Integrand size = 20, antiderivative size = 141 \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {2 \left (3 b d^2-a e^2\right ) p x}{3 b}-d e p x^2-\frac {2}{9} e^2 p x^3+\frac {2 \sqrt {a} \left (3 b d^2-a e^2\right ) p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}-\frac {d \left (b d^2-3 a e^2\right ) p \log \left (a+b x^2\right )}{3 b e}+\frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e} \]
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Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2513, 815, 649, 211, 266} \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {2 \sqrt {a} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 b d^2-a e^2\right )}{3 b^{3/2}}+\frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {d p \left (b d^2-3 a e^2\right ) \log \left (a+b x^2\right )}{3 b e}-\frac {2 p x \left (3 b d^2-a e^2\right )}{3 b}-d e p x^2-\frac {2}{9} e^2 p x^3 \]
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Rule 211
Rule 266
Rule 649
Rule 815
Rule 2513
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {(2 b p) \int \frac {x (d+e x)^3}{a+b x^2} \, dx}{3 e} \\ & = \frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {(2 b p) \int \left (\frac {e \left (3 b d^2-a e^2\right )}{b^2}+\frac {3 d e^2 x}{b}+\frac {e^3 x^2}{b}-\frac {a e \left (3 b d^2-a e^2\right )-b d \left (b d^2-3 a e^2\right ) x}{b^2 \left (a+b x^2\right )}\right ) \, dx}{3 e} \\ & = -\frac {2 \left (3 b d^2-a e^2\right ) p x}{3 b}-d e p x^2-\frac {2}{9} e^2 p x^3+\frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}+\frac {(2 p) \int \frac {a e \left (3 b d^2-a e^2\right )-b d \left (b d^2-3 a e^2\right ) x}{a+b x^2} \, dx}{3 b e} \\ & = -\frac {2 \left (3 b d^2-a e^2\right ) p x}{3 b}-d e p x^2-\frac {2}{9} e^2 p x^3+\frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {\left (2 d \left (b d^2-3 a e^2\right ) p\right ) \int \frac {x}{a+b x^2} \, dx}{3 e}+\frac {\left (2 a \left (3 b d^2-a e^2\right ) p\right ) \int \frac {1}{a+b x^2} \, dx}{3 b} \\ & = -\frac {2 \left (3 b d^2-a e^2\right ) p x}{3 b}-d e p x^2-\frac {2}{9} e^2 p x^3+\frac {2 \sqrt {a} \left (3 b d^2-a e^2\right ) p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}-\frac {d \left (b d^2-3 a e^2\right ) p \log \left (a+b x^2\right )}{3 b e}+\frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.50 \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {3 \left (-b^{3/2} d^3-3 \sqrt {-a} b d^2 e+3 a \sqrt {b} d e^2+\sqrt {-a} a e^3\right ) p \log \left (\sqrt {-a}-\sqrt {b} x\right )-3 \left (b^{3/2} d^3-3 \sqrt {-a} b d^2 e-3 a \sqrt {b} d e^2+\sqrt {-a} a e^3\right ) p \log \left (\sqrt {-a}+\sqrt {b} x\right )+\sqrt {b} \left (6 a e^3 p x-b e p x \left (18 d^2+9 d e x+2 e^2 x^2\right )+3 b (d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )\right )}{9 b^{3/2} e} \]
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Time = 1.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.33
method | result | size |
parts | \(\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e^{2} x^{3}}{3}+\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e d \,x^{2}+d^{2} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) x +\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) d^{3}}{3 e}-\frac {2 p b \left (-\frac {e \left (-\frac {1}{3} x^{3} b \,e^{2}-\frac {3}{2} b d e \,x^{2}+x a \,e^{2}-3 b \,d^{2} x \right )}{b^{2}}+\frac {\frac {\left (-3 a b d \,e^{2}+b^{2} d^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (a^{2} e^{3}-3 a b \,d^{2} e \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{b^{2}}\right )}{3 e}\) | \(187\) |
risch | \(-\frac {i e^{2} \pi \,x^{3} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{6}-\frac {i x \pi \,d^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}-\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}+\frac {2 x a p \,e^{2}}{3 b}+\frac {p \ln \left (-a^{2} e^{3}+3 a b \,d^{2} e -\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, x \right ) \sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}}{3 e \,b^{2}}-\frac {p \ln \left (-a^{2} e^{3}+3 a b \,d^{2} e +\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, x \right ) \sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}}{3 e \,b^{2}}+\frac {\left (e x +d \right )^{3} \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{3 e}-\frac {p \ln \left (-a^{2} e^{3}+3 a b \,d^{2} e -\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, x \right ) d^{3}}{3 e}-\frac {p \ln \left (-a^{2} e^{3}+3 a b \,d^{2} e +\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, x \right ) d^{3}}{3 e}+e \ln \left (c \right ) d \,x^{2}+\frac {i e^{2} \pi \,x^{3} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{6}+\frac {i x \pi \,d^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {i x \pi \,d^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i e \pi d \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{6}+\frac {\ln \left (c \right ) e^{2} x^{3}}{3}+\frac {e p \ln \left (-a^{2} e^{3}+3 a b \,d^{2} e -\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, x \right ) a d}{b}+\frac {e p \ln \left (-a^{2} e^{3}+3 a b \,d^{2} e +\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, x \right ) a d}{b}+\ln \left (c \right ) d^{2} x -\frac {i x \pi \,d^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{6}+\frac {i e \pi d \,x^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-d e p \,x^{2}-\frac {2 e^{2} p \,x^{3}}{9}-2 d^{2} p x\) | \(965\) |
