Optimal. Leaf size=184 \[ \frac {10 e^2 \left (c d^2-a e^2\right )^3 x}{c^5 d^5}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6} \]
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Rubi [A]
time = 0.14, antiderivative size = 184, 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 = {640, 45}
\begin {gather*} \frac {e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^7}{\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)^5}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac {\left (c d^2-a e^2\right )^5}{c^5 d^5 (a e+c d x)^2}+\frac {5 e \left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)}{c^5 d^5}+\frac {5 \left (c d^2 e^4-a e^6\right ) (a e+c d x)^2}{c^5 d^5}+\frac {e^5 (a e+c d x)^3}{c^5 d^5}\right ) \, dx\\ &=\frac {10 e^2 \left (c d^2-a e^2\right )^3 x}{c^5 d^5}-\frac {\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac {e^5 (a e+c d x)^4}{4 c^6 d^6}+\frac {5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 263, normalized size = 1.43 \begin {gather*} \frac {12 a^5 e^{10}-12 a^4 c d e^8 (5 d+4 e x)+30 a^3 c^2 d^2 e^6 \left (4 d^2+6 d e x-e^2 x^2\right )-10 a^2 c^3 d^3 e^4 \left (12 d^3+24 d^2 e x-12 d e^2 x^2-e^3 x^3\right )+5 a c^4 d^4 e^2 \left (12 d^4+24 d^3 e x-36 d^2 e^2 x^2-8 d e^3 x^3-e^4 x^4\right )+c^5 d^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )+60 e \left (c d^2-a e^2\right )^4 (a e+c d x) \log (a e+c d x)}{12 c^6 d^6 (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 298, normalized size = 1.62
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{4} c^{3} d^{3} e^{3} x^{4}+\frac {2}{3} a \,c^{2} d^{2} e^{4} x^{3}-\frac {5}{3} c^{3} d^{4} e^{2} x^{3}-\frac {3}{2} a^{2} c d \,e^{5} x^{2}+5 a \,c^{2} d^{3} e^{3} x^{2}-5 c^{3} d^{5} e \,x^{2}+4 e^{6} a^{3} x -15 e^{4} d^{2} a^{2} c x +20 d^{4} e^{2} c^{2} a x -10 d^{6} c^{3} x \right )}{c^{5} d^{5}}-\frac {-a^{5} e^{10}+5 a^{4} c \,d^{2} e^{8}-10 a^{3} c^{2} d^{4} e^{6}+10 a^{2} c^{3} d^{6} e^{4}-5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}}{c^{6} d^{6} \left (c d x +a e \right )}+\frac {5 e \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}\) | \(298\) |
risch | \(\frac {e^{5} x^{4}}{4 c^{2} d^{2}}-\frac {2 e^{6} a \,x^{3}}{3 c^{3} d^{3}}+\frac {5 e^{4} x^{3}}{3 c^{2} d}+\frac {3 e^{7} a^{2} x^{2}}{2 c^{4} d^{4}}-\frac {5 e^{5} a \,x^{2}}{c^{3} d^{2}}+\frac {5 e^{3} x^{2}}{c^{2}}-\frac {4 e^{8} a^{3} x}{c^{5} d^{5}}+\frac {15 e^{6} a^{2} x}{c^{4} d^{3}}-\frac {20 e^{4} a x}{c^{3} d}+\frac {10 e^{2} d x}{c^{2}}+\frac {a^{5} e^{10}}{c^{6} d^{6} \left (c d x +a e \right )}-\frac {5 a^{4} e^{8}}{c^{5} d^{4} \left (c d x +a e \right )}+\frac {10 a^{3} e^{6}}{c^{4} d^{2} \left (c d x +a e \right )}-\frac {10 a^{2} e^{4}}{c^{3} \left (c d