Integrand size = 28, antiderivative size = 161 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {3 e (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^3}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e^2 (b d-a e) (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 e^2 (a+b x) (b d-a e) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^3}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^3}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {e^3}{b^6}+\frac {(b d-a e)^3}{b^6 (a+b x)^3}+\frac {3 e (b d-a e)^2}{b^6 (a+b x)^2}+\frac {3 e^2 (b d-a e)}{b^6 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {3 e (b d-a e)^2}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^3}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e^2 (b d-a e) (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-5 a^3 e^3+a^2 b e^2 (9 d-4 e x)+a b^2 e \left (-3 d^2+12 d e x+4 e^2 x^2\right )-b^3 \left (d^3+6 d^2 e x-2 e^3 x^3\right )-6 e^2 (-b d+a e) (a+b x)^2 \log (a+b x)}{2 b^4 (a+b x) \sqrt {(a+b x)^2}} \]
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Time = 2.48 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} x}{\left (b x +a \right ) b^{3}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-3 a^{2} e^{3}+6 a b d \,e^{2}-3 b^{2} d^{2} e \right ) x -\frac {5 a^{3} e^{3}-9 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}}{2 b}\right )}{\left (b x +a \right )^{3} b^{3}}-\frac {3 \sqrt {\left (b x +a \right )^{2}}\, e^{2} \left (a e -b d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{4}}\) | \(154\) |
default | \(-\frac {\left (6 \ln \left (b x +a \right ) x^{2} a \,b^{2} e^{3}-6 \ln \left (b x +a \right ) b^{3} d \,e^{2} x^{2}-2 e^{3} x^{3} b^{3}+12 \ln \left (b x +a \right ) x \,a^{2} b \,e^{3}-12 \ln \left (b x +a \right ) x a \,b^{2} d \,e^{2}-4 x^{2} a \,b^{2} e^{3}+6 \ln \left (b x +a \right ) a^{3} e^{3}-6 \ln \left (b x +a \right ) a^{2} b d \,e^{2}+4 a^{2} b \,e^{3} x -12 x a \,b^{2} d \,e^{2}+6 b^{3} d^{2} e x +5 a^{3} e^{3}-9 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (b x +a \right )}{2 b^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(209\) |
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Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 \, b^{3} e^{3} x^{3} + 4 \, a b^{2} e^{3} x^{2} - b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - 2 \, {\left (3 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 2 \, a^{2} b e^{3}\right )} x + 6 \, {\left (a^{2} b d e^{2} - a^{3} e^{3} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
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\[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (115) = 230\).
Time = 0.21 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^{3} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {3 \, d e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {3 \, a e^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} - \frac {3 \, d^{2} e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {2 \, a^{2} e^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} + \frac {6 \, a d e^{2} x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {6 \, a^{2} e^{3} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {d^{3}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, a d^{2} e}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {9 \, a^{2} d e^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, a^{3} e^{3}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^{3} x}{b^{3} \mathrm {sgn}\left (b x + a\right )} + \frac {3 \, {\left (b d e^{2} - a e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 9 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 6 \, {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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