Optimal. Leaf size=131 \[ \frac {10 e^2 (b d-a e)^3 x}{b^5}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e^3 (b d-a e)^2 (a+b x)^2}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^3}{3 b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6} \]
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Rubi [A]
time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} \frac {5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac {5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {10 e^2 x (b d-a e)^3}{b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^5}{(a+b x)^2} \, dx\\ &=\int \left (\frac {10 e^2 (b d-a e)^3}{b^5}+\frac {(b d-a e)^5}{b^5 (a+b x)^2}+\frac {5 e (b d-a e)^4}{b^5 (a+b x)}+\frac {10 e^3 (b d-a e)^2 (a+b x)}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^2}{b^5}+\frac {e^5 (a+b x)^3}{b^5}\right ) \, dx\\ &=\frac {10 e^2 (b d-a e)^3 x}{b^5}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e^3 (b d-a e)^2 (a+b x)^2}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^3}{3 b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 230, normalized size = 1.76 \begin {gather*} \frac {12 a^5 e^5-12 a^4 b e^4 (5 d+4 e x)+30 a^3 b^2 e^3 \left (4 d^2+6 d e x-e^2 x^2\right )+10 a^2 b^3 e^2 \left (-12 d^3-24 d^2 e x+12 d e^2 x^2+e^3 x^3\right )-5 a b^4 e \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 )+b^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 (b d-a e)^4 (a+b x) \log (a+b x)}{12 b^6 (a+b 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. \(259\) vs.
\(2(127)=254\).
time = 0.71, size = 260, normalized size = 1.98
method | result | size |
norman | \(\frac {\frac {e^{5} x^{5}}{4 b}-\frac {5 e^{2} \left (e^{3} a^{3}-4 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -4 b^{3} d^{3}\right ) x^{2}}{2 b^{4}}+\frac {5 e^{3} \left (a^{2} e^{2}-4 a b d e +6 b^{2} d^{2}\right ) x^{3}}{6 b^{3}}-\frac {5 e^{4} \left (a e -4 b d \right ) x^{4}}{12 b^{2}}-\frac {\left (5 a^{5} e^{5}-20 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) x}{a \,b^{5}}}{b x +a}+\frac {5 e \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(256\) |
default | \(-\frac {e^{2} \left (-\frac {1}{4} b^{3} x^{4} e^{3}+\frac {2}{3} a \,b^{2} e^{3} x^{3}-\frac {5}{3} b^{3} d \,e^{2} x^{3}-\frac {3}{2} a^{2} b \,e^{3} x^{2}+5 a \,b^{2} d \,e^{2} x^{2}-5 b^{3} d^{2} e \,x^{2}+4 e^{3} a^{3} x -15 a^{2} b d \,e^{2} x +20 a \,b^{2} d^{2} e x -10 b^{3} d^{3} x \right )}{b^{5}}-\frac {-a^{5} e^{5}+5 a^{4} b d \,e^{4}-10 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e +b^{5} d^{5}}{b^{6} \left (b x +a \right )}+\frac {5 e \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(260\) |
risch | \(\frac {e^{5} x^{4}}{4 b^{2}}-\frac {2 e^{5} a \,x^{3}}{3 b^{3}}+\frac {5 e^{4} d \,x^{3}}{3 b^{2}}+\frac {3 e^{5} a^{2} x^{2}}{2 b^{4}}-\frac {5 e^{4} a d \,x^{2}}{b^{3}}+\frac {5 e^{3} d^{2} x^{2}}{b^{2}}-\frac {4 e^{5} a^{3} x}{b^{5}}+\frac {15 e^{4} a^{2} d x}{b^{4}}-\frac {20 e^{3} a \,d^{2} x}{b^{3}}+\frac {10 e^{2} d^{3} x}{b^{2}}+\frac {a^{5} e^{5}}{b^{6} \left (b x +a \right )}-\frac {5 a^{4} d \,e^{4}}{b^{5} \left (b x +a \right )}+\frac {10 a^{3} d^{2} e^{3}}{b^{4} \left (b x +a \right )}-\frac {10 a^{2} d^{3} e^{2}}{b^{3} \left (b x +a \right )}+\frac {5 a \,d^{4} e}{b^{2} \left (b x +a \right )}-\frac {d^{5}}{b \left (b x +a \right )}+\frac {5 e^{5} \ln \left (b x +a \right ) a^{4}}{b^{6}}-\frac {20 e^{4} \ln \left (b x +a \right ) a^{3} d}{b^{5}}+\frac {30 e^{3} \ln \left (b x +a \right ) a^{2} d^{2}}{b^{4}}-\frac {20 e^{2} \ln \left (b x +a \right ) a \,d^{3}}{b^{3}}+\frac {5 e \ln \left (b x +a \right ) d^{4}}{b^{2}}\) | \(326\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 249, normalized size = 1.90 \begin {gather*} -\frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} x^{4} e^{5} + 4 \, {\left (5 \, b^{3} d e^{4} - 2 \, a b^{2} e^{5}\right )} x^{3} + 6 \, {\left (10 \, b^{3} d^{2} e^{3} - 10 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{2} + 12 \, {\left (10 \, b^{3} d^{3} e^{2} - 20 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} - 4 \, a^{3} e^{5}\right )} x}{12 \, b^{5}} + \frac {5 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left (b x + a\right )}{b^{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. 340 vs.