x +a e \right )}+\frac {5 d^{2} a \,e^{2}}{c^{2} \left (c d x +a e \right )}-\frac {d^{4}}{c \left (c d x +a e \right )}+\frac {5 e^{9} \ln \left (c d x +a e \right ) a^{4}}{c^{6} d^{6}}-\frac {20 e^{7} \ln \left (c d x +a e \right ) a^{3}}{c^{5} d^{4}}+\frac {30 e^{5} \ln \left (c d x +a e \right ) a^{2}}{c^{4} d^{2}}-\frac {20 e^{3} \ln \left (c d x +a e \right ) a}{c^{3}}+\frac {5 d^{2} e \ln \left (c d x +a e \right )}{c^{2}}\) | \(378\) |
norman | \(\frac {\frac {10 a^{5} e^{10}-35 a^{4} c \,d^{2} e^{8}+40 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}-10 a \,c^{4} d^{8} e^{2}-2 c^{5} d^{10}}{2 c^{6} d^{5}}+\frac {e^{6} x^{6}}{4 c d}+\frac {\left (10 a^{5} e^{12}-35 a^{4} c \,d^{2} e^{10}+45 a^{3} c^{2} d^{4} e^{8}-30 a^{2} c^{3} d^{6} e^{6}+20 a \,c^{4} d^{8} e^{4}-22 c^{5} d^{10} e^{2}\right ) x}{2 c^{6} d^{6} e}-\frac {5 e^{3} \left (3 e^{6} a^{3}-13 e^{4} d^{2} a^{2} c +22 d^{4} e^{2} c^{2} a -18 d^{6} c^{3}\right ) x^{3}}{6 c^{4} d^{4}}+\frac {5 e^{4} \left (2 a^{2} e^{4}-9 a c \,d^{2} e^{2}+16 c^{2} d^{4}\right ) x^{4}}{12 c^{3} d^{3}}-\frac {e^{5} \left (5 e^{2} a -23 c \,d^{2}\right ) x^{5}}{12 c^{2} d^{2}}}{\left (c d x +a e \right ) \left (e x +d \right )}+\frac {5 e \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{6} d^{6}}\) | \(390\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 284, normalized size = 1.54 \begin {gather*} -\frac {c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{c^{7} d^{7} x + a c^{6} d^{6} e} + \frac {3 \, c^{3} d^{3} x^{4} e^{5} + 4 \, {\left (5 \, c^{3} d^{4} e^{4} - 2 \, a c^{2} d^{2} e^{6}\right )} x^{3} + 6 \, {\left (10 \, c^{3} d^{5} e^{3} - 10 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{2} + 12 \, {\left (10 \, c^{3} d^{6} e^{2} - 20 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} - 4 \, a^{3} e^{8}\right )} x}{12 \, c^{5} d^{5}} + \frac {5 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \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. 399 vs.
\(2 (177) = 354\).
time = 2.19, size = 399, normalized size = 2.17 \begin {gather*} -\frac {12 \, c^{5} d^{10} + 48 \, a^{4} c d x e^{9} - 12 \, a^{5} e^{10} + 30 \, {\left (a^{3} c^{2} d^{2} x^{2} + 2 \, a^{4} c d^{2}\right )} e^{8} - 10 \, {\left (a^{2} c^{3} d^{3} x^{3} + 18 \, a^{3} c^{2} d^{3} x\right )} e^{7} + 5 \, {\left (a c^{4} d^{4} x^{4} - 24 \, a^{2} c^{3} d^{4} x^{2} - 24 \, a^{3} c^{2} d^{4}\right )} e^{6} - {\left (3 \, c^{5} d^{5} x^{5} - 40 \, a c^{4} d^{5} x^{3} - 240 \, a^{2} c^{3} d^{5} x\right )} e^{5} - 20 \, {\left (c^{5} d^{6} x^{4} - 9 \, a c^{4} d^{6} x^{2} - 6 \, a^{2} c^{3} d^{6}\right )} e^{4} - 60 \, {\left (c^{5} d^{7} x^{3} + 2 \, a c^{4} d^{7} x\right )} e^{3} - 60 \, {\left (2 \, c^{5} d^{8} x^{2} + a