Optimal. Leaf size=106 \[ -a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {3 a^2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {1}{8} a \sqrt {a+c x^2} (8 A+3 B x)+\frac {1}{12} \left (a+c x^2\right )^{3/2} (4 A+3 B x) \]
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Rubi [A] time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {815, 844, 217, 206, 266, 63, 208} \begin {gather*} -a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {3 a^2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {1}{8} a \sqrt {a+c x^2} (8 A+3 B x)+\frac {1}{12} \left (a+c x^2\right )^{3/2} (4 A+3 B x) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{x} \, dx &=\frac {1}{12} (4 A+3 B x) \left (a+c x^2\right )^{3/2}+\frac {\int \frac {(4 a A c+3 a B c x) \sqrt {a+c x^2}}{x} \, dx}{4 c}\\ &=\frac {1}{8} a (8 A+3 B x) \sqrt {a+c x^2}+\frac {1}{12} (4 A+3 B x) \left (a+c x^2\right )^{3/2}+\frac {\int \frac {8 a^2 A c^2+3 a^2 B c^2 x}{x \sqrt {a+c x^2}} \, dx}{8 c^2}\\ &=\frac {1}{8} a (8 A+3 B x) \sqrt {a+c x^2}+\frac {1}{12} (4 A+3 B x) \left (a+c x^2\right )^{3/2}+\left (a^2 A\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\frac {1}{8} \left (3 a^2 B\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {1}{8} a (8 A+3 B x) \sqrt {a+c x^2}+\frac {1}{12} (4 A+3 B x) \left (a+c x^2\right )^{3/2}+\frac {1}{2} \left (a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\frac {1}{8} \left (3 a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=\frac {1}{8} a (8 A+3 B x) \sqrt {a+c x^2}+\frac {1}{12} (4 A+3 B x) \left (a+c x^2\right )^{3/2}+\frac {3 a^2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {\left (a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c}\\ &=\frac {1}{8} a (8 A+3 B x) \sqrt {a+c x^2}+\frac {1}{12} (4 A+3 B x) \left (a+c x^2\right )^{3/2}+\frac {3 a^2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}-a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 118, normalized size = 1.11 \begin {gather*} \frac {1}{24} \left (-24 a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {9 a^{5/2} B \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {c} \sqrt {a+c x^2}}+\sqrt {a+c x^2} \left (32 a A+15 a B x+8 A c x^2+6 B c x^3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 114, normalized size = 1.08 \begin {gather*} 2 a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {3 a^2 B \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 \sqrt {c}}+\frac {1}{24} \sqrt {a+c x^2} \left (32 a A+15 a B x+8 A c x^2+6 B c x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 439, normalized size = 4.14 \begin {gather*} \left [\frac {9 \, B a^{2} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 24 \, A a^{\frac {3}{2}} c \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (6 \, B c^{2} x^{3} + 8 \, A c^{2} x^{2} + 15 \, B a c x + 32 \, A a c\right )} \sqrt {c x^{2} + a}}{48 \, c}, -\frac {9 \, B a^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 12 \, A a^{\frac {3}{2}} c \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (6 \, B c^{2} x^{3} + 8 \, A c^{2} x^{2} + 15 \, B a c x + 32 \, A a c\right )} \sqrt {c x^{2} + a}}{24 \, c}, \frac {48 \, A \sqrt {-a} a c \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 9 \, B a^{2} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (6 \, B c^{2} x^{3} + 8 \, A c^{2} x^{2} + 15 \, B a c x + 32 \, A a c\right )} \sqrt {c x^{2} + a}}{48 \, c}, -\frac {9 \, B a^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 24 \, A \sqrt {-a} a c \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (6 \, B c^{2} x^{3} + 8 \, A c^{2} x^{2} + 15 \, B a c x + 32 \, A a c\right )} \sqrt {c x^{2} + a}}{24 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 100, normalized size = 0.94 \begin {gather*} \frac {2 \, A a^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3 \, B a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, \sqrt {c}} + \frac {1}{24} \, \sqrt {c x^{2} + a} {\left (32 \, A a + {\left (15 \, B a + 2 \, {\left (3 \, B c x + 4 \, A c\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 1.01 \begin {gather*} -A \,a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {3 B \,a^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 \sqrt {c}}+\frac {3 \sqrt {c \,x^{2}+a}\, B a x}{8}+\sqrt {c \,x^{2}+a}\, A a +\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B x}{4}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 88, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B x + \frac {3}{8} \, \sqrt {c x^{2} + a} B a x + \frac {3 \, B a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} - A a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A + \sqrt {c x^{2} + a} A a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 83, normalized size = 0.78 \begin {gather*} \frac {A\,{\left (c\,x^2+a\right )}^{3/2}}{3}-A\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )+A\,a\,\sqrt {c\,x^2+a}+\frac {B\,x\,{\left (c\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {c\,x^2}{a}\right )}{{\left (\frac {c\,x^2}{a}+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.92, size = 218, normalized size = 2.06 \begin {gather*} - A a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )} + \frac {A a^{2}}{\sqrt {c} x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {A a \sqrt {c} x}{\sqrt {\frac {a}{c x^{2}} + 1}} + A c \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + \frac {B a^{\frac {3}{2}} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {B a^{\frac {3}{2}} x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 B \sqrt {a} c x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + \frac {B c^{2} x^{5}}{4 \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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