Optimal. Leaf size=255 \[ -\frac {c^2 d x^2 \left (6 a^2 e^4+8 a c d^2 e^2+3 c^2 d^4\right )}{e^7}+\frac {c^2 x^3 \left (6 a^2 e^4+12 a c d^2 e^2+5 c^2 d^4\right )}{3 e^6}+\frac {c x \left (4 a^3 e^6+18 a^2 c d^2 e^4+20 a c^2 d^4 e^2+7 c^3 d^6\right )}{e^8}-\frac {c^3 d x^4 \left (2 a e^2+c d^2\right )}{e^5}+\frac {c^3 x^5 \left (4 a e^2+3 c d^2\right )}{5 e^4}-\frac {\left (a e^2+c d^2\right )^4}{e^9 (d+e x)}-\frac {8 c d \left (a e^2+c d^2\right )^3 \log (d+e x)}{e^9}-\frac {c^4 d x^6}{3 e^3}+\frac {c^4 x^7}{7 e^2} \]
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Rubi [A] time = 0.26, antiderivative size = 255, 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 = {697} \[ \frac {c^2 x^3 \left (6 a^2 e^4+12 a c d^2 e^2+5 c^2 d^4\right )}{3 e^6}-\frac {c^2 d x^2 \left (6 a^2 e^4+8 a c d^2 e^2+3 c^2 d^4\right )}{e^7}+\frac {c x \left (18 a^2 c d^2 e^4+4 a^3 e^6+20 a c^2 d^4 e^2+7 c^3 d^6\right )}{e^8}+\frac {c^3 x^5 \left (4 a e^2+3 c d^2\right )}{5 e^4}-\frac {c^3 d x^4 \left (2 a e^2+c d^2\right )}{e^5}-\frac {\left (a e^2+c d^2\right )^4}{e^9 (d+e x)}-\frac {8 c d \left (a e^2+c d^2\right )^3 \log (d+e x)}{e^9}-\frac {c^4 d x^6}{3 e^3}+\frac {c^4 x^7}{7 e^2} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx &=\int \left (\frac {c \left (7 c^3 d^6+20 a c^2 d^4 e^2+18 a^2 c d^2 e^4+4 a^3 e^6\right )}{e^8}-\frac {2 c^2 d \left (3 c^2 d^4+8 a c d^2 e^2+6 a^2 e^4\right ) x}{e^7}+\frac {c^2 \left (5 c^2 d^4+12 a c d^2 e^2+6 a^2 e^4\right ) x^2}{e^6}-\frac {4 c^3 d \left (c d^2+2 a e^2\right ) x^3}{e^5}+\frac {c^3 \left (3 c d^2+4 a e^2\right ) x^4}{e^4}-\frac {2 c^4 d x^5}{e^3}+\frac {c^4 x^6}{e^2}+\frac {\left (c d^2+a e^2\right )^4}{e^8 (d+e x)^2}-\frac {8 c d \left (c d^2+a e^2\right )^3}{e^8 (d+e x)}\right ) \, dx\\ &=\frac {c \left (7 c^3 d^6+20 a c^2 d^4 e^2+18 a^2 c d^2 e^4+4 a^3 e^6\right ) x}{e^8}-\frac {c^2 d \left (3 c^2 d^4+8 a c d^2 e^2+6 a^2 e^4\right ) x^2}{e^7}+\frac {c^2 \left (5 c^2 d^4+12 a c d^2 e^2+6 a^2 e^4\right ) x^3}{3 e^6}-\frac {c^3 d \left (c d^2+2 a e^2\right ) x^4}{e^5}+\frac {c^3 \left (3 c d^2+4 a e^2\right ) x^5}{5 e^4}-\frac {c^4 d x^6}{3 e^3}+\frac {c^4 x^7}{7 e^2}-\frac {\left (c d^2+a e^2\right )^4}{e^9 (d+e x)}-\frac {8 c d \left (c d^2+a e^2\right )^3 \log (d+e x)}{e^9}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 289, normalized size = 1.13 \[ \frac {-105 a^4 e^8+420 a^3 c e^6 \left (-d^2+d e x+e^2 x^2\right )+210 a^2 c^2 e^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+42 a c^3 e^2 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )-840 c d (d+e x) \left (a e^2+c d^2\right )^3 \log (d+e x)+c^4 \left (-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right )}{105 e^9 (d+e x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 421, normalized size = 1.65 \[ \frac {15 \, c^{4} e^{8} x^{8} - 20 \, c^{4} d e^{7} x^{7} - 105 \, c^{4} d^{8} - 420 \, a c^{3} d^{6} e^{2} - 630 \, a^{2} c^{2} d^{4} e^{4} - 420 \, a^{3} c d^{2} e^{6} - 105 \, a^{4} e^{8} + 28 \, {\left (c^{4} d^{2} e^{6} + 3 \, a