Optimal. Leaf size=137 \[ \frac {(b B-6 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac {A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac {b^2 (b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{3/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2059, 806, 678,
626, 634, 212} \begin {gather*} -\frac {b^2 (b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{3/2}}+\frac {\left (b x^2+c x^4\right )^{3/2} (b B-6 A c)}{6 b}+\frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (b B-6 A c)}{16 c}+\frac {A \left (b x^2+c x^4\right )^{5/2}}{b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 634
Rule 678
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^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=\frac {A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac {\left (-2 (-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} \, dx,x,x^2\right )}{b}\\ &=\frac {(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac {A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac {1}{4} (-b B+6 A c) \text {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {(b B-6 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac {A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac {\left (b^2 (b B-6 A c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{32 c}\\ &=\frac {(b B-6 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac {A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac {\left (b^2 (b B-6 A c)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c}\\ &=\frac {(b B-6 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {(b B-6 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b}+\frac {A \left (b x^2+c x^4\right )^{5/2}}{b x^4}-\frac {b^2 (b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 124, normalized size = 0.91 \begin {gather*} \frac {x \left (\sqrt {c} x \left (b+c x^2\right ) \left (3 b^2 B+4 c^2 x^2 \left (3 A+2 B x^2\right )+2 b c \left (15 A+7 B x^2\right )\right )+3 b^2 (b B-6 A c) \sqrt {b+c x^2} \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{48 c^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 162, normalized size = 1.18
method | result | size |
risch | \(\frac {\left (8 B \,c^{2} x^{4}+12 A \,c^{2} x^{2}+14 b B \,x^{2} c +30 A b c +3 b^{2} B \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{48 c}+\frac {\left (\frac {3 b^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) A}{8 \sqrt {c}}-\frac {b^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) B}{16 c^{\frac {3}{2}}}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x \sqrt {c \,x^{2}+b}}\) | \(135\) |
default | \(\frac {\left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} \left (8 B \sqrt {c}\, \left (c \,x^{2}+b \right )^{\frac {5}{2}} x +12 A \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} x -2 B \left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}\, b x +18 A \sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}} b x -3 B \sqrt {c \,x^{2}+b}\, \sqrt {c}\, b^{2} x +18 A \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{2} c -3 B \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{3}\right )}{48 x^{3} \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}}}\) | \(162\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 168, normalized size = 1.23 \begin {gather*} \frac {1}{16} \, {\left (\frac {3 \, b^{2} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{\sqrt {c}} + 6 \, \sqrt {c x^{4} + b x^{2}} b + \frac {4 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{2}}\right )} A + \frac {1}{96} \, {\left (12 \, \sqrt {c x^{4} + b x^{2}} b x^{2} - \frac {3 \, b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {3}{2}}} + 16 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} b^{2}}{c}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.11, size = 224, normalized size = 1.64 \begin {gather*} \left [-\frac {3 \, {\left (B b^{3} - 6 \, A b^{2} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (8 \, B c^{3} x^{4} + 3 \, B b^{2} c + 30 \, A b c^{2} + 2 \, {\left (7 \, B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, c^{2}}, \frac {3 \, {\left (B b^{3} - 6 \, A b^{2} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (8 \, B c^{3} x^{4} + 3 \, B b^{2} c + 30 \, A b c^{2} + 2 \, {\left (7 \, B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, c^{2}}\right ] \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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 142, normalized size = 1.04 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (4 \, B c x^{2} \mathrm {sgn}\left (x\right ) + \frac {7 \, B b c^{4} \mathrm {sgn}\left (x\right ) + 6 \, A c^{5} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x^{2} + \frac {3 \, {\left (B b^{2} c^{3} \mathrm {sgn}\left (x\right ) + 10 \, A b c^{4} \mathrm {sgn}\left (x\right )\right )}}{c^{4}}\right )} \sqrt {c x^{2} + b} x + \frac {{\left (B b^{3} \mathrm {sgn}\left (x\right ) - 6 \, A b^{2} c \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{16 \, c^{\frac {3}{2}}} - \frac {{\left (B b^{3} \log \left ({\left | b \right |}\right ) - 6 \, A b^{2} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{32 \, c^{\frac {3}{2}}} \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 (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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