Optimal. Leaf size=145 \[ \frac {e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac {e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}+\frac {e^4 x^3}{3 c^2 d^2} \]
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Rubi [A] time = 0.15, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \begin {gather*} \frac {e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}+\frac {e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac {e^4 x^3}{3 c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^4}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {6 c^2 d^4 e^2-8 a c d^2 e^4+3 a^2 e^6}{c^4 d^4}+\frac {2 e^3 \left (2 c d^2-a e^2\right ) x}{c^3 d^3}+\frac {e^4 x^2}{c^2 d^2}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^2}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)}\right ) \, dx\\ &=\frac {e^2 \left (6 c^2 d^4-8 a c d^2 e^2+3 a^2 e^4\right ) x}{c^4 d^4}+\frac {e^3 \left (2 c d^2-a e^2\right ) x^2}{c^3 d^3}+\frac {e^4 x^3}{3 c^2 d^2}-\frac {\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 196, normalized size = 1.35 \begin {gather*} \frac {-3 a^4 e^8+3 a^3 c d e^6 (4 d+3 e x)-6 a^2 c^2 d^2 e^4 \left (3 d^2+4 d e x-e^2 x^2\right )+2 a c^3 d^3 e^2 \left (6 d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right )-12 e \left (a e^2-c d^2\right )^3 (a e+c d x) \log (a e+c d x)+c^4 d^4 \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )}{3 c^5 d^5 (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 305, normalized size = 2.10 \begin {gather*} \frac {c^{4} d^{4} e^{4} x^{4} - 3 \, c^{4} d^{8} + 12 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 2 \, {\left (3 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (3 \, c^{4} d^{6} e^{2} - 3 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \, {\left (6 \, a c^{3} d^{5} e^{3} - 8 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\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 (c d x + a e\right )}{3 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 561, normalized size = 3.87 \begin {gather*} \frac {4 \, {\left (c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{7} d^{9} - 2 \, a c^{6} d^{7} e^{2} + a^{2} c^{5} d^{5} e^{4}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {2 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{c^{5} d^{5}} + \frac {{\left (c^{4} d^{4} x^{3} e^{10} + 6 \, c^{4} d^{5} x^{2} e^{9} + 18 \, c^{4} d^{6} x e^{8} - 3 \, a c^{3} d^{3} x^{2} e^{11} - 24 \, a c^{3} d^{4} x e^{10} + 9 \, a^{2} c^{2} d^{2} x e^{12}\right )} e^{\left (-6\right )}}{3 \, c^{6} d^{6}} - \frac {c^{6} d^{13} - 6 \, a c^{5} d^{11} e^{2} + 15 \, a^{2} c^{4} d^{9} e^{4} - 20 \, a^{3} c^{3} d^{7} e^{6} + 15 \, a^{4} c^{2} d^{5} e^{8} - 6 \, a^{5} c d^{3} e^{10} + a^{6} d e^{12} + {\left (c^{6} d^{12} e - 6 \, a c^{5} d^{10} e^{3} + 15 \, a^{2} c^{4} d^{8} e^{5} - 20 \, a^{3} c^{3} d^{6} e^{7} + 15 \, a^{4} c^{2} d^{4} e^{9} - 6 \, a^{5} c d^{2} e^{11} + a^{6} e^{13}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 275, normalized size = 1.90 \begin {gather*} \frac {e^{4} x^{3}}{3 c^{2} d^{2}}-\frac {a^{4} e^{8}}{\left (c d x +a e \right ) c^{5} d^{5}}+\frac {4 a^{3} e^{6}}{\left (c d x +a e \right ) c^{4} d^{3}}-\frac {6 a^{2} e^{4}}{\left (c d x +a e \right ) c^{3} d}+\frac {4 a d \,e^{2}}{\left (c d x +a e \right ) c^{2}}-\frac {a \,e^{5} x^{2}}{c^{3} d^{3}}-\frac {d^{3}}{\left (c d x +a e \right ) c}+\frac {2 e^{3} x^{2}}{c^{2} d}-\frac {4 a^{3} e^{7} \ln \left (c d x +a e \right )}{c^{5} d^{5}}+\frac {12 a^{2} e^{5} \ln \left (c d x +a e \right )}{c^{4} d^{3}}+\frac {3 a^{2} e^{6} x}{c^{4} d^{4}}-\frac {12 a \,e^{3} \ln \left (c d x +a e \right )}{c^{3} d}-\frac {8 a \,e^{4} x}{c^{3} d^{2}}+\frac {4 d e \ln \left (c d x +a e \right )}{c^{2}}+\frac {6 e^{2} x}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 214, normalized size = 1.48 \begin {gather*} -\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}}{c^{6} d^{6} x + a c^{5} d^{5} e} + \frac {c^{2} d^{2} e^{4} x^{3} + 3 \, {\left (2 \, c^{2} d^{3} e^{3} - a c d e^{5}\right )} x^{2} + 3 \, {\left (6 \, c^{2} d^{4} e^{2} - 8 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x}{3 \, c^{4} d^{4}} + \frac {4 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 242, normalized size = 1.67 \begin {gather*} x^2\,\left (\frac {2\,e^3}{c^2\,d}-\frac {a\,e^5}{c^3\,d^3}\right )-x\,\left (\frac {a^2\,e^6}{c^4\,d^4}-\frac {6\,e^2}{c^2}+\frac {2\,a\,e\,\left (\frac {4\,e^3}{c^2\,d}-\frac {2\,a\,e^5}{c^3\,d^3}\right )}{c\,d}\right )-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (4\,a^3\,e^7-12\,a^2\,c\,d^2\,e^5+12\,a\,c^2\,d^4\,e^3-4\,c^3\,d^6\,e\right )}{c^5\,d^5}-\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}{c\,d\,\left (x\,c^5\,d^5+a\,e\,c^4\,d^4\right )}+\frac {e^4\,x^3}{3\,c^2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.88, size = 185, normalized size = 1.28 \begin {gather*} x^{2} \left (- \frac {a e^{5}}{c^{3} d^{3}} + \frac {2 e^{3}}{c^{2} d}\right ) + x \left (\frac {3 a^{2} e^{6}}{c^{4} d^{4}} - \frac {8 a e^{4}}{c^{3} d^{2}} + \frac {6 e^{2}}{c^{2}}\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}}{a c^{5} d^{5} e + c^{6} d^{6} x} + \frac {e^{4} x^{3}}{3 c^{2} d^{2}} - \frac {4 e \left (a e^{2} - c d^{2}\right )^{3} \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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