Optimal. Leaf size=96 \[ -\frac {A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}-\frac {(9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}+\frac {2 c (9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}} \]
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Rubi [A]
time = 0.15, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2059, 806, 672,
664} \begin {gather*} \frac {2 c \left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{315 b^3 x^{10}}-\frac {\left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{63 b^2 x^{12}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rule 806
Rule 2059
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}+\frac {\left (-7 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \text {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{9 b}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}-\frac {(9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac {(c (9 b B-4 A c)) \text {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{63 b^2}\\ &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}-\frac {(9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}+\frac {2 c (9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 66, normalized size = 0.69 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (9 b B x^2 \left (-5 b+2 c x^2\right )+A \left (-35 b^2+20 b c x^2-8 c^2 x^4\right )\right )}{315 b^3 x^{14}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 70, normalized size = 0.73
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (8 A \,c^{2} x^{4}-18 x^{4} b B c -20 A b c \,x^{2}+45 b^{2} B \,x^{2}+35 b^{2} A \right ) \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}}}{315 b^{3} x^{12}}\) | \(70\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (8 A \,c^{2} x^{4}-18 x^{4} b B c -20 A b c \,x^{2}+45 b^{2} B \,x^{2}+35 b^{2} A \right ) \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}}}{315 b^{3} x^{12}}\) | \(70\) |
trager | \(-\frac {\left (8 A \,c^{4} x^{8}-18 B b \,c^{3} x^{8}-4 A b \,c^{3} x^{6}+9 B \,b^{2} c^{2} x^{6}+3 A \,b^{2} c^{2} x^{4}+72 B \,b^{3} c \,x^{4}+50 A \,b^{3} c \,x^{2}+45 B \,b^{4} x^{2}+35 A \,b^{4}\right ) \sqrt {x^{4} c +b \,x^{2}}}{315 b^{3} x^{10}}\) | \(111\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (8 A \,c^{4} x^{8}-18 B b \,c^{3} x^{8}-4 A b \,c^{3} x^{6}+9 B \,b^{2} c^{2} x^{6}+3 A \,b^{2} c^{2} x^{4}+72 B \,b^{3} c \,x^{4}+50 A \,b^{3} c \,x^{2}+45 B \,b^{4} x^{2}+35 A \,b^{4}\right )}{315 x^{10} b^{3}}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 241 vs.
\(2 (84) = 168\).
time = 0.29, size = 241, normalized size = 2.51 \begin {gather*} \frac {1}{140} \, B {\left (\frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{2} x^{2}} - \frac {4 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} c}{x^{6}} + \frac {15 \, \sqrt {c x^{4} + b x^{2}} b}{x^{8}} - \frac {35 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{10}}\right )} - \frac {1}{630} \, A {\left (\frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{3} x^{2}} - \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{2} x^{4}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b x^{6}} - \frac {5 \, \sqrt {c x^{4} + b x^{2}} c}{x^{8}} - \frac {35 \, \sqrt {c x^{4} + b x^{2}} b}{x^{10}} + \frac {105 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{12}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.75, size = 109, normalized size = 1.14 \begin {gather*} \frac {{\left (2 \, {\left (9 \, B b c^{3} - 4 \, A c^{4}\right )} x^{8} - {\left (9 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} x^{6} - 35 \, A b^{4} - 3 \, {\left (24 \, B b^{3} c + A b^{2} c^{2}\right )} x^{4} - 5 \, {\left (9 \, B b^{4} + 10 \, A b^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, b^{3} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{13}}\, dx \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. 430 vs.
\(2 (84) = 168\).
time = 2.77, size = 430, normalized size = 4.48 \begin {gather*} \frac {4 \, {\left (315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{14} B c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B b c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} A c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b^{2} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 1260 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A b c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 819 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{3} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 1764 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b^{2} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 441 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{4} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{3} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{5} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{4} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 81 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{6} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{5} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 9 \, B b^{7} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 4 \, A b^{6} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right )\right )}}{315 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.44, size = 206, normalized size = 2.15 \begin {gather*} \frac {4\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{315\,b^2\,x^4}-\frac {10\,A\,c\,\sqrt {c\,x^4+b\,x^2}}{63\,x^8}-\frac {B\,b\,\sqrt {c\,x^4+b\,x^2}}{7\,x^8}-\frac {8\,B\,c\,\sqrt {c\,x^4+b\,x^2}}{35\,x^6}-\frac {A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b\,x^6}-\frac {A\,b\,\sqrt {c\,x^4+b\,x^2}}{9\,x^{10}}-\frac {8\,A\,c^4\,\sqrt {c\,x^4+b\,x^2}}{315\,b^3\,x^2}-\frac {B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{35\,b\,x^4}+\frac {2\,B\,c^3\,\sqrt {c\,x^4+b\,x^2}}{35\,b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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