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Time = 0.36 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.27 \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\left [-\frac {2 \, b e^{2} p x^{3} + 9 \, b d e p x^{2} - 3 \, {\left (3 \, b d^{2} - a e^{2}\right )} p \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} p x - 3 \, {\left (b e^{2} p x^{3} + 3 \, b d e p x^{2} + 3 \, b d^{2} p x + 3 \, a d e p\right )} \log \left (b x^{2} + a\right ) - 3 \, {\left (b e^{2} x^{3} + 3 \, b d e x^{2} + 3 \, b d^{2} x\right )} \log \left (c\right )}{9 \, b}, -\frac {2 \, b e^{2} p x^{3} + 9 \, b d e p x^{2} - 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} p \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} p x - 3 \, {\left (b e^{2} p x^{3} + 3 \, b d e p x^{2} + 3 \, b d^{2} p x + 3 \, a d e p\right )} \log \left (b x^{2} + a\right ) - 3 \, {\left (b e^{2} x^{3} + 3 \, b d e x^{2} + 3 \, b d^{2} x\right )} \log \left (c\right )}{9 \, b}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (131) = 262\).
Time = 8.66 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.65 \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\begin {cases} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) \log {\left (0^{p} c \right )} & \text {for}\: a = 0 \wedge b = 0 \\\left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) \log {\left (a^{p} c \right )} & \text {for}\: b = 0 \\- 2 d^{2} p x + d^{2} x \log {\left (c \left (b x^{2}\right )^{p} \right )} - d e p x^{2} + d e x^{2} \log {\left (c \left (b x^{2}\right )^{p} \right )} - \frac {2 e^{2} p x^{3}}{9} + \frac {e^{2} x^{3} \log {\left (c \left (b x^{2}\right )^{p} \right )}}{3} & \text {for}\: a = 0 \\- \frac {2 a^{2} e^{2} p \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{3 b^{2} \sqrt {- \frac {a}{b}}} + \frac {a^{2} e^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{3 b^{2} \sqrt {- \frac {a}{b}}} + \frac {2 a d^{2} p \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{b \sqrt {- \frac {a}{b}}} - \frac {a d^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{b \sqrt {- \frac {a}{b}}} + \frac {a d e \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{b} + \frac {2 a e^{2} p x}{3 b} - 2 d^{2} p x + d^{2} x \log {\left (c \left (a + b x^{2}\right )^{p} \right )} - d e p x^{2} + d e x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )} - \frac {2 e^{2} p x^{3}}{9} + \frac {e^{2} x^{3} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93 \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {1}{9} \, {\left (\frac {9 \, a d e \log \left (b x^{2} + a\right )}{b^{2}} + \frac {6 \, {\left (3 \, a b d^{2} - a^{2} e^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} - \frac {2 \, b e^{2} x^{3} + 9 \, b d e x^{2} + 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} x}{b^{2}}\right )} b p + \frac {1}{3} \, {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \]
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Time = 0.44 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.08 \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {1}{9} \, {\left (2 \, e^{2} p - 3 \, e^{2} \log \left (c\right )\right )} x^{3} + \frac {a d e p \log \left (b x^{2} + a\right )}{b} - {\left (d e p - d e \log \left (c\right )\right )} x^{2} + \frac {1}{3} \, {\left (e^{2} p x^{3} + 3 \, d e p x^{2} + 3 \, d^{2} p x\right )} \log \left (b x^{2} + a\right ) - \frac {{\left (6 \, b d^{2} p - 2 \, a e^{2} p - 3 \, b d^{2} \log \left (c\right )\right )} x}{3 \, b} + \frac {2 \, {\left (3 \, a b d^{2} p - a^{2} e^{2} p\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} \]
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Time = 4.45 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.87 \[ \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {e^2\,x^3\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{3}-2\,d^2\,p\,x-\frac {2\,e^2\,p\,x^3}{9}+d^2\,x\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )+d\,e\,x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )-d\,e\,p\,x^2+\frac {2\,a\,e^2\,p\,x}{3\,b}-\frac {2\,\sqrt {a}\,d^2\,p\,\mathrm {atan}\left (\frac {3\,\sqrt {a}\,b^{3/2}\,d^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}-\frac {a^{3/2}\,\sqrt {b}\,e^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}\right )}{\sqrt {b}}+\frac {2\,a^{3/2}\,e^2\,p\,\mathrm {atan}\left (\frac {3\,\sqrt {a}\,b^{3/2}\,d^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}-\frac {a^{3/2}\,\sqrt {b}\,e^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}\right )}{3\,b^{3/2}}+\frac {a\,d\,e\,p\,\ln \left (b\,x^2+a\right )}{b} \]
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