Optimal. Leaf size=120 \[ \frac {3 B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {8 A c \sqrt {a+c x^2}}{3 a^3 x}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}-\frac {3 B \sqrt {a+c x^2}}{2 a^2 x^2}+\frac {A+B x}{a x^3 \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 120, 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} \begin {gather*} \frac {8 A c \sqrt {a+c x^2}}{3 a^3 x}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}-\frac {3 B \sqrt {a+c x^2}}{2 a^2 x^2}+\frac {3 B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {A+B x}{a x^3 \sqrt {a+c x^2}} \end {gather*}
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^4 \left (a+c x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a x^3 \sqrt {a+c x^2}}-\frac {\int \frac {-4 a A c-3 a B c x}{x^4 \sqrt {a+c x^2}} \, dx}{a^2 c}\\ &=\frac {A+B x}{a x^3 \sqrt {a+c x^2}}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}+\frac {\int \frac {9 a^2 B c-8 a A c^2 x}{x^3 \sqrt {a+c x^2}} \, dx}{3 a^3 c}\\ &=\frac {A+B x}{a x^3 \sqrt {a+c x^2}}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}-\frac {3 B \sqrt {a+c x^2}}{2 a^2 x^2}-\frac {\int \frac {16 a^2 A c^2+9 a^2 B c^2 x}{x^2 \sqrt {a+c x^2}} \, dx}{6 a^4 c}\\ &=\frac {A+B x}{a x^3 \sqrt {a+c x^2}}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}-\frac {3 B \sqrt {a+c x^2}}{2 a^2 x^2}+\frac {8 A c \sqrt {a+c x^2}}{3 a^3 x}-\frac {(3 B c) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{2 a^2}\\ &=\frac {A+B x}{a x^3 \sqrt {a+c x^2}}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}-\frac {3 B \sqrt {a+c x^2}}{2 a^2 x^2}+\frac {8 A c \sqrt {a+c x^2}}{3 a^3 x}-\frac {(3 B c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac {A+B x}{a x^3 \sqrt {a+c x^2}}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}-\frac {3 B \sqrt {a+c x^2}}{2 a^2 x^2}+\frac {8 A c \sqrt {a+c x^2}}{3 a^3 x}-\frac {(3 B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 a^2}\\ &=\frac {A+B x}{a x^3 \sqrt {a+c x^2}}-\frac {4 A \sqrt {a+c x^2}}{3 a^2 x^3}-\frac {3 B \sqrt {a+c x^2}}{2 a^2 x^2}+\frac {8 A c \sqrt {a+c x^2}}{3 a^3 x}+\frac {3 B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 90, normalized size = 0.75 \begin {gather*} \frac {-\frac {a^2 (2 A+3 B x)}{x^3}+a \left (\frac {8 A c}{x}-9 B c\right )+9 a B c \sqrt {\frac {c x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )+16 A c^2 x}{6 a^3 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 102, normalized size = 0.85 \begin {gather*} \frac {-2 a^2 A-3 a^2 B x+8 a A c x^2-9 a B c x^3+16 A c^2 x^4}{6 a^3 x^3 \sqrt {a+c x^2}}-\frac {3 B c \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 232, normalized size = 1.93 \begin {gather*} \left [\frac {9 \, {\left (B c^{2} x^{5} + B a c x^{3}\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 \, A c^{2} x^{4} - 9 \, B a c x^{3} + 8 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (a^{3} c x^{5} + a^{4} x^{3}\right )}}, -\frac {9 \, {\left (B c^{2} x^{5} + B a c x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (16 \, A c^{2} x^{4} - 9 \, B a c x^{3} + 8 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{3} c x^{5} + a^{4} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 203, normalized size = 1.69 \begin {gather*} \frac {\frac {A c^{2} x}{a^{3}} - \frac {B c}{a^{2}}}{\sqrt {c x^{2} + a}} - \frac {3 \, B c \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B c - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A c^{\frac {3}{2}} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{2} c - 10 \, A a^{2} c^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 122, normalized size = 1.02 \begin {gather*} \frac {8 A \,c^{2} x}{3 \sqrt {c \,x^{2}+a}\, a^{3}}+\frac {3 B c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}}}-\frac {3 B c}{2 \sqrt {c \,x^{2}+a}\, a^{2}}+\frac {4 A c}{3 \sqrt {c \,x^{2}+a}\, a^{2} x}-\frac {B}{2 \sqrt {c \,x^{2}+a}\, a \,x^{2}}-\frac {A}{3 \sqrt {c \,x^{2}+a}\, a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 110, normalized size = 0.92 \begin {gather*} \frac {8 \, A c^{2} x}{3 \, \sqrt {c x^{2} + a} a^{3}} + \frac {3 \, B c \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {3 \, B c}{2 \, \sqrt {c x^{2} + a} a^{2}} + \frac {4 \, A c}{3 \, \sqrt {c x^{2} + a} a^{2} x} - \frac {B}{2 \, \sqrt {c x^{2} + a} a x^{2}} - \frac {A}{3 \, \sqrt {c x^{2} + a} a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.85, size = 95, normalized size = 0.79 \begin {gather*} \frac {3\,B\,c\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {B}{2\,a\,x^2\,\sqrt {c\,x^2+a}}-\frac {3\,B\,c}{2\,a^2\,\sqrt {c\,x^2+a}}+\frac {A\,\left (-a^2+4\,a\,c\,x^2+8\,c^2\,x^4\right )}{3\,a^3\,x^3\,\sqrt {c\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 28.08, size = 311, normalized size = 2.59 \begin {gather*} A \left (- \frac {a^{3} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}} + \frac {3 a^{2} c^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}} + \frac {12 a c^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}} + \frac {8 c^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} x^{2} + 6 a^{4} c^{5} x^{4} + 3 a^{3} c^{6} x^{6}}\right ) + B \left (- \frac {1}{2 a \sqrt {c} x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 \sqrt {c}}{2 a^{2} x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {3 c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{2 a^{\frac {5}{2}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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