Optimal. Leaf size=71 \[ \frac {(b c-a d)^2 \left (c+d x^2\right )^4}{8 d^3}-\frac {b (b c-a d) \left (c+d x^2\right )^5}{5 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{12 d^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 45}
\begin {gather*} -\frac {b \left (c+d x^2\right )^5 (b c-a d)}{5 d^3}+\frac {\left (c+d x^2\right )^4 (b c-a d)^2}{8 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{12 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\frac {1}{2} \text {Subst}\left (\int (a+b x)^2 (c+d x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {(-b c+a d)^2 (c+d x)^3}{d^2}-\frac {2 b (b c-a d) (c+d x)^4}{d^2}+\frac {b^2 (c+d x)^5}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2 \left (c+d x^2\right )^4}{8 d^3}-\frac {b (b c-a d) \left (c+d x^2\right )^5}{5 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{12 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 119, normalized size = 1.68 \begin {gather*} \frac {1}{120} x^2 \left (60 a^2 c^3+30 a c^2 (2 b c+3 a d) x^2+20 c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+15 d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+12 b d^2 (3 b c+2 a d) x^8+10 b^2 d^3 x^{10}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 128, normalized size = 1.80
method | result | size |
norman | \(\frac {b^{2} d^{3} x^{12}}{12}+\left (\frac {1}{5} a b \,d^{3}+\frac {3}{10} b^{2} c \,d^{2}\right ) x^{10}+\left (\frac {1}{8} a^{2} d^{3}+\frac {3}{4} a b c \,d^{2}+\frac {3}{8} b^{2} c^{2} d \right ) x^{8}+\left (\frac {1}{2} a^{2} c \,d^{2}+a b \,c^{2} d +\frac {1}{6} b^{2} c^{3}\right ) x^{6}+\left (\frac {3}{4} a^{2} c^{2} d +\frac {1}{2} a b \,c^{3}\right ) x^{4}+\frac {a^{2} c^{3} x^{2}}{2}\) | \(125\) |
default | \(\frac {b^{2} d^{3} x^{12}}{12}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{10}}{10}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{8}}{8}+\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{6}}{6}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{4}}{4}+\frac {a^{2} c^{3} x^{2}}{2}\) | \(128\) |
gosper | \(\frac {1}{12} b^{2} d^{3} x^{12}+\frac {1}{5} x^{10} a b \,d^{3}+\frac {3}{10} x^{10} b^{2} c \,d^{2}+\frac {1}{8} x^{8} a^{2} d^{3}+\frac {3}{4} x^{8} a b c \,d^{2}+\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {1}{2} x^{6} a^{2} c \,d^{2}+x^{6} a b \,c^{2} d +\frac {1}{6} x^{6} b^{2} c^{3}+\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {1}{2} a^{2} c^{3} x^{2}\) | \(135\) |
risch | \(\frac {1}{12} b^{2} d^{3} x^{12}+\frac {1}{5} x^{10} a b \,d^{3}+\frac {3}{10} x^{10} b^{2} c \,d^{2}+\frac {1}{8} x^{8} a^{2} d^{3}+\frac {3}{4} x^{8} a b c \,d^{2}+\frac {3}{8} x^{8} b^{2} c^{2} d +\frac {1}{2} x^{6} a^{2} c \,d^{2}+x^{6} a b \,c^{2} d +\frac {1}{6} x^{6} b^{2} c^{3}+\frac {3}{4} x^{4} a^{2} c^{2} d +\frac {1}{2} x^{4} a b \,c^{3}+\frac {1}{2} a^{2} c^{3} x^{2}\) | \(135\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.40, size = 127, normalized size = 1.79 \begin {gather*} \frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {1}{10} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + \frac {1}{2} \, a^{2} c^{3} x^{2} + \frac {1}{6} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 127, normalized size = 1.79 \begin {gather*} \frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {1}{10} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + \frac {1}{2} \, a^{2} c^{3} x^{2} + \frac {1}{6} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \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. 136 vs.
\(2 (60) = 120\).
time = 0.02, size = 136, normalized size = 1.92 \begin {gather*} \frac {a^{2} c^{3} x^{2}}{2} + \frac {b^{2} d^{3} x^{12}}{12} + x^{10} \left (\frac {a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{10}\right ) + x^{8} \left (\frac {a^{2} d^{3}}{8} + \frac {3 a b c d^{2}}{4} + \frac {3 b^{2} c^{2} d}{8}\right ) + x^{6} \left (\frac {a^{2} c d^{2}}{2} + a b c^{2} d + \frac {b^{2} c^{3}}{6}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c^{2} d}{4} + \frac {a b c^{3}}{2}\right ) \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. 134 vs.
\(2 (65) = 130\).
time = 1.32, size = 134, normalized size = 1.89 \begin {gather*} \frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {3}{10} \, b^{2} c d^{2} x^{10} + \frac {1}{5} \, a b d^{3} x^{10} + \frac {3}{8} \, b^{2} c^{2} d x^{8} + \frac {3}{4} \, a b c d^{2} x^{8} + \frac {1}{8} \, a^{2} d^{3} x^{8} + \frac {1}{6} \, b^{2} c^{3} x^{6} + a b c^{2} d x^{6} + \frac {1}{2} \, a^{2} c d^{2} x^{6} + \frac {1}{2} \, a b c^{3} x^{4} + \frac {3}{4} \, a^{2} c^{2} d x^{4} + \frac {1}{2} \, a^{2} c^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 118, normalized size = 1.66 \begin {gather*} x^6\,\left (\frac {a^2\,c\,d^2}{2}+a\,b\,c^2\,d+\frac {b^2\,c^3}{6}\right )+x^8\,\left (\frac {a^2\,d^3}{8}+\frac {3\,a\,b\,c\,d^2}{4}+\frac {3\,b^2\,c^2\,d}{8}\right )+\frac {a^2\,c^3\,x^2}{2}+\frac {b^2\,d^3\,x^{12}}{12}+\frac {a\,c^2\,x^4\,\left (3\,a\,d+2\,b\,c\right )}{4}+\frac {b\,d^2\,x^{10}\,\left (2\,a\,d+3\,b\,c\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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