c^{4} d^{8}\right )} e^{2} - 60 \, {\left (c^{5} d^{9} x e - 4 \, a c^{4} d^{7} x e^{3} + a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{5} x e^{5} - 4 \, a^{2} c^{3} d^{6} e^{4} - 4 \, a^{3} c^{2} d^{3} x e^{7} + 6 \, a^{3} c^{2} d^{4} e^{6} + a^{4} c d x e^{9} - 4 \, a^{4} c d^{2} e^{8} + a^{5} e^{10}\right )} \log \left (c d x + a e\right )}{12 \, {\left (c^{7} d^{7} x + a c^{6} d^{6} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.12, size = 264, normalized size = 1.43 \begin {gather*} x^{3} \left (- \frac {2 a e^{6}}{3 c^{3} d^{3}} + \frac {5 e^{4}}{3 c^{2} d}\right ) + x^{2} \cdot \left (\frac {3 a^{2} e^{7}}{2 c^{4} d^{4}} - \frac {5 a e^{5}}{c^{3} d^{2}} + \frac {5 e^{3}}{c^{2}}\right ) + x \left (- \frac {4 a^{3} e^{8}}{c^{5} d^{5}} + \frac {15 a^{2} e^{6}}{c^{4} d^{3}} - \frac {20 a e^{4}}{c^{3} d} + \frac {10 d e^{2}}{c^{2}}\right ) + \frac {a^{5} e^{10} - 5 a^{4} c d^{2} e^{8} + 10 a^{3} c^{2} d^{4} e^{6} - 10 a^{2} c^{3} d^{6} e^{4} + 5 a c^{4} d^{8} e^{2} - c^{5} d^{10}}{a c^{6} d^{6} e + c^{7} d^{7} x} + \frac {e^{5} x^{4}}{4 c^{2} d^{2}} + \frac {5 e \left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{6} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 296, normalized size = 1.61 \begin {gather*} \frac {5 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{6} d^{6}} - \frac {c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{{\left (c d x + a e\right )} c^{6} d^{6}} + \frac {3 \, c^{6} d^{6} x^{4} e^{5} + 20 \, c^{6} d^{7} x^{3} e^{4} + 60 \, c^{6} d^{8} x^{2} e^{3} + 120 \, c^{6} d^{9} x e^{2} - 8 \, a c^{5} d^{5} x^{3} e^{6} - 60 \, a c^{5} d^{6} x^{2} e^{5} - 240 \, a c^{5} d^{7} x e^{4} + 18 \, a^{2} c^{4} d^{4} x^{2} e^{7} + 180 \, a^{2} c^{4} d^{5} x e^{6} - 48 \, a^{3} c^{3} d^{3} x e^{8}}{12 \, c^{8} d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 387, normalized size = 2.10 \begin {gather*} x\,\left (\frac {10\,d\,e^2}{c^2}+\frac {2\,a\,e\,\left (\frac {a^2\,e^7}{c^4\,d^4}-\frac {10\,e^3}{c^2}+\frac {2\,a\,e\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c^2\,d^2}\right )+x^3\,\left (\frac {5\,e^4}{3\,c^2\,d}-\frac {2\,a\,e^6}{3\,c^3\,d^3}\right )-x^2\,\left (\frac {a^2\,e^7}{2\,c^4\,d^4}-\frac {5\,e^3}{c^2}+\frac {a\,e\,\left (\frac {5\,e^4}{c^2\,d}-\frac {2\,a\,e^6}{c^3\,d^3}\right )}{c\,d}\right )+\frac {e^5\,x^4}{4\,c^2\,d^2}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (5\,a^4\,e^9-20\,a^3\,c\,d^2\,e^7+30\,a^2\,c^2\,d^4\,e^5-20\,a\,c^3\,d^6\,e^3+5\,c^4\,d^8\,e\right )}{c^6\,d^6}+\frac {a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}}{c\,d\,\left (x\,c^6\,d^6+a\,e\,c^5\,d^5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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