Optimal. Leaf size=17 \[ \frac {\left (a+b x^2\right )^3}{c+d x} \]
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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {1604}
\begin {gather*} \frac {\left (a+b x^2\right )^3}{c+d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 1604
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 (-a d+b x (6 c+5 d x))}{(c+d x)^2} \, dx &=\frac {\left (a+b x^2\right )^3}{c+d x}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(17)=34\).
time = 0.02, size = 90, normalized size = 5.29 \begin {gather*} \frac {a^3 d^6+3 a^2 b d^4 \left (c^2+c d x+d^2 x^2\right )+3 a b^2 d^2 \left (c^4+c^3 d x+d^4 x^4\right )+b^3 \left (c^6+c^5 d x+d^6 x^6\right )}{d^6 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs.
\(2(17)=34\).
time = 0.07, size = 157, normalized size = 9.24
method | result | size |
gosper | \(\frac {b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}{d x +c}\) | \(38\) |
norman | \(\frac {b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}-\frac {d \,a^{3} x}{c}}{d x +c}\) | \(45\) |
default | \(\frac {b \left (b^{2} x^{5} d^{4}-b^{2} c \,x^{4} d^{3}+3 a b \,d^{4} x^{3}+b^{2} c^{2} d^{2} x^{3}-3 a b c \,d^{3} x^{2}-b^{2} c^{3} d \,x^{2}+3 a^{2} d^{4} x +3 a b \,c^{2} d^{2} x +b^{2} c^{4} x \right )}{d^{5}}-\frac {-a^{3} d^{6}-3 a^{2} b \,c^{2} d^{4}-3 a \,b^{2} c^{4} d^{2}-b^{3} c^{6}}{d^{6} \left (d x +c \right )}\) | \(157\) |
risch | \(\frac {b^{3} x^{5}}{d}-\frac {b^{3} c \,x^{4}}{d^{2}}+\frac {3 b^{2} a \,x^{3}}{d}+\frac {b^{3} c^{2} x^{3}}{d^{3}}-\frac {3 b^{2} a c \,x^{2}}{d^{2}}-\frac {b^{3} c^{3} x^{2}}{d^{4}}+\frac {3 b \,a^{2} x}{d}+\frac {3 b^{2} a \,c^{2} x}{d^{3}}+\frac {b^{3} c^{4} x}{d^{5}}+\frac {a^{3}}{d x +c}+\frac {3 a^{2} b \,c^{2}}{d^{2} \left (d x +c \right )}+\frac {3 a \,b^{2} c^{4}}{d^{4} \left (d x +c \right )}+\frac {b^{3} c^{6}}{d^{6} \left (d x +c \right )}\) | \(176\) |
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. 160 vs.
\(2 (17) = 34\).
time = 0.28, size = 160, normalized size = 9.41 \begin {gather*} \frac {b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}}{d^{7} x + c d^{6}} + \frac {b^{3} d^{4} x^{5} - b^{3} c d^{3} x^{4} + {\left (b^{3} c^{2} d^{2} + 3 \, a b^{2} d^{4}\right )} x^{3} - {\left (b^{3} c^{3} d + 3 \, a b^{2} c d^{3}\right )} x^{2} + {\left (b^{3} c^{4} + 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b d^{4}\right )} x}{d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (17) = 34\).
time = 0.33, size = 120, normalized size = 7.06 \begin {gather*} \frac {b^{3} d^{6} x^{6} + 3 \, a b^{2} d^{6} x^{4} + 3 \, a^{2} b d^{6} x^{2} + b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6} + {\left (b^{3} c^{5} d + 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c d^{5}\right )} x}{d^{7} x + c d^{6}} \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. 153 vs.
\(2 (12) = 24\).
time = 0.28, size = 153, normalized size = 9.00 \begin {gather*} - \frac {b^{3} c x^{4}}{d^{2}} + \frac {b^{3} x^{5}}{d} + x^{3} \cdot \left (\frac {3 a b^{2}}{d} + \frac {b^{3} c^{2}}{d^{3}}\right ) + x^{2} \left (- \frac {3 a b^{2} c}{d^{2}} - \frac {b^{3} c^{3}}{d^{4}}\right ) + x \left (\frac {3 a^{2} b}{d} + \frac {3 a b^{2} c^{2}}{d^{3}} + \frac {b^{3} c^{4}}{d^{5}}\right ) + \frac {a^{3} d^{6} + 3 a^{2} b c^{2} d^{4} + 3 a b^{2} c^{4} d^{2} + b^{3} c^{6}}{c d^{6} + d^{7} x} \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. 216 vs.
\(2 (17) = 34\).
time = 3.62, size = 216, normalized size = 12.71 \begin {gather*} \frac {{\left (b^{3} - \frac {6 \, b^{3} c}{d x + c} + \frac {15 \, b^{3} c^{2}}{{\left (d x + c\right )}^{2}} - \frac {20 \, b^{3} c^{3}}{{\left (d x + c\right )}^{3}} + \frac {15 \, b^{3} c^{4}}{{\left (d x + c\right )}^{4}} + \frac {3 \, a b^{2} d^{2}}{{\left (d x + c\right )}^{2}} - \frac {12 \, a b^{2} c d^{2}}{{\left (d x + c\right )}^{3}} + \frac {18 \, a b^{2} c^{2} d^{2}}{{\left (d x + c\right )}^{4}} + \frac {3 \, a^{2} b d^{4}}{{\left (d x + c\right )}^{4}}\right )} {\left (d x + c\right )}^{5}}{d^{6}} + \frac {\frac {b^{3} c^{6} d^{5}}{d x + c} + \frac {3 \, a b^{2} c^{4} d^{7}}{d x + c} + \frac {3 \, a^{2} b c^{2} d^{9}}{d x + c} + \frac {a^{3} d^{11}}{d x + c}}{d^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 252, normalized size = 14.82 \begin {gather*} x^3\,\left (\frac {3\,a\,b^2}{d}+\frac {b^3\,c^2}{d^3}\right )-x\,\left (\frac {2\,c\,\left (\frac {4\,b^3\,c^3}{d^4}-\frac {2\,c\,\left (\frac {9\,a\,b^2}{d}+\frac {3\,b^3\,c^2}{d^3}\right )}{d}+\frac {12\,a\,b^2\,c}{d^2}\right )}{d}+\frac {c^2\,\left (\frac {9\,a\,b^2}{d}+\frac {3\,b^3\,c^2}{d^3}\right )}{d^2}-\frac {3\,a^2\,b}{d}\right )+x^2\,\left (\frac {2\,b^3\,c^3}{d^4}-\frac {c\,\left (\frac {9\,a\,b^2}{d}+\frac {3\,b^3\,c^2}{d^3}\right )}{d}+\frac {6\,a\,b^2\,c}{d^2}\right )+\frac {a^3\,d^6+3\,a^2\,b\,c^2\,d^4+3\,a\,b^2\,c^4\,d^2+b^3\,c^6}{d\,\left (x\,d^6+c\,d^5\right )}+\frac {b^3\,x^5}{d}-\frac {b^3\,c\,x^4}{d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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