Integrand size = 15, antiderivative size = 78 \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=-\frac {3 a x}{b^3}+\frac {(a+b x)^2}{2 b^4}-\frac {a \left (a^2-3 c\right ) \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{b^4 \sqrt {c}}+\frac {\left (3 a^2-c\right ) \log \left (c+(a+b x)^2\right )}{2 b^4} \]
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Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {378, 716, 649, 209, 266} \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=-\frac {a \left (a^2-3 c\right ) \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{b^4 \sqrt {c}}+\frac {\left (3 a^2-c\right ) \log \left ((a+b x)^2+c\right )}{2 b^4}+\frac {(a+b x)^2}{2 b^4}-\frac {3 a x}{b^3} \]
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Rule 209
Rule 266
Rule 378
Rule 649
Rule 716
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-a+x)^3}{c+x^2} \, dx,x,a+b x\right )}{b^4} \\ & = \frac {\text {Subst}\left (\int \left (-3 a+x-\frac {a^3-3 a c-\left (3 a^2-c\right ) x}{c+x^2}\right ) \, dx,x,a+b x\right )}{b^4} \\ & = -\frac {3 a x}{b^3}+\frac {(a+b x)^2}{2 b^4}-\frac {\text {Subst}\left (\int \frac {a^3-3 a c-\left (3 a^2-c\right ) x}{c+x^2} \, dx,x,a+b x\right )}{b^4} \\ & = -\frac {3 a x}{b^3}+\frac {(a+b x)^2}{2 b^4}-\frac {\left (a \left (a^2-3 c\right )\right ) \text {Subst}\left (\int \frac {1}{c+x^2} \, dx,x,a+b x\right )}{b^4}+\frac {\left (3 a^2-c\right ) \text {Subst}\left (\int \frac {x}{c+x^2} \, dx,x,a+b x\right )}{b^4} \\ & = -\frac {3 a x}{b^3}+\frac {(a+b x)^2}{2 b^4}-\frac {a \left (a^2-3 c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b^4 \sqrt {c}}+\frac {\left (3 a^2-c\right ) \log \left (c+(a+b x)^2\right )}{2 b^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=\frac {b x (-4 a+b x)-\frac {2 \left (a^3-3 a c\right ) \arctan \left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}+\left (3 a^2-c\right ) \log \left (a^2+c+2 a b x+b^2 x^2\right )}{2 b^4} \]
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Time = 0.91 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.42
method | result | size |
default | \(-\frac {-\frac {1}{2} b \,x^{2}+2 a x}{b^{3}}+\frac {\frac {\left (3 a^{2} b -b c \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{2 b^{2}}+\frac {\left (2 a^{3}+2 a c -\frac {\left (3 a^{2} b -b c \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 \sqrt {c}\, b}\right )}{\sqrt {c}\, b}}{b^{3}}\) | \(111\) |
risch | \(\frac {x^{2}}{2 b^{2}}-\frac {2 a x}{b^{3}}+\frac {3 \ln \left (-a^{3} c -\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, b x +3 a \,c^{2}-\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, a \right ) a^{2}}{2 b^{4}}-\frac {c \ln \left (-a^{3} c -\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, b x +3 a \,c^{2}-\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, a \right )}{2 b^{4}}+\frac {\ln \left (-a^{3} c -\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, b x +3 a \,c^{2}-\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, a \right ) \sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}}{2 b^{4} c}+\frac {3 \ln \left (-a^{3} c +\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, b x +3 a \,c^{2}+\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, a \right ) a^{2}}{2 b^{4}}-\frac {c \ln \left (-a^{3} c +\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, b x +3 a \,c^{2}+\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, a \right )}{2 b^{4}}-\frac {\ln \left (-a^{3} c +\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, b x +3 a \,c^{2}+\sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}\, a \right ) \sqrt {-c \,a^{2} \left (a^{2}-3 c \right )^{2}}}{2 b^{4} c}\) | \(419\) |
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Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.54 \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=\left [\frac {b^{2} c x^{2} - 4 \, a b c x + {\left (a^{3} - 3 \, a c\right )} \sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) + {\left (3 \, a^{2} c - c^{2}\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4} c}, \frac {b^{2} c x^{2} - 4 \, a b c x - 2 \, {\left (a^{3} - 3 \, a c\right )} \sqrt {c} \arctan \left (\frac {b x + a}{\sqrt {c}}\right ) + {\left (3 \, a^{2} c - c^{2}\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4} c}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (71) = 142\).
Time = 0.40 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.68 \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=- \frac {2 a x}{b^{3}} + \left (- \frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) \log {\left (x + \frac {a^{4} - 2 b^{4} c \left (- \frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) - c^{2}}{a^{3} b - 3 a b c} \right )} + \left (\frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) \log {\left (x + \frac {a^{4} - 2 b^{4} c \left (\frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) - c^{2}}{a^{3} b - 3 a b c} \right )} + \frac {x^{2}}{2 b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.04 \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=\frac {b x^{2} - 4 \, a x}{2 \, b^{3}} + \frac {{\left (3 \, a^{2} - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4}} - \frac {{\left (a^{3} - 3 \, a c\right )} \arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{b^{4} \sqrt {c}} \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99 \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=\frac {{\left (3 \, a^{2} - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4}} - \frac {{\left (a^{3} - 3 \, a c\right )} \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b^{4} \sqrt {c}} + \frac {b^{2} x^{2} - 4 \, a b x}{2 \, b^{4}} \]
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Time = 9.96 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12 \[ \int \frac {x^3}{c+(a+b x)^2} \, dx=\frac {x^2}{2\,b^2}-\frac {2\,a\,x}{b^3}-\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+c\right )\,\left (4\,b^4\,c^2-12\,a^2\,b^4\,c\right )}{8\,b^8\,c}+\frac {a\,\mathrm {atan}\left (\frac {a+b\,x}{\sqrt {c}}\right )\,\left (3\,c-a^2\right )}{b^4\,\sqrt {c}} \]
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