Optimal. Leaf size=163 \[ -\frac {c^2 d \left (10 c d^2+9 a e^2\right ) x}{e^6}+\frac {3 c^2 \left (2 c d^2+a e^2\right ) x^2}{2 e^5}-\frac {c^3 d x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {6 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.11, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711}
\begin {gather*} -\frac {c^2 d x \left (9 a e^2+10 c d^2\right )}{e^6}+\frac {3 c^2 x^2 \left (a e^2+2 c d^2\right )}{2 e^5}+\frac {6 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)}-\frac {\left (a e^2+c d^2\right )^3}{2 e^7 (d+e x)^2}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac {c^3 d x^3}{e^4}+\frac {c^3 x^4}{4 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx &=\int \left (-\frac {c^2 d \left (10 c d^2+9 a e^2\right )}{e^6}+\frac {3 c^2 \left (2 c d^2+a e^2\right ) x}{e^5}-\frac {3 c^3 d x^2}{e^4}+\frac {c^3 x^3}{e^3}+\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^3}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^2}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {c^2 d \left (10 c d^2+9 a e^2\right ) x}{e^6}+\frac {3 c^2 \left (2 c d^2+a e^2\right ) x^2}{2 e^5}-\frac {c^3 d x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {6 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 198, normalized size = 1.21 \begin {gather*} \frac {-2 a^3 e^6+6 a^2 c d e^4 (3 d+4 e x)+6 a c^2 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+c^3 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )+12 c \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right ) (d+e x)^2 \log (d+e x)}{4 e^7 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 191, normalized size = 1.17
method | result | size |
default | \(-\frac {c^{2} \left (-\frac {1}{4} c \,x^{4} e^{3}+c d \,e^{2} x^{3}-\frac {3}{2} a \,e^{3} x^{2}-3 c \,d^{2} e \,x^{2}+9 a d \,e^{2} x +10 c \,d^{3} x \right )}{e^{6}}+\frac {6 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{7} \left (e x +d \right )}+\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{2 e^{7} \left (e x +d \right )^{2}}\) | \(191\) |
norman | \(\frac {-\frac {e^{6} a^{3}-9 e^{4} d^{2} a^{2} c -54 d^{4} e^{2} c^{2} a -45 d^{6} c^{3}}{2 e^{7}}+\frac {c^{3} x^{6}}{4 e}+\frac {c^{2} \left (6 e^{2} a +5 c \,d^{2}\right ) x^{4}}{4 e^{3}}-\frac {c^{3} d \,x^{5}}{2 e^{2}}+\frac {2 d \left (3 e^{4} a^{2} c +18 d^{2} e^{2} c^{2} a +15 d^{4} c^{3}\right ) x}{e^{6}}-\frac {c^{2} d \left (6 e^{2} a +5 c \,d^{2}\right ) x^{3}}{e^{4}}}{\left (e x +d \right )^{2}}+\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(203\) |
risch | \(\frac {c^{3} x^{4}}{4 e^{3}}-\frac {c^{3} d \,x^{3}}{e^{4}}+\frac {3 c^{2} a \,x^{2}}{2 e^{3}}+\frac {3 c^{3} d^{2} x^{2}}{e^{5}}-\frac {9 c^{2} a d x}{e^{4}}-\frac {10 c^{3} d^{3} x}{e^{6}}+\frac {\left (6 d \,e^{4} a^{2} c +12 d^{3} e^{2} c^{2} a +6 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}-9 e^{4} d^{2} a^{2} c -21 d^{4} e^{2} c^{2} a -11 d^{6} c^{3}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {3 c \ln \left (e x +d \right ) a^{2}}{e^{3}}+\frac {18 c^{2} \ln \left (e x +d \right ) a \,d^{2}}{e^{5}}+\frac {15 c^{3} \ln \left (e x +d \right ) d^{4}}{e^{7}}\) | \(214\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 201, normalized size = 1.23 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{4} \, {\left (c^{3} x^{4} e^{3} - 4 \, c^{3} d x^{3} e^{2} + 6 \, {\left (2 \, c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{2} - 4 \, {\left (10 \, c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x\right )} e^{\left (-6\right )} + \frac {11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} e^{7}\right )}} \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. 297 vs.
\(2 (147) = 294\).
time = 2.13, size = 297, normalized size = 1.82 \begin {gather*} -\frac {16 \, c^{3} d^{5} x e - 22 \, c^{3} d^{6} - {\left (c^{3} x^{6} + 6 \, a c^{2} x^{4} - 2 \, a^{3}\right )} e^{6} + 2 \, {\left (c^{3} d x^{5} + 12 \, a c^{2} d x^{3} - 12 \, a^{2} c d x\right )} e^{5} - {\left (5 \, c^{3} d^{2} x^{4} - 66 \, a c^{2} d^{2} x^{2} + 18 \, a^{2} c d^{2}\right )} e^{4} + 4 \, {\left (5 \, c^{3} d^{3} x^{3} - 3 \, a c^{2} d^{3} x\right )} e^{3} + 2 \, {\left (34 \, c^{3} d^{4} x^{2} - 21 \, a c^{2} d^{4}\right )} e^{2} - 12 \, {\left (10 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 12 \, a c^{2} d^{3} x e^{3} + a^{2} c x^{2} e^{6} + 2 \, a^{2} c d x e^{5} + {\left (6 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{4} + {\left (5 \, c^{3} d^{4} x^{2} + 6 \, a c^{2} d^{4}\right )} e^{2}\right )} \log \left (x e + d\right )}{4 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.64, size = 218, normalized size = 1.34 \begin {gather*} - \frac {c^{3} d x^{3}}{e^{4}} + \frac {c^{3} x^{4}}{4 e^{3}} + \frac {3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x^{2} \cdot \left (\frac {3 a c^{2}}{2 e^{3}} + \frac {3 c^{3} d^{2}}{e^{5}}\right ) + x \left (- \frac {9 a c^{2} d}{e^{4}} - \frac {10 c^{3} d^{3}}{e^{6}}\right ) + \frac {- a^{3} e^{6} + 9 a^{2} c d^{2} e^{4} + 21 a c^{2} d^{4} e^{2} + 11 c^{3} d^{6} + x \left (12 a^{2} c d e^{5} + 24 a c^{2} d^{3} e^{3} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 192, normalized size = 1.18 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{4} \, {\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 6 \, a c^{2} x^{2} e^{9} - 36 \, a c^{2} d x e^{8}\right )} e^{\left (-12\right )} + \frac {{\left (11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 235, normalized size = 1.44 \begin {gather*} x^2\,\left (\frac {3\,a\,c^2}{2\,e^3}+\frac {3\,c^3\,d^2}{e^5}\right )+\frac {\frac {-a^3\,e^6+9\,a^2\,c\,d^2\,e^4+21\,a\,c^2\,d^4\,e^2+11\,c^3\,d^6}{2\,e}+x\,\left (6\,a^2\,c\,d\,e^4+12\,a\,c^2\,d^3\,e^2+6\,c^3\,d^5\right )}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x\,\left (\frac {8\,c^3\,d^3}{e^6}-\frac {3\,d\,\left (\frac {3\,a\,c^2}{e^3}+\frac {6\,c^3\,d^2}{e^5}\right )}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,e^4+18\,a\,c^2\,d^2\,e^2+15\,c^3\,d^4\right )}{e^7}+\frac {c^3\,x^4}{4\,e^3}-\frac {c^3\,d\,x^3}{e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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