Optimal. Leaf size=140 \[ -\frac {c^2 \sqrt {a+c x^2} (5 A+16 B x)}{16 x^2}-\frac {\left (a+c x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}-\frac {c \left (a+c x^2\right )^{3/2} (5 A+8 B x)}{24 x^4}-\frac {5 A c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}}+B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {811, 844, 217, 206, 266, 63, 208} \begin {gather*} -\frac {c^2 \sqrt {a+c x^2} (5 A+16 B x)}{16 x^2}-\frac {c \left (a+c x^2\right )^{3/2} (5 A+8 B x)}{24 x^4}-\frac {\left (a+c x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}-\frac {5 A c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}}+B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 811
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^7} \, dx &=-\frac {(5 A+6 B x) \left (a+c x^2\right )^{5/2}}{30 x^6}-\frac {\int \frac {(-10 a A c-12 a B c x) \left (a+c x^2\right )^{3/2}}{x^5} \, dx}{12 a}\\ &=-\frac {c (5 A+8 B x) \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {(5 A+6 B x) \left (a+c x^2\right )^{5/2}}{30 x^6}+\frac {\int \frac {\left (60 a^2 A c^2+96 a^2 B c^2 x\right ) \sqrt {a+c x^2}}{x^3} \, dx}{96 a^2}\\ &=-\frac {c^2 (5 A+16 B x) \sqrt {a+c x^2}}{16 x^2}-\frac {c (5 A+8 B x) \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {(5 A+6 B x) \left (a+c x^2\right )^{5/2}}{30 x^6}-\frac {\int \frac {-120 a^3 A c^3-384 a^3 B c^3 x}{x \sqrt {a+c x^2}} \, dx}{384 a^3}\\ &=-\frac {c^2 (5 A+16 B x) \sqrt {a+c x^2}}{16 x^2}-\frac {c (5 A+8 B x) \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {(5 A+6 B x) \left (a+c x^2\right )^{5/2}}{30 x^6}+\frac {1}{16} \left (5 A c^3\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\left (B c^3\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=-\frac {c^2 (5 A+16 B x) \sqrt {a+c x^2}}{16 x^2}-\frac {c (5 A+8 B x) \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {(5 A+6 B x) \left (a+c x^2\right )^{5/2}}{30 x^6}+\frac {1}{32} \left (5 A c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\left (B c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=-\frac {c^2 (5 A+16 B x) \sqrt {a+c x^2}}{16 x^2}-\frac {c (5 A+8 B x) \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {(5 A+6 B x) \left (a+c x^2\right )^{5/2}}{30 x^6}+B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {1}{16} \left (5 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 )\\ &=-\frac {c^2 (5 A+16 B x) \sqrt {a+c x^2}}{16 x^2}-\frac {c (5 A+8 B x) \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {(5 A+6 B x) \left (a+c x^2\right )^{5/2}}{30 x^6}+B c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {5 A c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 125, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {a+c x^2} \left (48 a^3 B x \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};-\frac {c x^2}{a}\right )+5 a A \sqrt {\frac {c x^2}{a}+1} \left (8 a^2+26 a c x^2+33 c^2 x^4\right )+75 A c^3 x^6 \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )\right )}{240 a x^6 \sqrt {\frac {c x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.83, size = 141, normalized size = 1.01 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-40 a^2 A-48 a^2 B x-130 a A c x^2-176 a B c x^3-165 A c^2 x^4-368 B c^2 x^5\right )}{240 x^6}+\frac {5 A c^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}-B c^{5/2} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 574, normalized size = 4.10 \begin {gather*} \left [\frac {240 \, B a c^{\frac {5}{2}} x^{6} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 75 \, 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 (368 \, B a c^{2} x^{5} + 165 \, A a c^{2} x^{4} + 176 \, B a^{2} c x^{3} + 130 \, A a^{2} c x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{480 \, a x^{6}}, -\frac {480 \, B a \sqrt {-c} c^{2} x^{6} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 75 \, 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 (368 \, B a c^{2} x^{5} + 165 \, A a c^{2} x^{4} + 176 \, B a^{2} c x^{3} + 130 \, A a^{2} c x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{480 \, a x^{6}}, \frac {75 \, A \sqrt {-a} c^{3} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 120 \, B a c^{\frac {5}{2}} x^{6} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - {\left (368 \, B a c^{2} x^{5} + 165 \, A a c^{2} x^{4} + 176 \, B a^{2} c x^{3} + 130 \, A a^{2} c x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{240 \, a x^{6}}, -\frac {240 \, B a \sqrt {-c} c^{2} x^{6} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 75 \, A \sqrt {-a} c^{3} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (368 \, B a c^{2} x^{5} + 165 \, A a c^{2} x^{4} + 176 \, B a^{2} c x^{3} + 130 \, A a^{2} c x^{2} + 48 \, B a^{3} x + 40 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{240 \, a x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 397, normalized size = 2.84 \begin {gather*} \frac {5 \, A c^{3} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a}} - B c^{\frac {5}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {165 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} A c^{3} + 720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{10} B a c^{\frac {5}{2}} + 25 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} A a c^{3} - 2160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} B a^{2} c^{\frac {5}{2}} + 450 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A a^{2} c^{3} + 3680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} B a^{3} c^{\frac {5}{2}} + 450 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a^{3} c^{3} - 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{4} c^{\frac {5}{2}} + 25 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{4} c^{3} + 1488 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{5} c^{\frac {5}{2}} + 165 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{5} c^{3} - 368 \, B a^{6} c^{\frac {5}{2}}}{120 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 281, normalized size = 2.01 \begin {gather*} -\frac {5 A \,c^{3} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{16 \sqrt {a}}+B \,c^{\frac {5}{2}} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )+\frac {\sqrt {c \,x^{2}+a}\, B \,c^{3} x}{a}+\frac {5 \sqrt {c \,x^{2}+a}\, A \,c^{3}}{16 a}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,c^{3} x}{3 a^{2}}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{3}}{48 a^{2}}+\frac {8 \left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,c^{3} x}{15 a^{3}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,c^{3}}{16 a^{3}}-\frac {8 \left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,c^{2}}{15 a^{3} x}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A \,c^{2}}{16 a^{3} x^{2}}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {7}{2}} B c}{15 a^{2} x^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A c}{24 a^{2} x^{4}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{5 a \,x^{5}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{6 a \,x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 243, normalized size = 1.74 \begin {gather*} \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{3} x}{3 \, a^{2}} + \frac {\sqrt {c x^{2} + a} B c^{3} x}{a} + B c^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - \frac {5 \, A c^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{16 \, \sqrt {a}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A c^{3}}{16 \, a^{3}} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c^{3}}{48 \, a^{2}} + \frac {5 \, \sqrt {c x^{2} + a} A c^{3}}{16 \, a} - \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B c^{2}}{15 \, a^{2} x} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c^{2}}{16 \, a^{3} x^{2}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} B c}{15 \, a^{2} x^{3}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{24 \, a^{2} x^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{5 \, a x^{5}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{6 \, a x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.16, size = 299, normalized size = 2.14 \begin {gather*} - \frac {A a^{3}}{6 \sqrt {c} x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {17 A a^{2} \sqrt {c}}{24 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {35 A a c^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{2 x} - \frac {3 A c^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 A c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{16 \sqrt {a}} - \frac {B \sqrt {a} c^{2}}{x \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {B a^{2} \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {11 B a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 x^{2}} - \frac {8 B c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15} + B c^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )} - \frac {B c^{3} x}{\sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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