c^{3} e^{8}\right )} x^{6} - 42 \, {\left (c^{4} d^{3} e^{5} + 3 \, a c^{3} d e^{7}\right )} x^{5} + 70 \, {\left (c^{4} d^{4} e^{4} + 3 \, a c^{3} d^{2} e^{6} + 3 \, a^{2} c^{2} e^{8}\right )} x^{4} - 140 \, {\left (c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5} + 3 \, a^{2} c^{2} d e^{7}\right )} x^{3} + 420 \, {\left (c^{4} d^{6} e^{2} + 3 \, a c^{3} d^{4} e^{4} + 3 \, a^{2} c^{2} d^{2} e^{6} + a^{3} c e^{8}\right )} x^{2} + 105 \, {\left (7 \, c^{4} d^{7} e + 20 \, a c^{3} d^{5} e^{3} + 18 \, a^{2} c^{2} d^{3} e^{5} + 4 \, a^{3} c d e^{7}\right )} x - 840 \, {\left (c^{4} d^{8} + 3 \, a c^{3} d^{6} e^{2} + 3 \, a^{2} c^{2} d^{4} e^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (e x + d\right )}{105 \, {\left (e^{10} x + d e^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 396, normalized size = 1.55 \[ \frac {1}{105} \, {\left (15 \, c^{4} - \frac {140 \, c^{4} d}{x e + d} + \frac {84 \, {\left (7 \, c^{4} d^{2} e^{2} + a c^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {210 \, {\left (7 \, c^{4} d^{3} e^{3} + 3 \, a c^{3} d e^{5}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {70 \, {\left (35 \, c^{4} d^{4} e^{4} + 30 \, a c^{3} d^{2} e^{6} + 3 \, a^{2} c^{2} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac {420 \, {\left (7 \, c^{4} d^{5} e^{5} + 10 \, a c^{3} d^{3} e^{7} + 3 \, a^{2} c^{2} d e^{9}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} + \frac {420 \, {\left (7 \, c^{4} d^{6} e^{6} + 15 \, a c^{3} d^{4} e^{8} + 9 \, a^{2} c^{2} d^{2} e^{10} + a^{3} c e^{12}\right )} e^{\left (-6\right )}}{{\left (x e + d\right )}^{6}}\right )} {\left (x e + d\right )}^{7} e^{\left (-9\right )} + 8 \, {\left (c^{4} d^{7} + 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{4} d^{8} e^{7}}{x e + d} + \frac {4 \, a c^{3} d^{6} e^{9}}{x e + d} + \frac {6 \, a^{2} c^{2} d^{4} e^{11}}{x e + d} + \frac {4 \, a^{3} c d^{2} e^{13}}{x e + d} + \frac {a^{4} e^{15}}{x e + d}\right )} e^{\left (-16\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 378, normalized size = 1.48 \[ \frac {c^{4} x^{7}}{7 e^{2}}-\frac {c^{4} d \,x^{6}}{3 e^{3}}+\frac {4 a \,c^{3} x^{5}}{5 e^{2}}+\frac {3 c^{4} d^{2} x^{5}}{5 e^{4}}-\frac {2 a \,c^{3} d \,x^{4}}{e^{3}}-\frac {c^{4} d^{3} x^{4}}{e^{5}}+\frac {2 a^{2} c^{2} x^{3}}{e^{2}}+\frac {4 a \,c^{3} d^{2} x^{3}}{e^{4}}+\frac {5 c^{4} d^{4} x^{3}}{3 e^{6}}-\frac {6 a^{2} c^{2} d \,x^{2}}{e^{3}}-\frac {8 a \,c^{3} d^{3} x^{2}}{e^{5}}-\frac {3 c^{4} d^{5} x^{2}}{e^{7}}-\frac {a^{4}}{\left (e x +d \right ) e}-\frac {4 a^{3} c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {8 a^{3} c d \ln \left (e x +d \right )}{e^{3}}+\frac {4 a^{3} c x}{e^{2}}-\frac {6 a^{2} c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {24 a^{2} c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {18 a^{2} c^{2} d^{2} x}{e^{4}}-\frac {4 a \,c^{3} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {24 a \,c^{3} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {20 a \,c^{3} d^{4} x}{e^{6}}-\frac {c^{4} d^{8}}{\left (e x +d \right ) e^{9}}-\frac {8 c^{4} d^{7} \ln \left (e x +d \right )}{e^{9}}+\frac {7 c^{4} d^{6} x}{e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 330, normalized size = 1.29 \[ -\frac {c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{e^{10} x + d e^{9}} + \frac {15 \, c^{4} e^{6} x^{7} - 35 \, c^{4} d e^{5} x^{6} + 21 \, {\left (3 \, c^{4} d^{2} e^{4} + 4 \, a c^{3} e^{6}\right )} x^{5} - 105 \, {\left (c^{4} d^{3} e^{3} + 2 \, a c^{3} d e^{5}\right )} x^{4} + 35 \, {\left (5 \, c^{4} d^{4} e^{2} + 12 \, a c^{3} d^{2} e^{4} + 6 \, a^{2} c^{2} e^{6}\right )} x^{3} - 105 \, {\left (3 \, c^{4} d^{5} e + 8 \, a c^{3} d^{3} e^{3} + 6 \, a^{2} c^{2} d e^{5}\right )} x^{2} + 105 \, {\left (7 \, c^{4} d^{6} + 20 \, a c^{3} d^{4} e^{2} + 18 \, a^{2} c^{2} d^{2} e^{4} + 4 \, a^{3} c e^{6}\right )} x}{105 \, e^{8}} - \frac {8 \, {\left (c^{4} d^{7} + 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 701, normalized size = 2.75 \[ x^4\,\left (\frac {c^4\,d^3}{2\,e^5}-\frac {d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{2\,e}\right )+x^5\,\left (\frac {4\,a\,c^3}{5\,e^2}+\frac {3\,c^4\,d^2}{5\,e^4}\right )+x^2\,\left (\frac {d\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e}-\frac {6\,a^2\,c^2}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{2\,e^2}\right )-x^3\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{3\,e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{3\,e}-\frac {2\,a^2\,c^2}{e^2}\right )+x\,\left (\frac {4\,a^3\,c}{e^2}+\frac {d^2\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e}-\frac {6\,a^2\,c^2}{e^2}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e}-\frac {6\,a^2\,c^2}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e^2}\right )}{e}\right )-\frac {a^4\,e^8+4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8}{e\,\left (x\,e^9+d\,e^8\right )}+\frac {c^4\,x^7}{7\,e^2}-\frac {\ln \left (d+e\,x\right )\,\left (8\,a^3\,c\,d\,e^6+24\,a^2\,c^2\,d^3\,e^4+24\,a\,c^3\,d^5\,e^2+8\,c^4\,d^7\right )}{e^9}-\frac {c^4\,d\,x^6}{3\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.02, size = 314, normalized size = 1.23 \[ - \frac {c^{4} d x^{6}}{3 e^{3}} + \frac {c^{4} x^{7}}{7 e^{2}} - \frac {8 c d \left (a e^{2} + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{9}} + x^{5} \left (\frac {4 a c^{3}}{5 e^{2}} + \frac {3 c^{4} d^{2}}{5 e^{4}}\right ) + x^{4} \left (- \frac {2 a c^{3} d}{e^{3}} - \frac {c^{4} d^{3}}{e^{5}}\right ) + x^{3} \left (\frac {2 a^{2} c^{2}}{e^{2}} + \frac {4 a c^{3} d^{2}}{e^{4}} + \frac {5 c^{4} d^{4}}{3 e^{6}}\right ) + x^{2} \left (- \frac {6 a^{2} c^{2} d}{e^{3}} - \frac {8 a c^{3} d^{3}}{e^{5}} - \frac {3 c^{4} d^{5}}{e^{7}}\right ) + x \left (\frac {4 a^{3} c}{e^{2}} + \frac {18 a^{2} c^{2} d^{2}}{e^{4}} + \frac {20 a c^{3} d^{4}}{e^{6}} + \frac {7 c^{4} d^{6}}{e^{8}}\right ) + \frac {- a^{4} e^{8} - 4 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8}}{d e^{9} + e^{10} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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