Optimal. Leaf size=143 \[ -\frac {5 c \sqrt {a+c x^2} (4 a B-3 A c x)}{8 x}-\frac {\left (a+c x^2\right )^{5/2} (A-2 B x)}{4 x^4}-\frac {5 \left (a+c x^2\right )^{3/2} (4 a B+3 A c x)}{24 x^3}-\frac {15}{8} \sqrt {a} A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {5}{2} a B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 143, 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, 811, 844, 217, 206, 266, 63, 208} \begin {gather*} -\frac {\left (a+c x^2\right )^{5/2} (A-2 B x)}{4 x^4}-\frac {5 \left (a+c x^2\right )^{3/2} (4 a B+3 A c x)}{24 x^3}-\frac {5 c \sqrt {a+c x^2} (4 a B-3 A c x)}{8 x}-\frac {15}{8} \sqrt {a} A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {5}{2} a B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 811
Rule 813
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^5} \, dx &=-\frac {(A-2 B x) \left (a+c x^2\right )^{5/2}}{4 x^4}-\frac {5}{16} \int \frac {(-8 a B-4 A c x) \left (a+c x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {5 (4 a B+3 A c x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac {(A-2 B x) \left (a+c x^2\right )^{5/2}}{4 x^4}+\frac {5 \int \frac {\left (32 a^2 B c+24 a A c^2 x\right ) \sqrt {a+c x^2}}{x^2} \, dx}{64 a}\\ &=-\frac {5 c (4 a B-3 A c x) \sqrt {a+c x^2}}{8 x}-\frac {5 (4 a B+3 A c x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac {(A-2 B x) \left (a+c x^2\right )^{5/2}}{4 x^4}-\frac {5 \int \frac {-48 a^2 A c^2-64 a^2 B c^2 x}{x \sqrt {a+c x^2}} \, dx}{128 a}\\ &=-\frac {5 c (4 a B-3 A c x) \sqrt {a+c x^2}}{8 x}-\frac {5 (4 a B+3 A c x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac {(A-2 B x) \left (a+c x^2\right )^{5/2}}{4 x^4}+\frac {1}{8} \left (15 a A c^2\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\frac {1}{2} \left (5 a B c^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=-\frac {5 c (4 a B-3 A c x) \sqrt {a+c x^2}}{8 x}-\frac {5 (4 a B+3 A c x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac {(A-2 B x) \left (a+c x^2\right )^{5/2}}{4 x^4}+\frac {1}{16} \left (15 a A c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\frac {1}{2} \left (5 a B c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=-\frac {5 c (4 a B-3 A c x) \sqrt {a+c x^2}}{8 x}-\frac {5 (4 a B+3 A c x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac {(A-2 B x) \left (a+c x^2\right )^{5/2}}{4 x^4}+\frac {5}{2} a B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {1}{8} (15 a A c) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )\\ &=-\frac {5 c (4 a B-3 A c x) \sqrt {a+c x^2}}{8 x}-\frac {5 (4 a B+3 A c x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac {(A-2 B x) \left (a+c x^2\right )^{5/2}}{4 x^4}+\frac {5}{2} a B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {15}{8} \sqrt {a} A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 96, normalized size = 0.67 \begin {gather*} -\frac {A c^2 \left (a+c x^2\right )^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {c x^2}{a}+1\right )}{7 a^3}-\frac {a^2 B \sqrt {a+c x^2} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x^2}{a}\right )}{3 x^3 \sqrt {\frac {c x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 144, normalized size = 1.01 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-6 a^2 A-8 a^2 B x-27 a A c x^2-56 a B c x^3+24 A c^2 x^4+12 B c^2 x^5\right )}{24 x^4}+\frac {15}{4} \sqrt {a} A c^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {5}{2} a B c^{3/2} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 534, normalized size = 3.73 \begin {gather*} \left [\frac {60 \, B a c^{\frac {3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 45 \, A \sqrt {a} c^{2} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{48 \, x^{4}}, -\frac {120 \, B a \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 45 \, A \sqrt {a} c^{2} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{48 \, x^{4}}, \frac {45 \, A \sqrt {-a} c^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 30 \, B a c^{\frac {3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{24 \, x^{4}}, -\frac {60 \, B a \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 45 \, A \sqrt {-a} c^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{24 \, x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 316, normalized size = 2.21 \begin {gather*} \frac {15 \, A a c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} - \frac {5}{2} \, B a c^{\frac {3}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {1}{2} \, {\left (B c^{2} x + 2 \, A c^{2}\right )} \sqrt {c x^{2} + a} + \frac {27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A a c^{2} + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} B a^{2} c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a^{2} c^{2} - 168 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{3} c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{3} c^{2} + 152 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{4} c^{\frac {3}{2}} + 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{4} c^{2} - 56 \, B a^{5} c^{\frac {3}{2}}}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 236, normalized size = 1.65 \begin {gather*} -\frac {15 A \sqrt {a}\, c^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8}+\frac {5 B a \,c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2}+\frac {5 \sqrt {c \,x^{2}+a}\, B \,c^{2} x}{2}+\frac {15 \sqrt {c \,x^{2}+a}\, A \,c^{2}}{8}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,c^{2} x}{3 a}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{2}}{8 a}+\frac {4 \left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,c^{2} x}{3 a^{2}}+\frac {3 \left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,c^{2}}{8 a^{2}}-\frac {4 \left (c \,x^{2}+a \right )^{\frac {7}{2}} B c}{3 a^{2} x}-\frac {3 \left (c \,x^{2}+a \right )^{\frac {7}{2}} A c}{8 a^{2} x^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{3 a \,x^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 198, normalized size = 1.38 \begin {gather*} \frac {5}{2} \, \sqrt {c x^{2} + a} B c^{2} x + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{2} x}{3 \, a} + \frac {5}{2} \, B a c^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - \frac {15}{8} \, A \sqrt {a} c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \frac {15}{8} \, \sqrt {c x^{2} + a} A c^{2} + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} A c^{2}}{8 \, a^{2}} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c^{2}}{8 \, a} - \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B c}{3 \, a x} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{8 \, a^{2} x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{3 \, a x^{3}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.86, size = 299, normalized size = 2.09 \begin {gather*} - \frac {15 A \sqrt {a} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8} - \frac {A a^{3}}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 A a^{2} \sqrt {c}}{8 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {A a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{x} + \frac {7 A a c^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {A c^{\frac {5}{2}} x}{\sqrt {\frac {a}{c x^{2}} + 1}} - \frac {2 B a^{\frac {3}{2}} c}{x \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {B \sqrt {a} c^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} - \frac {2 B \sqrt {a} c^{2} x}{\sqrt {1 + \frac {c x^{2}}{a}}} - \frac {B a^{2} \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 x^{2}} - \frac {B a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3} + \frac {5 B a c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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