Integrand size = 16, antiderivative size = 103 \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=\frac {3 d (b c-a d)^2 x}{b^4}+\frac {d^2 (3 b c-2 a d) x^2}{2 b^3}+\frac {d^3 x^3}{3 b^2}+\frac {a (b c-a d)^3}{b^5 (a+b x)}+\frac {(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5} \]
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Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=\frac {a (b c-a d)^3}{b^5 (a+b x)}+\frac {(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5}+\frac {3 d x (b c-a d)^2}{b^4}+\frac {d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac {d^3 x^3}{3 b^2} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 d (b c-a d)^2}{b^4}+\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^2}{b^2}+\frac {a (-b c+a d)^3}{b^4 (a+b x)^2}+\frac {(b c-4 a d) (b c-a d)^2}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {3 d (b c-a d)^2 x}{b^4}+\frac {d^2 (3 b c-2 a d) x^2}{2 b^3}+\frac {d^3 x^3}{3 b^2}+\frac {a (b c-a d)^3}{b^5 (a+b x)}+\frac {(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.97 \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=\frac {18 b d (b c-a d)^2 x+3 b^2 d^2 (3 b c-2 a d) x^2+2 b^3 d^3 x^3-\frac {6 a (-b c+a d)^3}{a+b x}+6 (b c-4 a d) (b c-a d)^2 \log (a+b x)}{6 b^5} \]
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Time = 0.45 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.54
method | result | size |
default | \(\frac {d \left (\frac {1}{3} d^{2} x^{3} b^{2}-x^{2} a b \,d^{2}+\frac {3}{2} x^{2} b^{2} c d +3 a^{2} d^{2} x -6 a b c d x +3 b^{2} c^{2} x \right )}{b^{4}}+\frac {\left (-4 a^{3} d^{3}+9 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{5} \left (b x +a \right )}\) | \(159\) |
norman | \(\frac {\frac {\left (4 a^{4} d^{3}-9 a^{3} b c \,d^{2}+6 a^{2} b^{2} c^{2} d -a \,b^{3} c^{3}\right ) x}{b^{4} a}+\frac {d^{3} x^{4}}{3 b}+\frac {d \left (4 a^{2} d^{2}-9 a b c d +6 b^{2} c^{2}\right ) x^{2}}{2 b^{3}}-\frac {d^{2} \left (4 a d -9 b c \right ) x^{3}}{6 b^{2}}}{b x +a}-\frac {\left (4 a^{3} d^{3}-9 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(170\) |
risch | \(\frac {d^{3} x^{3}}{3 b^{2}}-\frac {d^{3} x^{2} a}{b^{3}}+\frac {3 d^{2} x^{2} c}{2 b^{2}}+\frac {3 d^{3} a^{2} x}{b^{4}}-\frac {6 d^{2} a c x}{b^{3}}+\frac {3 d \,c^{2} x}{b^{2}}-\frac {a^{4} d^{3}}{b^{5} \left (b x +a \right )}+\frac {3 a^{3} c \,d^{2}}{b^{4} \left (b x +a \right )}-\frac {3 a^{2} c^{2} d}{b^{3} \left (b x +a \right )}+\frac {a \,c^{3}}{b^{2} \left (b x +a \right )}-\frac {4 \ln \left (b x +a \right ) a^{3} d^{3}}{b^{5}}+\frac {9 \ln \left (b x +a \right ) a^{2} c \,d^{2}}{b^{4}}-\frac {6 \ln \left (b x +a \right ) a \,c^{2} d}{b^{3}}+\frac {\ln \left (b x +a \right ) c^{3}}{b^{2}}\) | \(205\) |
parallelrisch | \(-\frac {-2 d^{3} x^{4} b^{4}+4 x^{3} a \,b^{3} d^{3}-9 x^{3} b^{4} c \,d^{2}+24 \ln \left (b x +a \right ) x \,a^{3} b \,d^{3}-54 \ln \left (b x +a \right ) x \,a^{2} b^{2} c \,d^{2}+36 \ln \left (b x +a \right ) x a \,b^{3} c^{2} d -6 \ln \left (b x +a \right ) x \,b^{4} c^{3}-12 x^{2} a^{2} b^{2} d^{3}+27 x^{2} a \,b^{3} c \,d^{2}-18 x^{2} b^{4} c^{2} d +24 \ln \left (b x +a \right ) a^{4} d^{3}-54 \ln \left (b x +a \right ) a^{3} b c \,d^{2}+36 \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d -6 \ln \left (b x +a \right ) a \,b^{3} c^{3}+24 a^{4} d^{3}-54 a^{3} b c \,d^{2}+36 a^{2} b^{2} c^{2} d -6 a \,b^{3} c^{3}}{6 b^{5} \left (b x +a \right )}\) | \(257\) |
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (99) = 198\).
Time = 0.22 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.39 \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=\frac {2 \, b^{4} d^{3} x^{4} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 3 \, {\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{2} + 18 \, {\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x + 6 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{6} x + a b^{5}\right )}} \]
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Time = 0.34 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.44 \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=x^{2} \left (- \frac {a d^{3}}{b^{3}} + \frac {3 c d^{2}}{2 b^{2}}\right ) + x \left (\frac {3 a^{2} d^{3}}{b^{4}} - \frac {6 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{b^{2}}\right ) + \frac {- a^{4} d^{3} + 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + a b^{3} c^{3}}{a b^{5} + b^{6} x} + \frac {d^{3} x^{3}}{3 b^{2}} - \frac {\left (a d - b c\right )^{2} \cdot \left (4 a d - b c\right ) \log {\left (a + b x \right )}}{b^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.61 \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=\frac {a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{b^{6} x + a b^{5}} + \frac {2 \, b^{2} d^{3} x^{3} + 3 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{2} + 18 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{4}} + \frac {{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (99) = 198\).
Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.24 \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=\frac {\frac {{\left (2 \, d^{3} + \frac {3 \, {\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {18 \, {\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{3}}{b^{4}} - \frac {6 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {6 \, {\left (\frac {a b^{6} c^{3}}{b x + a} - \frac {3 \, a^{2} b^{5} c^{2} d}{b x + a} + \frac {3 \, a^{3} b^{4} c d^{2}}{b x + a} - \frac {a^{4} b^{3} d^{3}}{b x + a}\right )}}{b^{7}}}{6 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.83 \[ \int \frac {x (c+d x)^3}{(a+b x)^2} \, dx=x\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )-x^2\,\left (\frac {a\,d^3}{b^3}-\frac {3\,c\,d^2}{2\,b^2}\right )-\frac {a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}{b\,\left (x\,b^5+a\,b^4\right )}+\frac {d^3\,x^3}{3\,b^2}-\frac {\ln \left (a+b\,x\right )\,\left (4\,a^3\,d^3-9\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b^5} \]
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