Optimal. Leaf size=141 \[ \frac {2 \sqrt {a} p \left (3 b d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}+\frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {d p \left (b d^2-3 a e^2\right ) \log \left (a+b x^2\right )}{3 b e}-\frac {2 p x \left (3 b d^2-a e^2\right )}{3 b}-d e p x^2-\frac {2}{9} e^2 p x^3 \]
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Rubi [A] time = 0.13, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2463, 801, 635, 205, 260} \[ \frac {2 \sqrt {a} p \left (3 b d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2}}+\frac {(d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {d p \left (b d^2-3 a e^2\right ) \log \left (a+b x^2\right )}{3 b e}-\frac {2 p x \left (3 b d^2-a e^2\right )}{3 b}-d e p x^2-\frac {2}{9} e^2 p x^3 \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rule 2463
Rubi steps
\begin {align*} \int (d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\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*}
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Mathematica [A] time = 0.48, size = 211, normalized size = 1.50 \[ \frac {3 p \left (-3 \sqrt {-a} b d^2 e+3 a \sqrt {b} d e^2+\sqrt {-a} a e^3-b^{3/2} d^3\right ) \log \left (\sqrt {-a}-\sqrt {b} x\right )-3 p \left (-3 \sqrt {-a} b d^2 e-3 a \sqrt {b} d e^2+\sqrt {-a} a e^3+b^{3/2} d^3\right ) \log \left (\sqrt {-a}+\sqrt {b} x\right )+\sqrt {b} \left (3 b (d+e x)^3 \log \left (c \left (a+b x^2\right )^p\right )+6 a e^3 p x-b e p x \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )}{9 b^{3/2} e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 320, normalized size = 2.27 \[ \left [-\frac {2 \, b e^{2} p x^{3} + 9 \, b d e p x^{2} - 3 \, {\left (3 \, b d^{2} - a e^{2}\right )} p \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} p x - 3 \, {\left (b e^{2} p x^{3} + 3 \, b d e p x^{2} + 3 \, b d^{2} p x + 3 \, a d e p\right )} \log \left (b x^{2} + a\right ) - 3 \, {\left (b e^{2} x^{3} + 3 \, b d e x^{2} + 3 \, b d^{2} x\right )} \log \relax (c)}{9 \, b}, -\frac {2 \, b e^{2} p x^{3} + 9 \, b d e p x^{2} - 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} p \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} p x - 3 \, {\left (b e^{2} p x^{3} + 3 \, b d e p x^{2} + 3 \, b d^{2} p x + 3 \, a d e p\right )} \log \left (b x^{2} + a\right ) - 3 \, {\left (b e^{2} x^{3} + 3 \, b d e x^{2} + 3 \, b d^{2} x\right )} \log \relax (c)}{9 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 173, normalized size = 1.23 \[ \frac {2 \, {\left (3 \, a b d^{2} p - a^{2} p e^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} + \frac {3 \, b p x^{3} e^{2} \log \left (b x^{2} + a\right ) + 9 \, b d p x^{2} e \log \left (b x^{2} + a\right ) - 2 \, b p x^{3} e^{2} - 9 \, b d p x^{2} e + 9 \, b d^{2} p x \log \left (b x^{2} + a\right ) + 3 \, b x^{3} e^{2} \log \relax (c) + 9 \, b d x^{2} e \log \relax (c) - 18 \, b d^{2} p x + 9 \, a d p e \log \left (b x^{2} + a\right ) + 9 \, b d^{2} x \log \relax (c) + 6 \, a p x e^{2}}{9 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 965, normalized size = 6.84 \[ \frac {\left (e x +d \right )^{3} \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{3 e}+d e \,x^{2} \ln \relax (c )-\frac {d^{3} 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 )}{3 e}-\frac {d^{3} 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 )}{3 e}+\frac {e^{2} x^{3} \ln \relax (c )}{3}+d^{2} x \ln \relax (c )-\frac {2 e^{2} p \,x^{3}}{9}-2 d^{2} p x -d e p \,x^{2}-\frac {i \pi d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}{2}+\frac {i \pi \,e^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{6}+\frac {i \pi \,e^{2} x^{3} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{6}-\frac {i \pi d e \,x^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{2}+\frac {i \pi \,d^{2} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2}+\frac {i \pi \,d^{2} x \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2}+\frac {2 a \,e^{2} p x}{3 b}-\frac {i \pi \,e^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}{6}+\frac {i \pi d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2}+\frac {i \pi d e \,x^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,d^{2} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}{2}-\frac {i \pi \,e^{2} x^{3} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{6}-\frac {i \pi \,d^{2} x \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{2}+\frac {\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, 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 )}{3 b^{2} e}-\frac {\sqrt {-a^{3} b \,e^{6}+6 a^{2} b^{2} d^{2} e^{4}-9 a \,b^{3} d^{4} e^{2}}\, 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 )}{3 b^{2} e}+\frac {a d 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 )}{b}+\frac {a d 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 )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 131, normalized size = 0.93 \[ \frac {1}{9} \, {\left (\frac {9 \, a d e \log \left (b x^{2} + a\right )}{b^{2}} + \frac {6 \, {\left (3 \, a b d^{2} - a^{2} e^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} - \frac {2 \, b e^{2} x^{3} + 9 \, b d e x^{2} + 6 \, {\left (3 \, b d^{2} - a e^{2}\right )} x}{b^{2}}\right )} b p + \frac {1}{3} \, {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.28, size = 263, normalized size = 1.87 \[ \frac {e^2\,x^3\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{3}-2\,d^2\,p\,x-\frac {2\,e^2\,p\,x^3}{9}+d^2\,x\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )+d\,e\,x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )-d\,e\,p\,x^2+\frac {2\,a\,e^2\,p\,x}{3\,b}-\frac {2\,\sqrt {a}\,d^2\,p\,\mathrm {atan}\left (\frac {3\,\sqrt {a}\,b^{3/2}\,d^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}-\frac {a^{3/2}\,\sqrt {b}\,e^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}\right )}{\sqrt {b}}+\frac {2\,a^{3/2}\,e^2\,p\,\mathrm {atan}\left (\frac {3\,\sqrt {a}\,b^{3/2}\,d^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}-\frac {a^{3/2}\,\sqrt {b}\,e^2\,p\,x}{a^2\,e^2\,p-3\,a\,b\,d^2\,p}\right )}{3\,b^{3/2}}+\frac {a\,d\,e\,p\,\ln \left (b\,x^2+a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.02, size = 309, normalized size = 2.19 \[ \begin {cases} - \frac {i a^{\frac {3}{2}} e^{2} p \log {\left (a + b x^{2} \right )}}{3 b^{2} \sqrt {\frac {1}{b}}} + \frac {2 i a^{\frac {3}{2}} e^{2} p \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{3 b^{2} \sqrt {\frac {1}{b}}} + \frac {i \sqrt {a} d^{2} p \log {\left (a + b x^{2} \right )}}{b \sqrt {\frac {1}{b}}} - \frac {2 i \sqrt {a} d^{2} p \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{b \sqrt {\frac {1}{b}}} + \frac {a d e p \log {\left (a + b x^{2} \right )}}{b} + \frac {2 a e^{2} p x}{3 b} + d^{2} p x \log {\left (a + b x^{2} \right )} - 2 d^{2} p x + d^{2} x \log {\relax (c )} + d e p x^{2} \log {\left (a + b x^{2} \right )} - d e p x^{2} + d e x^{2} \log {\relax (c )} + \frac {e^{2} p x^{3} \log {\left (a + b x^{2} \right )}}{3} - \frac {2 e^{2} p x^{3}}{9} + \frac {e^{2} x^{3} \log {\relax (c )}}{3} & \text {for}\: b \neq 0 \\\left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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