Optimal. Leaf size=136 \[ a^{5/2} (-B) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {15}{8} a^2 A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {1}{8} a \sqrt {a+c x^2} (8 a B+15 A c x)-\frac {\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac {1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \]
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Rubi [A] time = 0.12, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {813, 815, 844, 217, 206, 266, 63, 208} \begin {gather*} \frac {15}{8} a^2 A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+a^{5/2} (-B) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {1}{8} a \sqrt {a+c x^2} (8 a B+15 A c x)-\frac {\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac {1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 813
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^2} \, dx &=-\frac {(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}-\frac {1}{2} \int \frac {(-2 a B-10 A c x) \left (a+c x^2\right )^{3/2}}{x} \, dx\\ &=\frac {1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}-\frac {\int \frac {\left (-8 a^2 B c-30 a A c^2 x\right ) \sqrt {a+c x^2}}{x} \, dx}{8 c}\\ &=\frac {1}{8} a (8 a B+15 A c x) \sqrt {a+c x^2}+\frac {1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}-\frac {\int \frac {-16 a^3 B c^2-30 a^2 A c^3 x}{x \sqrt {a+c x^2}} \, dx}{16 c^2}\\ &=\frac {1}{8} a (8 a B+15 A c x) \sqrt {a+c x^2}+\frac {1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\left (a^3 B\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\frac {1}{8} \left (15 a^2 A c\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=\frac {1}{8} a (8 a B+15 A c x) \sqrt {a+c x^2}+\frac {1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\frac {1}{2} \left (a^3 B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\frac {1}{8} \left (15 a^2 A c\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 B+15 A c x) \sqrt {a+c x^2}+\frac {1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\frac {15}{8} a^2 A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {\left (a^3 B\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 B+15 A c x) \sqrt {a+c x^2}+\frac {1}{12} (4 a B+15 A c x) \left (a+c x^2\right )^{3/2}-\frac {(5 A-B x) \left (a+c x^2\right )^{5/2}}{5 x}+\frac {15}{8} a^2 A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-a^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.21, size = 117, normalized size = 0.86 \begin {gather*} -a^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {a^3 A \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};-\frac {c x^2}{a}\right )}{x \sqrt {a+c x^2}}+\frac {1}{15} B \sqrt {a+c x^2} \left (23 a^2+11 a c x^2+3 c^2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 141, normalized size = 1.04 \begin {gather*} 2 a^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {\sqrt {a+c x^2} \left (-120 a^2 A+184 a^2 B x+135 a A c x^2+88 a B c x^3+30 A c^2 x^4+24 B c^2 x^5\right )}{120 x}-\frac {15}{8} a^2 A \sqrt {c} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 519, normalized size = 3.82 \begin {gather*} \left [\frac {225 \, A a^{2} \sqrt {c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 120 \, B a^{\frac {5}{2}} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, x}, -\frac {225 \, A a^{2} \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 60 \, B a^{\frac {5}{2}} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, x}, \frac {240 \, B \sqrt {-a} a^{2} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 225 \, A a^{2} \sqrt {c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, x}, -\frac {225 \, A a^{2} \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 120 \, B \sqrt {-a} a^{2} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 150, normalized size = 1.10 \begin {gather*} \frac {2 \, B a^{3} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {15}{8} \, A a^{2} \sqrt {c} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {2 \, A a^{3} \sqrt {c}}{{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a} + \frac {1}{120} \, {\left (184 \, B a^{2} + {\left (135 \, A a c + 2 \, {\left (44 \, B a c + 3 \, {\left (4 \, B c^{2} x + 5 \, A c^{2}\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 158, normalized size = 1.16 \begin {gather*} \frac {15 A \,a^{2} \sqrt {c}\, \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8}-B \,a^{\frac {5}{2}} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )+\frac {15 \sqrt {c \,x^{2}+a}\, A a c x}{8}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A c x}{4}+\sqrt {c \,x^{2}+a}\, B \,a^{2}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A c x}{a}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B a}{3}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B}{5}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 120, normalized size = 0.88 \begin {gather*} \frac {5}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c x + \frac {15}{8} \, \sqrt {c x^{2} + a} A a c x + \frac {15}{8} \, A a^{2} \sqrt {c} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - B a^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \frac {1}{5} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B + \frac {1}{3} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B a + \sqrt {c x^{2} + a} B a^{2} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.23, size = 104, normalized size = 0.76 \begin {gather*} \frac {B\,{\left (c\,x^2+a\right )}^{5/2}}{5}+B\,a^2\,\sqrt {c\,x^2+a}+\frac {B\,a\,{\left (c\,x^2+a\right )}^{3/2}}{3}-\frac {A\,{\left (c\,x^2+a\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {c\,x^2}{a}\right )}{x\,{\left (\frac {c\,x^2}{a}+1\right )}^{5/2}}+B\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.72, size = 318, normalized size = 2.34 \begin {gather*} - \frac {A a^{\frac {5}{2}}}{x \sqrt {1 + \frac {c x^{2}}{a}}} + A a^{\frac {3}{2}} c x \sqrt {1 + \frac {c x^{2}}{a}} - \frac {7 A a^{\frac {3}{2}} c x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A \sqrt {a} c^{2} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {15 A a^{2} \sqrt {c} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8} + \frac {A c^{3} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} - B a^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )} + \frac {B a^{3}}{\sqrt {c} x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B a^{2} \sqrt {c} x}{\sqrt {\frac {a}{c x^{2}} + 1}} + 2 B 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 ) + B c^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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