Optimal. Leaf size=129 \[ \frac {5 A c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}-\frac {8 B \sqrt {a+c x^2}}{3 a^3 x}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}+\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 129, 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 = {823, 835, 807, 266, 63, 208} \[ \frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}+\frac {5 A c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {8 B \sqrt {a+c x^2}}{3 a^3 x}+\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 835
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx &=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-5 a A c-4 a B c x}{x^3 \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}+\frac {\int \frac {15 a^2 A c^2+8 a^2 B c^2 x}{x^3 \sqrt {a+c x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}-\frac {\int \frac {-16 a^3 B c^2+15 a^2 A c^3 x}{x^2 \sqrt {a+c x^2}} \, dx}{6 a^5 c^2}\\ &=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}-\frac {8 B \sqrt {a+c x^2}}{3 a^3 x}-\frac {(5 A c) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{2 a^3}\\ &=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}-\frac {8 B \sqrt {a+c x^2}}{3 a^3 x}-\frac {(5 A c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a^3}\\ &=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}-\frac {8 B \sqrt {a+c x^2}}{3 a^3 x}-\frac {(5 A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 a^3}\\ &=\frac {A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}}+\frac {5 A+4 B x}{3 a^2 x^2 \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 x^2}-\frac {8 B \sqrt {a+c x^2}}{3 a^3 x}+\frac {5 A c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 106, normalized size = 0.82 \[ \frac {-\frac {3 a^3 (A+2 B x)}{x^2}-4 a^2 c (5 A+6 B x)-a c^2 x^2 (15 A+16 B x)+\frac {15 A c \left (a+c x^2\right )^2 \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )}{\sqrt {\frac {c x^2}{a}+1}}}{6 a^4 \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 307, normalized size = 2.38 \[ \left [\frac {15 \, {\left (A c^{3} x^{6} + 2 \, A a c^{2} x^{4} + A a^{2} c x^{2}\right )} \sqrt {a} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (16 \, B a c^{2} x^{5} + 15 \, A a c^{2} x^{4} + 24 \, B a^{2} c x^{3} + 20 \, A a^{2} c x^{2} + 6 \, B a^{3} x + 3 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (a^{4} c^{2} x^{6} + 2 \, a^{5} c x^{4} + a^{6} x^{2}\right )}}, -\frac {15 \, {\left (A c^{3} x^{6} + 2 \, A a c^{2} x^{4} + A a^{2} c x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (16 \, B a c^{2} x^{5} + 15 \, A a c^{2} x^{4} + 24 \, B a^{2} c x^{3} + 20 \, A a^{2} c x^{2} + 6 \, B a^{3} x + 3 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{4} c^{2} x^{6} + 2 \, a^{5} c x^{4} + a^{6} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 197, normalized size = 1.53 \[ -\frac {{\left ({\left (\frac {5 \, B c^{2} x}{a^{3}} + \frac {6 \, A c^{2}}{a^{3}}\right )} x + \frac {6 \, B c}{a^{2}}\right )} x + \frac {7 \, A c}{a^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} - \frac {5 \, A c \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a \sqrt {c} + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a c - 2 \, B a^{2} \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 134, normalized size = 1.04 \[ -\frac {4 B c x}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}-\frac {5 A c}{6 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}-\frac {8 B c x}{3 \sqrt {c \,x^{2}+a}\, a^{3}}+\frac {5 A c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {7}{2}}}-\frac {5 A c}{2 \sqrt {c \,x^{2}+a}\, a^{3}}-\frac {B}{\left (c \,x^{2}+a \right )^{\frac {3}{2}} a x}-\frac {A}{2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 122, normalized size = 0.95 \[ -\frac {8 \, B c x}{3 \, \sqrt {c x^{2} + a} a^{3}} - \frac {4 \, B c x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {5 \, A c \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{2 \, a^{\frac {7}{2}}} - \frac {5 \, A c}{2 \, \sqrt {c x^{2} + a} a^{3}} - \frac {5 \, A c}{6 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {B}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} a x} - \frac {A}{2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 123, normalized size = 0.95 \[ \frac {B\,a^2-8\,B\,{\left (c\,x^2+a\right )}^2+4\,B\,a\,\left (c\,x^2+a\right )}{3\,a^3\,x\,{\left (c\,x^2+a\right )}^{3/2}}-\frac {10\,A\,c}{3\,a^2\,{\left (c\,x^2+a\right )}^{3/2}}-\frac {A}{2\,a\,x^2\,{\left (c\,x^2+a\right )}^{3/2}}+\frac {5\,A\,c\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{7/2}}-\frac {5\,A\,c^2\,x^2}{2\,a^3\,{\left (c\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 22.82, size = 1034, normalized size = 8.02 \[ A \left (- \frac {6 a^{17} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {46 a^{16} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {15 a^{16} c x^{2} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {30 a^{16} c x^{2} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {70 a^{15} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {45 a^{15} c^{2} x^{4} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {90 a^{15} c^{2} x^{4} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {30 a^{14} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {45 a^{14} c^{3} x^{6} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {90 a^{14} c^{3} x^{6} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} - \frac {15 a^{13} c^{4} x^{8} \log {\left (\frac {c x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}} + \frac {30 a^{13} c^{4} x^{8} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} c x^{4} + 36 a^{\frac {35}{2}} c^{2} x^{6} + 12 a^{\frac {33}{2}} c^{3} x^{8}}\right ) + B \left (- \frac {3 a^{2} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}} - \frac {12 a c^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}} - \frac {8 c^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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