Integrand size = 18, antiderivative size = 94 \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=-\frac {a (b c-a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^2}{2 b^3}+\frac {d (2 b c-a d) x^3}{3 b^2}+\frac {d^2 x^4}{4 b}+\frac {a^2 (b c-a d)^2 \log (a+b x)}{b^5} \]
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Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=\frac {a^2 (b c-a d)^2 \log (a+b x)}{b^5}-\frac {a x (b c-a d)^2}{b^4}+\frac {x^2 (b c-a d)^2}{2 b^3}+\frac {d x^3 (2 b c-a d)}{3 b^2}+\frac {d^2 x^4}{4 b} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a (-b c+a d)^2}{b^4}+\frac {(b c-a d)^2 x}{b^3}+\frac {d (2 b c-a d) x^2}{b^2}+\frac {d^2 x^3}{b}+\frac {a^2 (-b c+a d)^2}{b^4 (a+b x)}\right ) \, dx \\ & = -\frac {a (b c-a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^2}{2 b^3}+\frac {d (2 b c-a d) x^3}{3 b^2}+\frac {d^2 x^4}{4 b}+\frac {a^2 (b c-a d)^2 \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=\frac {b x \left (-12 a^3 d^2+6 a^2 b d (4 c+d x)-4 a b^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^3 x \left (6 c^2+8 c d x+3 d^2 x^2\right )\right )+12 a^2 (b c-a d)^2 \log (a+b x)}{12 b^5} \]
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Time = 1.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29
method | result | size |
norman | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 b^{3}}+\frac {d^{2} x^{4}}{4 b}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{b^{4}}-\frac {d \left (a d -2 b c \right ) x^{3}}{3 b^{2}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(121\) |
default | \(-\frac {-\frac {d^{2} x^{4} b^{3}}{4}+\frac {\left (\left (a d -b c \right ) b^{2} d -b^{3} d c \right ) x^{3}}{3}+\frac {\left (\left (a d -b c \right ) b^{2} c -b d \left (a^{2} d -a b c \right )\right ) x^{2}}{2}+\left (a d -b c \right ) \left (a^{2} d -a b c \right ) x}{b^{4}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(134\) |
risch | \(\frac {d^{2} x^{4}}{4 b}-\frac {x^{3} a \,d^{2}}{3 b^{2}}+\frac {2 x^{3} d c}{3 b}-\frac {x^{2} a c d}{b^{2}}+\frac {x^{2} c^{2}}{2 b}+\frac {x^{2} a^{2} d^{2}}{2 b^{3}}-\frac {a^{3} d^{2} x}{b^{4}}+\frac {2 a^{2} c d x}{b^{3}}-\frac {a \,c^{2} x}{b^{2}}+\frac {a^{4} \ln \left (b x +a \right ) d^{2}}{b^{5}}-\frac {2 a^{3} \ln \left (b x +a \right ) c d}{b^{4}}+\frac {a^{2} \ln \left (b x +a \right ) c^{2}}{b^{3}}\) | \(152\) |
parallelrisch | \(\frac {3 d^{2} x^{4} b^{4}-4 x^{3} a \,b^{3} d^{2}+8 x^{3} b^{4} c d +6 x^{2} a^{2} b^{2} d^{2}-12 x^{2} a \,b^{3} c d +6 x^{2} b^{4} c^{2}+12 \ln \left (b x +a \right ) a^{4} d^{2}-24 \ln \left (b x +a \right ) a^{3} b c d +12 \ln \left (b x +a \right ) a^{2} b^{2} c^{2}-12 x \,a^{3} b \,d^{2}+24 x \,a^{2} b^{2} c d -12 x a \,b^{3} c^{2}}{12 b^{5}}\) | \(152\) |
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Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.43 \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=\frac {3 \, b^{4} d^{2} x^{4} + 4 \, {\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} - 12 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x + 12 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]
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Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=\frac {a^{2} \left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{5}} + x^{3} \left (- \frac {a d^{2}}{3 b^{2}} + \frac {2 c d}{3 b}\right ) + x^{2} \left (\frac {a^{2} d^{2}}{2 b^{3}} - \frac {a c d}{b^{2}} + \frac {c^{2}}{2 b}\right ) + x \left (- \frac {a^{3} d^{2}}{b^{4}} + \frac {2 a^{2} c d}{b^{3}} - \frac {a c^{2}}{b^{2}}\right ) + \frac {d^{2} x^{4}}{4 b} \]
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Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.40 \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=\frac {3 \, b^{3} d^{2} x^{4} + 4 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2} - 12 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{12 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x + a\right )}{b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.48 \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=\frac {3 \, b^{3} d^{2} x^{4} + 8 \, b^{3} c d x^{3} - 4 \, a b^{2} d^{2} x^{3} + 6 \, b^{3} c^{2} x^{2} - 12 \, a b^{2} c d x^{2} + 6 \, a^{2} b d^{2} x^{2} - 12 \, a b^{2} c^{2} x + 24 \, a^{2} b c d x - 12 \, a^{3} d^{2} x}{12 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \]
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Time = 0.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.50 \[ \int \frac {x^2 (c+d x)^2}{a+b x} \, dx=x^2\,\left (\frac {c^2}{2\,b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{2\,b}\right )-x^3\,\left (\frac {a\,d^2}{3\,b^2}-\frac {2\,c\,d}{3\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2\right )}{b^5}+\frac {d^2\,x^4}{4\,b}-\frac {a\,x\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{b} \]
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