Optimal. Leaf size=104 \[ -\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}+\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {823, 807, 266, 63, 208} \begin {gather*} \frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \left (a+c x^2\right )^{5/2}} \, dx &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-4 a A c-3 a B c x}{x^2 \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}+\frac {\int \frac {8 a^2 A c^2+3 a^2 B c^2 x}{x^2 \sqrt {a+c x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {B \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{a^2}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {B \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {B \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a^2 c}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 95, normalized size = 0.91 \begin {gather*} \frac {a^2 (4 B x-3 A)+3 a c x^2 (B x-4 A)-3 \sqrt {a} B x \left (a+c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-8 A c^2 x^4}{3 a^3 x \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.58, size = 101, normalized size = 0.97 \begin {gather*} \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {-3 a^2 A+4 a^2 B x-12 a A c x^2+3 a B c x^3-8 A c^2 x^4}{3 a^3 x \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 264, normalized size = 2.54 \begin {gather*} \left [\frac {3 \, {\left (B c^{2} x^{5} + 2 \, B a c x^{3} + B a^{2} x\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (8 \, A c^{2} x^{4} - 3 \, B a c x^{3} + 12 \, A a c x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )}}, \frac {3 \, {\left (B c^{2} x^{5} + 2 \, B a c x^{3} + B a^{2} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (8 \, A c^{2} x^{4} - 3 \, B a c x^{3} + 12 \, A a c x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 119, normalized size = 1.14 \begin {gather*} -\frac {{\left ({\left (\frac {5 \, A c^{2} x}{a^{3}} - \frac {3 \, B c}{a^{2}}\right )} x + \frac {6 \, A c}{a^{2}}\right )} x - \frac {4 \, B}{a}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, A \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 112, normalized size = 1.08 \begin {gather*} -\frac {4 A c x}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}-\frac {8 A c x}{3 \sqrt {c \,x^{2}+a}\, a^{3}}+\frac {B}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a}-\frac {B \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {5}{2}}}-\frac {A}{\left (c \,x^{2}+a \right )^{\frac {3}{2}} a x}+\frac {B}{\sqrt {c \,x^{2}+a}\, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 100, normalized size = 0.96 \begin {gather*} -\frac {8 \, A c x}{3 \, \sqrt {c x^{2} + a} a^{3}} - \frac {4 \, A c x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {B}{\sqrt {c x^{2} + a} a^{2}} + \frac {B}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 96, normalized size = 0.92 \begin {gather*} \frac {\frac {B}{3\,a}+\frac {B\,\left (c\,x^2+a\right )}{a^2}}{{\left (c\,x^2+a\right )}^{3/2}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A\,a^2-8\,A\,{\left (c\,x^2+a\right )}^2+4\,A\,a\,\left (c\,x^2+a\right )}{3\,a^3\,x\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 22.26, size = 910, normalized size = 8.75 \begin {gather*} A \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 ) + B \left (\frac {8 a^{7} \sqrt {1 + \frac {c x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {3 a^{7} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {14 a^{6} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {9 a^{6} c x^{2} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {18 a^{6} c x^{2} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {6 a^{5} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {9 a^{5} c^{2} x^{4} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {18 a^{5} c^{2} x^{4} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {3 a^{4} c^{3} x^{6} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {6 a^{4} c^{3} x^{6} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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