\(2 (129) = 258\).
time = 2.67, size = 340, normalized size = 2.60 \begin {gather*} -\frac {12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e - {\left (3 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} - 48 \, a^{4} b x + 12 \, a^{5}\right )} e^{5} - 20 \, {\left (b^{5} d x^{4} - 2 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 9 \, a^{3} b^{2} d x - 3 \, a^{4} b d\right )} e^{4} - 60 \, {\left (b^{5} d^{2} x^{3} - 3 \, a b^{4} d^{2} x^{2} - 4 \, a^{2} b^{3} d^{2} x + 2 \, a^{3} b^{2} d^{2}\right )} e^{3} - 120 \, {\left (b^{5} d^{3} x^{2} + a b^{4} d^{3} x - a^{2} b^{3} d^{3}\right )} e^{2} - 60 \, {\left ({\left (a^{4} b x + a^{5}\right )} e^{5} - 4 \, {\left (a^{3} b^{2} d x + a^{4} b d\right )} e^{4} + 6 \, {\left (a^{2} b^{3} d^{2} x + a^{3} b^{2} d^{2}\right )} e^{3} - 4 \, {\left (a b^{4} d^{3} x + a^{2} b^{3} d^{3}\right )} e^{2} + {\left (b^{5} d^{4} x + a b^{4} d^{4}\right )} e\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 231, normalized size = 1.76 \begin {gather*} x^{3} \left (- \frac {2 a e^{5}}{3 b^{3}} + \frac {5 d e^{4}}{3 b^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} e^{5}}{2 b^{4}} - \frac {5 a d e^{4}}{b^{3}} + \frac {5 d^{2} e^{3}}{b^{2}}\right ) + x \left (- \frac {4 a^{3} e^{5}}{b^{5}} + \frac {15 a^{2} d e^{4}}{b^{4}} - \frac {20 a d^{2} e^{3}}{b^{3}} + \frac {10 d^{3} e^{2}}{b^{2}}\right ) + \frac {a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}}{a b^{6} + b^{7} x} + \frac {e^{5} x^{4}}{4 b^{2}} + \frac {5 e \left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.42, size = 257, normalized size = 1.96 \begin {gather*} \frac {5 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{{\left (b x + a\right )} b^{6}} + \frac {3 \, b^{6} x^{4} e^{5} + 20 \, b^{6} d x^{3} e^{4} + 60 \, b^{6} d^{2} x^{2} e^{3} + 120 \, b^{6} d^{3} x e^{2} - 8 \, a b^{5} x^{3} e^{5} - 60 \, a b^{5} d x^{2} e^{4} - 240 \, a b^{5} d^{2} x e^{3} + 18 \, a^{2} b^{4} x^{2} e^{5} + 180 \, a^{2} b^{4} d x e^{4} - 48 \, a^{3} b^{3} x e^{5}}{12 \, b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 326, normalized size = 2.49 \begin {gather*} x\,\left (\frac {10\,d^3\,e^2}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{b^4}+\frac {10\,d^2\,e^3}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,e^5}{3\,b^3}-\frac {5\,d\,e^4}{3\,b^2}\right )+x^2\,\left (\frac {a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{2\,b^4}+\frac {5\,d^2\,e^3}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,e^5-20\,a^3\,b\,d\,e^4+30\,a^2\,b^2\,d^2\,e^3-20\,a\,b^3\,d^3\,e^2+5\,b^4\,d^4\,e\right )}{b^6}+\frac {a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {e^5\,x^4}{4\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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