Integrand size = 20, antiderivative size = 141 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {e^2 (3 b B d+A b e-3 a B e) x}{b^4}+\frac {B e^3 x^2}{2 b^3}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}-\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^5 (a+b x)}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) \log (a+b x)}{b^5} \]
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Time = 0.12 (sec) , antiderivative size = 141, 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} \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=-\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^5 (a+b x)}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}+\frac {3 e (b d-a e) \log (a+b x) (-2 a B e+A b e+b B d)}{b^5}+\frac {e^2 x (-3 a B e+A b e+3 b B d)}{b^4}+\frac {B e^3 x^2}{2 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2 (3 b B d+A b e-3 a B e)}{b^4}+\frac {B e^3 x}{b^3}+\frac {(A b-a B) (b d-a e)^3}{b^4 (a+b x)^3}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^4 (a+b x)^2}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e)}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {e^2 (3 b B d+A b e-3 a B e) x}{b^4}+\frac {B e^3 x^2}{2 b^3}-\frac {(A b-a B) (b d-a e)^3}{2 b^5 (a+b x)^2}-\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^5 (a+b x)}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.74 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {-A b \left (5 a^3 e^3+a^2 b e^2 (-9 d+4 e x)+a b^2 e \left (3 d^2-12 d e x-4 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )+B \left (7 a^4 e^3+a^3 b e^2 (-15 d+2 e x)+a^2 b^2 e \left (9 d^2-12 d e x-11 e^2 x^2\right )+b^4 x \left (-2 d^3+6 d e^2 x^2+e^3 x^3\right )-a b^3 \left (d^3-12 d^2 e x-12 d e^2 x^2+4 e^3 x^3\right )\right )+6 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^2 \log (a+b x)}{2 b^5 (a+b x)^2} \]
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Time = 0.75 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.90
method | result | size |
default | \(\frac {e^{2} \left (\frac {1}{2} B b e \,x^{2}+A b e x -3 B a e x +3 B b d x \right )}{b^{4}}-\frac {3 e \left (A a b \,e^{2}-A \,b^{2} d e -2 B \,a^{2} e^{2}+3 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {-A \,a^{3} b \,e^{3}+3 A \,a^{2} b^{2} d \,e^{2}-3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}+B \,a^{4} e^{3}-3 B \,a^{3} b d \,e^{2}+3 B \,a^{2} b^{2} d^{2} e -B a \,b^{3} d^{3}}{2 b^{5} \left (b x +a \right )^{2}}-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -4 B \,a^{3} e^{3}+9 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}}{b^{5} \left (b x +a \right )}\) | \(268\) |
norman | \(\frac {\frac {e^{2} \left (A b e -2 B a e +3 B b d \right ) x^{3}}{b^{2}}-\frac {9 A \,a^{3} b \,e^{3}-9 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}-18 B \,a^{4} e^{3}+27 B \,a^{3} b d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}}{2 b^{5}}-\frac {\left (6 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -12 B \,a^{3} e^{3}+18 B \,a^{2} b d \,e^{2}-6 B a \,b^{2} d^{2} e +b^{3} B \,d^{3}\right ) x}{b^{4}}+\frac {B \,e^{3} x^{4}}{2 b}}{\left (b x +a \right )^{2}}-\frac {3 e \left (A a b \,e^{2}-A \,b^{2} d e -2 B \,a^{2} e^{2}+3 B a b d e -b^{2} B \,d^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(268\) |
risch | \(\frac {B \,e^{3} x^{2}}{2 b^{3}}+\frac {e^{3} A x}{b^{3}}-\frac {3 e^{3} B a x}{b^{4}}+\frac {3 e^{2} B d x}{b^{3}}+\frac {\left (-3 A \,a^{2} b \,e^{3}+6 A a \,b^{2} d \,e^{2}-3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-9 B \,a^{2} b d \,e^{2}+6 B a \,b^{2} d^{2} e -b^{3} B \,d^{3}\right ) x -\frac {5 A \,a^{3} b \,e^{3}-9 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e +A \,b^{4} d^{3}-7 B \,a^{4} e^{3}+15 B \,a^{3} b d \,e^{2}-9 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}}{2 b}}{b^{4} \left (b x +a \right )^{2}}-\frac {3 e^{3} \ln \left (b x +a \right ) A a}{b^{4}}+\frac {3 e^{2} \ln \left (b x +a \right ) A d}{b^{3}}+\frac {6 e^{3} \ln \left (b x +a \right ) B \,a^{2}}{b^{5}}-\frac {9 e^{2} \ln \left (b x +a \right ) B a d}{b^{4}}+\frac {3 e \ln \left (b x +a \right ) B \,d^{2}}{b^{3}}\) | \(304\) |
parallelrisch | \(-\frac {36 B x \,a^{2} b^{2} d \,e^{2}-12 B x a \,b^{3} d^{2} e -6 B \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e -12 A \ln \left (b x +a \right ) x a \,b^{3} d \,e^{2}+36 B \ln \left (b x +a \right ) x \,a^{2} b^{2} d \,e^{2}+18 B \ln \left (b x +a \right ) x^{2} a \,b^{3} d \,e^{2}+6 A \ln \left (b x +a \right ) x^{2} a \,b^{3} e^{3}-6 A \ln \left (b x +a \right ) x^{2} b^{4} d \,e^{2}+12 A \ln \left (b x +a \right ) x \,a^{2} b^{2} e^{3}+A \,b^{4} d^{3}+9 A \,a^{3} b \,e^{3}-24 B \ln \left (b x +a \right ) x \,a^{3} b \,e^{3}-12 B \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{3}-6 B \ln \left (b x +a \right ) x^{2} b^{4} d^{2} e -18 B \,a^{4} e^{3}-9 B \,a^{2} b^{2} d^{2} e -9 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e +27 B \,a^{3} b d \,e^{2}-12 B \ln \left (b x +a \right ) a^{4} e^{3}-2 A \,x^{3} b^{4} e^{3}+2 B x \,b^{4} d^{3}-B \,x^{4} e^{3} b^{4}+12 A x \,a^{2} b^{2} e^{3}+6 A x \,b^{4} d^{2} e -24 B x \,a^{3} b \,e^{3}+6 A \ln \left (b x +a \right ) a^{3} b \,e^{3}+4 B \,x^{3} a \,b^{3} e^{3}-6 B \,x^{3} b^{4} d \,e^{2}-6 A \ln \left (b x +a \right ) a^{2} b^{2} d \,e^{2}-12 B \ln \left (b x +a \right ) x a \,b^{3} d^{2} e +B a \,b^{3} d^{3}-12 A x a \,b^{3} d \,e^{2}+18 B \ln \left (b x +a \right ) a^{3} b d \,e^{2}}{2 b^{5} \left (b x +a \right )^{2}}\) | \(507\) |
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (137) = 274\).
