Integrand size = 20, antiderivative size = 99 \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=\frac {e (2 b B d+A b e-2 a B e) x}{b^3}+\frac {B e^2 x^2}{2 b^2}-\frac {(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) \log (a+b x)}{b^4} \]
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Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=-\frac {(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac {(b d-a e) \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^4}+\frac {e x (-2 a B e+A b e+2 b B d)}{b^3}+\frac {B e^2 x^2}{2 b^2} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e (2 b B d+A b e-2 a B e)}{b^3}+\frac {B e^2 x}{b^2}+\frac {(A b-a B) (b d-a e)^2}{b^3 (a+b x)^2}+\frac {(b d-a e) (b B d+2 A b e-3 a B e)}{b^3 (a+b x)}\right ) \, dx \\ & = \frac {e (2 b B d+A b e-2 a B e) x}{b^3}+\frac {B e^2 x^2}{2 b^2}-\frac {(A b-a B) (b d-a e)^2}{b^4 (a+b x)}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) \log (a+b x)}{b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=\frac {e (2 b B d+A b e-2 a B e) x}{b^3}+\frac {B e^2 x^2}{2 b^2}+\frac {-A b^3 d^2+a b^2 B d^2+2 a A b^2 d e-2 a^2 b B d e-a^2 A b e^2+a^3 B e^2}{b^4 (a+b x)}+\frac {\left (b^2 B d^2+2 A b^2 d e-4 a b B d e-2 a A b e^2+3 a^2 B e^2\right ) \log (a+b x)}{b^4} \]
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Time = 0.72 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.53
method | result | size |
default | \(\frac {e \left (\frac {1}{2} B b e \,x^{2}+A b e x -2 B a e x +2 B b d x \right )}{b^{3}}+\frac {\left (-2 A a b \,e^{2}+2 A \,b^{2} d e +3 B \,a^{2} e^{2}-4 B a b d e +b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{4}}-\frac {A \,a^{2} b \,e^{2}-2 A a \,b^{2} d e +A \,b^{3} d^{2}-B \,a^{3} e^{2}+2 B \,a^{2} b d e -B a \,b^{2} d^{2}}{b^{4} \left (b x +a \right )}\) | \(151\) |
norman | \(\frac {\frac {\left (2 A \,a^{2} b \,e^{2}-2 A a \,b^{2} d e +A \,b^{3} d^{2}-3 B \,a^{3} e^{2}+4 B \,a^{2} b d e -B a \,b^{2} d^{2}\right ) x}{b^{3} a}+\frac {B \,e^{2} x^{3}}{2 b}+\frac {e \left (2 A b e -3 B a e +4 B b d \right ) x^{2}}{2 b^{2}}}{b x +a}-\frac {\left (2 A a b \,e^{2}-2 A \,b^{2} d e -3 B \,a^{2} e^{2}+4 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\) | \(165\) |
risch | \(\frac {B \,e^{2} x^{2}}{2 b^{2}}+\frac {e^{2} A x}{b^{2}}-\frac {2 e^{2} B a x}{b^{3}}+\frac {2 e B d x}{b^{2}}-\frac {A \,a^{2} e^{2}}{b^{3} \left (b x +a \right )}+\frac {2 A a d e}{b^{2} \left (b x +a \right )}-\frac {A \,d^{2}}{b \left (b x +a \right )}+\frac {B \,a^{3} e^{2}}{b^{4} \left (b x +a \right )}-\frac {2 B \,a^{2} d e}{b^{3} \left (b x +a \right )}+\frac {B a \,d^{2}}{b^{2} \left (b x +a \right )}-\frac {2 \ln \left (b x +a \right ) A a \,e^{2}}{b^{3}}+\frac {2 \ln \left (b x +a \right ) A d e}{b^{2}}+\frac {3 \ln \left (b x +a \right ) B \,a^{2} e^{2}}{b^{4}}-\frac {4 \ln \left (b x +a \right ) B a d e}{b^{3}}+\frac {\ln \left (b x +a \right ) B \,d^{2}}{b^{2}}\) | \(223\) |
parallelrisch | \(-\frac {-B \,e^{2} x^{3} b^{3}+4 A \ln \left (b x +a \right ) x a \,b^{2} e^{2}-4 A \ln \left (b x +a \right ) x \,b^{3} d e -2 A \,x^{2} b^{3} e^{2}-6 B \ln \left (b x +a \right ) x \,a^{2} b \,e^{2}+8 B \ln \left (b x +a \right ) x a \,b^{2} d e -2 B \ln \left (b x +a \right ) x \,b^{3} d^{2}+3 B \,x^{2} a \,b^{2} e^{2}-4 B \,x^{2} b^{3} d e +4 A \ln \left (b x +a \right ) a^{2} b \,e^{2}-4 A \ln \left (b x +a \right ) a \,b^{2} d e -6 B \ln \left (b x +a \right ) a^{3} e^{2}+8 B \ln \left (b x +a \right ) a^{2} b d e -2 B \ln \left (b x +a \right ) a \,b^{2} d^{2}+4 A \,a^{2} b \,e^{2}-4 A a \,b^{2} d e +2 A \,b^{3} d^{2}-6 B \,a^{3} e^{2}+8 B \,a^{2} b d e -2 B a \,b^{2} d^{2}}{2 b^{4} \left (b x +a \right )}\) | \(276\) |
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Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (96) = 192\).
Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.52 \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=\frac {B b^{3} e^{2} x^{3} + 2 \, {\left (B a b^{2} - A b^{3}\right )} d^{2} - 4 \, {\left (B a^{2} b - A a b^{2}\right )} d e + 2 \, {\left (B a^{3} - A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, B a b^{2} d e - {\left (2 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x + 2 \, {\left (B a b^{2} d^{2} - 2 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} d e + {\left (3 \, B a^{3} - 2 \, A a^{2} b\right )} e^{2} + {\left (B b^{3} d^{2} - 2 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d e + {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \]
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Time = 0.50 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=\frac {B e^{2} x^{2}}{2 b^{2}} + x \left (\frac {A e^{2}}{b^{2}} - \frac {2 B a e^{2}}{b^{3}} + \frac {2 B d e}{b^{2}}\right ) + \frac {- A a^{2} b e^{2} + 2 A a b^{2} d e - A b^{3} d^{2} + B a^{3} e^{2} - 2 B a^{2} b d e + B a b^{2} d^{2}}{a b^{4} + b^{5} x} + \frac {\left (a e - b d\right ) \left (- 2 A b e + 3 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=\frac {{\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e + {\left (B a^{3} - A a^{2} b\right )} e^{2}}{b^{5} x + a b^{4}} + \frac {B b e^{2} x^{2} + 2 \, {\left (2 \, B b d e - {\left (2 \, B a - A b\right )} e^{2}\right )} x}{2 \, b^{3}} + \frac {{\left (B b^{2} d^{2} - 2 \, {\left (2 \, B a b - A b^{2}\right )} d e + {\left (3 \, B a^{2} - 2 \, A a b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (96) = 192\).
Time = 0.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.31 \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=\frac {{\left (B e^{2} + \frac {2 \, {\left (2 \, B b^{2} d e - 3 \, B a b e^{2} + A b^{2} e^{2}\right )}}{{\left (b x + a\right )} b}\right )} {\left (b x + a\right )}^{2}}{2 \, b^{4}} - \frac {{\left (B b^{2} d^{2} - 4 \, B a b d e + 2 \, A b^{2} d e + 3 \, B a^{2} e^{2} - 2 \, A a b e^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {\frac {B a b^{4} d^{2}}{b x + a} - \frac {A b^{5} d^{2}}{b x + a} - \frac {2 \, B a^{2} b^{3} d e}{b x + a} + \frac {2 \, A a b^{4} d e}{b x + a} + \frac {B a^{3} b^{2} e^{2}}{b x + a} - \frac {A a^{2} b^{3} e^{2}}{b x + a}}{b^{6}} \]
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Time = 1.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.67 \[ \int \frac {(A+B x) (d+e x)^2}{(a+b x)^2} \, dx=x\,\left (\frac {A\,e^2+2\,B\,d\,e}{b^2}-\frac {2\,B\,a\,e^2}{b^3}\right )+\frac {\ln \left (a+b\,x\right )\,\left (3\,B\,a^2\,e^2-4\,B\,a\,b\,d\,e-2\,A\,a\,b\,e^2+B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{b^4}-\frac {-B\,a^3\,e^2+2\,B\,a^2\,b\,d\,e+A\,a^2\,b\,e^2-B\,a\,b^2\,d^2-2\,A\,a\,b^2\,d\,e+A\,b^3\,d^2}{b\,\left (x\,b^4+a\,b^3\right )}+\frac {B\,e^2\,x^2}{2\,b^2} \]
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