Optimal. Leaf size=68 \[ \frac {b (b B-A c) \left (b+c x^2\right )^4}{8 c^3}-\frac {(2 b B-A c) \left (b+c x^2\right )^5}{10 c^3}+\frac {B \left (b+c x^2\right )^6}{12 c^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1598, 457, 77}
\begin {gather*} -\frac {\left (b+c x^2\right )^5 (2 b B-A c)}{10 c^3}+\frac {b \left (b+c x^2\right )^4 (b B-A c)}{8 c^3}+\frac {B \left (b+c x^2\right )^6}{12 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^3} \, dx &=\int x^3 \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int x (A+B x) (b+c x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {b (b B-A c) (b+c x)^3}{c^2}+\frac {(-2 b B+A c) (b+c x)^4}{c^2}+\frac {B (b+c x)^5}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {b (b B-A c) \left (b+c x^2\right )^4}{8 c^3}-\frac {(2 b B-A c) \left (b+c x^2\right )^5}{10 c^3}+\frac {B \left (b+c x^2\right )^6}{12 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 69, normalized size = 1.01 \begin {gather*} \frac {1}{120} x^4 \left (30 A b^3+20 b^2 (b B+3 A c) x^2+45 b c (b B+A c) x^4+12 c^2 (3 b B+A c) x^6+10 B c^3 x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 76, normalized size = 1.12
method | result | size |
default | \(\frac {B \,c^{3} x^{12}}{12}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{10}}{10}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{8}}{8}+\frac {\left (3 A \,b^{2} c +B \,b^{3}\right ) x^{6}}{6}+\frac {A \,b^{3} x^{4}}{4}\) | \(76\) |
risch | \(\frac {1}{12} B \,c^{3} x^{12}+\frac {1}{10} x^{10} A \,c^{3}+\frac {3}{10} x^{10} B b \,c^{2}+\frac {3}{8} x^{8} A b \,c^{2}+\frac {3}{8} x^{8} B \,b^{2} c +\frac {1}{2} x^{6} A \,b^{2} c +\frac {1}{6} b^{3} B \,x^{6}+\frac {1}{4} A \,b^{3} x^{4}\) | \(78\) |
norman | \(\frac {\left (\frac {1}{10} A \,c^{3}+\frac {3}{10} B b \,c^{2}\right ) x^{12}+\left (\frac {3}{8} A b \,c^{2}+\frac {3}{8} B \,b^{2} c \right ) x^{10}+\left (\frac {1}{2} A \,b^{2} c +\frac {1}{6} B \,b^{3}\right ) x^{8}+\frac {A \,b^{3} x^{6}}{4}+\frac {B \,c^{3} x^{14}}{12}}{x^{2}}\) | \(79\) |
gosper | \(\frac {x^{4} \left (10 B \,c^{3} x^{8}+12 A \,c^{3} x^{6}+36 x^{6} B b \,c^{2}+45 A b \,c^{2} x^{4}+45 x^{4} B \,b^{2} c +60 A \,b^{2} c \,x^{2}+20 x^{2} B \,b^{3}+30 A \,b^{3}\right )}{120}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 73, normalized size = 1.07 \begin {gather*} \frac {1}{12} \, B c^{3} x^{12} + \frac {1}{10} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac {3}{8} \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + \frac {1}{4} \, A b^{3} x^{4} + \frac {1}{6} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.84, size = 73, normalized size = 1.07 \begin {gather*} \frac {1}{12} \, B c^{3} x^{12} + \frac {1}{10} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac {3}{8} \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + \frac {1}{4} \, A b^{3} x^{4} + \frac {1}{6} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 82, normalized size = 1.21 \begin {gather*} \frac {A b^{3} x^{4}}{4} + \frac {B c^{3} x^{12}}{12} + x^{10} \left (\frac {A c^{3}}{10} + \frac {3 B b c^{2}}{10}\right ) + x^{8} \cdot \left (\frac {3 A b c^{2}}{8} + \frac {3 B b^{2} c}{8}\right ) + x^{6} \left (\frac {A b^{2} c}{2} + \frac {B b^{3}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 77, normalized size = 1.13 \begin {gather*} \frac {1}{12} \, B c^{3} x^{12} + \frac {3}{10} \, B b c^{2} x^{10} + \frac {1}{10} \, A c^{3} x^{10} + \frac {3}{8} \, B b^{2} c x^{8} + \frac {3}{8} \, A b c^{2} x^{8} + \frac {1}{6} \, B b^{3} x^{6} + \frac {1}{2} \, A b^{2} c x^{6} + \frac {1}{4} \, A b^{3} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 69, normalized size = 1.01 \begin {gather*} x^6\,\left (\frac {B\,b^3}{6}+\frac {A\,c\,b^2}{2}\right )+x^{10}\,\left (\frac {A\,c^3}{10}+\frac {3\,B\,b\,c^2}{10}\right )+\frac {A\,b^3\,x^4}{4}+\frac {B\,c^3\,x^{12}}{12}+\frac {3\,b\,c\,x^8\,\left (A\,c+B\,b\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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