Time = 0.23 (sec) , antiderivative size = 442, normalized size of antiderivative = 3.13 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {B b^{4} e^{3} x^{4} - {\left (B a b^{3} + A b^{4}\right )} d^{3} + 3 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e - 3 \, {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} + {\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \, {\left (3 \, B b^{4} d e^{2} - {\left (2 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + {\left (12 \, B a b^{3} d e^{2} - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \, {\left (B b^{4} d^{3} - 3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{2} - {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \, {\left (B a^{2} b^{2} d^{2} e - {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} + {\left (2 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{2} e - {\left (3 \, B a b^{3} - A b^{4}\right )} d e^{2} + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (B a b^{3} d^{2} e - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} + {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (141) = 282\).
Time = 2.09 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.12 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {B e^{3} x^{2}}{2 b^{3}} + x \left (\frac {A e^{3}}{b^{3}} - \frac {3 B a e^{3}}{b^{4}} + \frac {3 B d e^{2}}{b^{3}}\right ) + \frac {- 5 A a^{3} b e^{3} + 9 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e - A b^{4} d^{3} + 7 B a^{4} e^{3} - 15 B a^{3} b d e^{2} + 9 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3} + x \left (- 6 A a^{2} b^{2} e^{3} + 12 A a b^{3} d e^{2} - 6 A b^{4} d^{2} e + 8 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 12 B a b^{3} d^{2} e - 2 B b^{4} d^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {3 e \left (a e - b d\right ) \left (- A b e + 2 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (137) = 274\).
Time = 0.23 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.00 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=-\frac {{\left (B a b^{3} + A b^{4}\right )} d^{3} - 3 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \, {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{2} - {\left (7 \, B a^{4} - 5 \, A a^{3} b\right )} e^{3} + 2 \, {\left (B b^{4} d^{3} - 3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} - {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {B b e^{3} x^{2} + 2 \, {\left (3 \, B b d e^{2} - {\left (3 \, B a - A b\right )} e^{3}\right )} x}{2 \, b^{4}} + \frac {3 \, {\left (B b^{2} d^{2} e - {\left (3 \, B a b - A b^{2}\right )} d e^{2} + {\left (2 \, B a^{2} - A a b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (137) = 274\).
Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.01 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=\frac {3 \, {\left (B b^{2} d^{2} e - 3 \, B a b d e^{2} + A b^{2} d e^{2} + 2 \, B a^{2} e^{3} - A a b e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {B b^{3} e^{3} x^{2} + 6 \, B b^{3} d e^{2} x - 6 \, B a b^{2} e^{3} x + 2 \, A b^{3} e^{3} x}{2 \, b^{6}} - \frac {B a b^{3} d^{3} + A b^{4} d^{3} - 9 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 15 \, B a^{3} b d e^{2} - 9 \, A a^{2} b^{2} d e^{2} - 7 \, B a^{4} e^{3} + 5 \, A a^{3} b e^{3} + 2 \, {\left (B b^{4} d^{3} - 6 \, B a b^{3} d^{2} e + 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} - 6 \, A a b^{3} d e^{2} - 4 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \]
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Time = 1.32 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.06 \[ \int \frac {(A+B x) (d+e x)^3}{(a+b x)^3} \, dx=x\,\left (\frac {A\,e^3+3\,B\,d\,e^2}{b^3}-\frac {3\,B\,a\,e^3}{b^4}\right )-\frac {\frac {-7\,B\,a^4\,e^3+15\,B\,a^3\,b\,d\,e^2+5\,A\,a^3\,b\,e^3-9\,B\,a^2\,b^2\,d^2\,e-9\,A\,a^2\,b^2\,d\,e^2+B\,a\,b^3\,d^3+3\,A\,a\,b^3\,d^2\,e+A\,b^4\,d^3}{2\,b}+x\,\left (-4\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3-6\,B\,a\,b^2\,d^2\,e-6\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+\frac {\ln \left (a+b\,x\right )\,\left (6\,B\,a^2\,e^3-9\,B\,a\,b\,d\,e^2-3\,A\,a\,b\,e^3+3\,B\,b^2\,d^2\,e+3\,A\,b^2\,d\,e^2\right )}{b^5}+\frac {B\,e^3\,x^2}{2\,b^3} \]
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