Optimal. Leaf size=115 \[ -\frac {5 B c^2 \sqrt {a+c x^2}}{16 x^2}-\frac {5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {5 B c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \]
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Rubi [A]
time = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {821, 272, 43,
65, 214} \begin {gather*} -\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {5 B c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {5 B c^2 \sqrt {a+c x^2}}{16 x^2}-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {5 B c \left (a+c x^2\right )^{3/2}}{24 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx &=-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}+B \int \frac {\left (a+c x^2\right )^{5/2}}{x^7} \, dx\\ &=-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac {1}{2} B \text {Subst}\left (\int \frac {(a+c x)^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac {1}{12} (5 B c) \text {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac {1}{16} \left (5 B c^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {5 B c^2 \sqrt {a+c x^2}}{16 x^2}-\frac {5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac {1}{32} \left (5 B c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )\\ &=-\frac {5 B c^2 \sqrt {a+c x^2}}{16 x^2}-\frac {5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac {1}{16} \left (5 B c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )\\ &=-\frac {5 B c^2 \sqrt {a+c x^2}}{16 x^2}-\frac {5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {5 B c^3 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 120, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {a+c x^2} \left (48 A c^3 x^6+8 a^3 (6 A+7 B x)+3 a c^2 x^4 (48 A+77 B x)+2 a^2 c x^2 (72 A+91 B x)\right )}{336 a x^7}+\frac {5 B c^3 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 160, normalized size = 1.39
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (48 A \,c^{3} x^{6}+231 a B \,c^{2} x^{5}+144 a A \,c^{2} x^{4}+182 a^{2} B c \,x^{3}+144 a^{2} A c \,x^{2}+56 B \,a^{3} x +48 A \,a^{3}\right )}{336 x^{7} a}-\frac {5 B \,c^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{16 \sqrt {a}}\) | \(114\) |
default | \(B \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 c \left (\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {c \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )-\frac {A \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 a \,x^{7}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 152, normalized size = 1.32 \begin {gather*} -\frac {5 \, B c^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{16 \, \sqrt {a}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c^{3}}{16 \, a^{3}} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{3}}{48 \, a^{2}} + \frac {5 \, \sqrt {c x^{2} + a} B c^{3}}{16 \, a} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{2}}{16 \, a^{3} x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c}{24 \, a^{2} x^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{6 \, a x^{6}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{7 \, a x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.53, size = 238, normalized size = 2.07 \begin {gather*} \left [\frac {105 \, B \sqrt {a} c^{3} x^{7} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{672 \, a x^{7}}, \frac {105 \, B \sqrt {-a} c^{3} x^{7} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{336 \, a x^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 605 vs.
\(2 (109) = 218\).
time = 11.17, size = 605, normalized size = 5.26 \begin {gather*} - \frac {15 A a^{7} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {33 A a^{6} c^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {17 A a^{5} c^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {3 A a^{4} c^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {12 A a^{3} c^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {8 A a^{2} c^{\frac {19}{2}} x^{10} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {2 A a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {7 A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 x^{2}} - \frac {A c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a} - \frac {B a^{3}}{6 \sqrt {c} x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {17 B a^{2} \sqrt {c}}{24 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {35 B a c^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {B c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{2 x} - \frac {3 B c^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 B c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{16 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 316 vs.
\(2 (91) = 182\).
time = 0.94, size = 316, normalized size = 2.75 \begin {gather*} \frac {5 \, B c^{3} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a}} + \frac {231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{13} B c^{3} + 336 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{12} A c^{\frac {7}{2}} - 196 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} B a c^{3} + 595 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} B a^{2} c^{3} + 1680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} A a^{2} c^{\frac {7}{2}} - 595 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B a^{4} c^{3} + 1008 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{4} c^{\frac {7}{2}} + 196 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{5} c^{3} - 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{6} c^{3} + 48 \, A a^{6} c^{\frac {7}{2}}}{168 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.39, size = 150, normalized size = 1.30 \begin {gather*} \frac {5\,B\,a\,{\left (c\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {11\,B\,{\left (c\,x^2+a\right )}^{5/2}}{16\,x^6}-\frac {A\,a^2\,\sqrt {c\,x^2+a}}{7\,x^7}-\frac {5\,B\,a^2\,\sqrt {c\,x^2+a}}{16\,x^6}-\frac {3\,A\,c^2\,\sqrt {c\,x^2+a}}{7\,x^3}-\frac {A\,c^3\,\sqrt {c\,x^2+a}}{7\,a\,x}-\frac {3\,A\,a\,c\,\sqrt {c\,x^2+a}}{7\,x^5}+\frac {B\,c^3\,\mathrm {atan}\left (\frac {\sqrt {c\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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