x^2}-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac {A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5} \]
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Rubi [A] time = 0.08, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {A c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {A c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5} \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^7} \, dx &=-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}-\frac {\int \frac {(-6 a B+A c x) \left (a+c x^2\right )^{3/2}}{x^6} \, dx}{6 a}\\ &=-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {(A 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}}{6 a x^6}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {(A c) \operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )}{12 a}\\ &=\frac {A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {\left (A c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{16 a}\\ &=\frac {A c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {\left (A c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac {A c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {\left (A 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 {A c^2 \sqrt {a+c x^2}}{16 a x^2}+\frac {A c \left (a+c x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{6 a x^6}-\frac {B \left (a+c x^2\right )^{5/2}}{5 a x^5}+\frac {A 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 = 0.02, size = 54, normalized size = 0.44 \begin {gather*} \frac {\left (a+c x^2\right )^{5/2} \left (A c^3 x^5 \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {c x^2}{a}+1\right )-a^3 B\right )}{5 a^4 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.78, size = 115, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-40 a^2 A-48 a^2 B x-70 a A c x^2-96 a B c x^3-15 A c^2 x^4-48 B c^2 x^5\right )}{240 a x^6}-\frac {A 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 = 0.47, size = 219, normalized size = 1.77 \begin {gather*} \left [\frac {15 \, A \sqrt {a} c^{3} x^{6} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (48 \, B a c^{2} x^{5} + 15 \, A a c^{2} x^{4} + 96 \, B a^{2} c x^{3} + 70 \, A a^{2} c x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{480 \, a^{2} x^{6}}, -\frac {15 \, A \sqrt {-a} c^{3} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (48 \, B a c^{2} x^{5} + 15 \, A a c^{2} x^{4} + 96 \, B a^{2} c x^{3} + 70 \, A a^{2} c x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{240 \, a^{2} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 379, normalized size = 3.06 \begin {gather*} -\frac {A c^{3} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} A c^{3} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{10} B a c^{\frac {5}{2}} + 235 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} A a c^{3} - 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} B a^{2} c^{\frac {5}{2}} + 390 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A a^{2} c^{3} + 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} B a^{3} c^{\frac {5}{2}} + 390 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a^{3} c^{3} - 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{4} c^{\frac {5}{2}} + 235 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{4} c^{3} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{5} c^{\frac {5}{2}} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{5} c^{3} - 48 \, B a^{6} c^{\frac {5}{2}}}{120 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{6} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 146, normalized size = 1.18 \begin {gather*} \frac {A \,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}\, A \,c^{3}}{16 a^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{3}}{48 a^{3}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,c^{2}}{48 a^{3} x^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A c}{24 a^{2} x^{4}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B}{5 a \,x^{5}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A}{6 a \,x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 134, normalized size = 1.08 \begin {gather*} \frac {A 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}} A c^{3}}{48 \, a^{3}} - \frac {\sqrt {c x^{2} + a} A c^{3}}{16 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A c^{2}}{48 \, a^{3} x^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A c}{24 \, a^{2} x^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B}{5 \, a x^{5}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A}{6 \, a x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 94, normalized size = 0.76 \begin {gather*} \frac {A\,a\,\sqrt {c\,x^2+a}}{16\,x^6}-\frac {A\,{\left (c\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {A\,{\left (c\,x^2+a\right )}^{5/2}}{16\,a\,x^6}-\frac {B\,{\left (c\,x^2+a\right )}^{5/2}}{5\,a\,x^5}-\frac {A\,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 [A] time = 13.39, size = 201, normalized size = 1.62 \begin {gather*} - \frac {A a^{2}}{6 \sqrt {c} x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {11 A a \sqrt {c}}{24 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {17 A c^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {A c^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {A c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{16 a^{\frac {3}{2}}} - \frac {B a \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {2 B c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{2}} - \frac {B c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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