Optimal. Leaf size=178 \[ -\frac {p \left (a^2 e^4-6 a b d^2 e^2+b^2 d^4\right ) \log \left (a+b x^2\right )}{4 b^2 e}+\frac {2 \sqrt {a} d p \left (b d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+\frac {(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac {e p x^2 \left (6 b d^2-a e^2\right )}{4 b}-\frac {2 d p x \left (b d^2-a e^2\right )}{b}-\frac {2}{3} d e^2 p x^3-\frac {1}{8} e^3 p x^4 \]
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Rubi [A] time = 0.16, antiderivative size = 178, 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} \[ -\frac {p \left (a^2 e^4-6 a b d^2 e^2+b^2 d^4\right ) \log \left (a+b x^2\right )}{4 b^2 e}+\frac {2 \sqrt {a} d p \left (b d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+\frac {(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac {e p x^2 \left (6 b d^2-a e^2\right )}{4 b}-\frac {2 d p x \left (b d^2-a e^2\right )}{b}-\frac {2}{3} d e^2 p x^3-\frac {1}{8} e^3 p x^4 \]
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)^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac {(b p) \int \frac {x (d+e x)^4}{a+b x^2} \, dx}{2 e}\\ &=\frac {(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}-\frac {(b p) \int \left (\frac {4 d e \left (b d^2-a e^2\right )}{b^2}+\frac {e^2 \left (6 b d^2-a e^2\right ) x}{b^2}+\frac {4 d e^3 x^2}{b}+\frac {e^4 x^3}{b}-\frac {4 a d e \left (b d^2-a e^2\right )-\left (b^2 d^4-6 a b d^2 e^2+a^2 e^4\right ) x}{b^2 \left (a+b x^2\right )}\right ) \, dx}{2 e}\\ &=-\frac {2 d \left (b d^2-a e^2\right ) p x}{b}-\frac {e \left (6 b d^2-a e^2\right ) p x^2}{4 b}-\frac {2}{3} d e^2 p x^3-\frac {1}{8} e^3 p x^4+\frac {(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}+\frac {p \int \frac {4 a d e \left (b d^2-a e^2\right )-\left (b^2 d^4-6 a b d^2 e^2+a^2 e^4\right ) x}{a+b x^2} \, dx}{2 b e}\\ &=-\frac {2 d \left (b d^2-a e^2\right ) p x}{b}-\frac {e \left (6 b d^2-a e^2\right ) p x^2}{4 b}-\frac {2}{3} d e^2 p x^3-\frac {1}{8} e^3 p x^4+\frac {(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}+\frac {\left (2 a d \left (b d^2-a e^2\right ) p\right ) \int \frac {1}{a+b x^2} \, dx}{b}+\frac {\left (\left (-b^2 d^4+6 a b d^2 e^2-a^2 e^4\right ) p\right ) \int \frac {x}{a+b x^2} \, dx}{2 b e}\\ &=-\frac {2 d \left (b d^2-a e^2\right ) p x}{b}-\frac {e \left (6 b d^2-a e^2\right ) p x^2}{4 b}-\frac {2}{3} d e^2 p x^3-\frac {1}{8} e^3 p x^4+\frac {2 \sqrt {a} d \left (b d^2-a e^2\right ) p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {\left (b^2 d^4-6 a b d^2 e^2+a^2 e^4\right ) p \log \left (a+b x^2\right )}{4 b^2 e}+\frac {(d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )}{4 e}\\ \end {align*}
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Mathematica [A] time = 0.78, size = 249, normalized size = 1.40 \[ \frac {-6 p \left (a^2 e^4+4 \sqrt {-a} b^{3/2} d^3 e-6 a b d^2 e^2+4 (-a)^{3/2} \sqrt {b} d e^3+b^2 d^4\right ) \log \left (\sqrt {-a}-\sqrt {b} x\right )-6 p \left (a^2 e^4-4 \sqrt {-a} b^{3/2} d^3 e-6 a b d^2 e^2+4 \sqrt {-a} a \sqrt {b} d e^3+b^2 d^4\right ) \log \left (\sqrt {-a}+\sqrt {b} x\right )+b \left (6 b (d+e x)^4 \log \left (c \left (a+b x^2\right )^p\right )+6 a e^3 p x (8 d+e x)-b e p x \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )}{24 b^2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 498, normalized size = 2.80 \[ \left [-\frac {3 \, b^{2} e^{3} p x^{4} + 16 \, b^{2} d e^{2} p x^{3} + 6 \, {\left (6 \, b^{2} d^{2} e - a b e^{3}\right )} p x^{2} - 24 \, {\left (b^{2} d^{3} - a b d e^{2}\right )} p \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 48 \, {\left (b^{2} d^{3} - a b d e^{2}\right )} p x - 6 \, {\left (b^{2} e^{3} p x^{4} + 4 \, b^{2} d e^{2} p x^{3} + 6 \, b^{2} d^{2} e p x^{2} + 4 \, b^{2} d^{3} p x + {\left (6 \, a b d^{2} e - a^{2} e^{3}\right )} p\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} d^{3} x\right )} \log \relax (c)}{24 \, b^{2}}, -\frac {3 \, b^{2} e^{3} p x^{4} + 16 \, b^{2} d e^{2} p x^{3} + 6 \, {\left (6 \, b^{2} d^{2} e - a b e^{3}\right )} p x^{2} - 48 \, {\left (b^{2} d^{3} - a b d e^{2}\right )} p \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 48 \, {\left (b^{2} d^{3} - a b d e^{2}\right )} p x - 6 \, {\left (b^{2} e^{3} p x^{4} + 4 \, b^{2} d e^{2} p x^{3} + 6 \, b^{2} d^{2} e p x^{2} + 4 \, b^{2} d^{3} p x + {\left (6 \, a b d^{2} e - a^{2} e^{3}\right )} p\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} d^{3} x\right )} \log \relax (c)}{24 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 273, normalized size = 1.53 \[ \frac {2 \, {\left (a b d^{3} p - a^{2} d p e^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {6 \, b^{2} p x^{4} e^{3} \log \left (b x^{2} + a\right ) + 24 \, b^{2} d p x^{3} e^{2} \log \left (b x^{2} + a\right ) + 36 \, b^{2} d^{2} p x^{2} e \log \left (b x^{2} + a\right ) - 3 \, b^{2} p x^{4} e^{3} - 16 \, b^{2} d p x^{3} e^{2} - 36 \, b^{2} d^{2} p x^{2} e + 24 \, b^{2} d^{3} p x \log \left (b x^{2} + a\right ) + 6 \, b^{2} x^{4} e^{3} \log \relax (c) + 24 \, b^{2} d x^{3} e^{2} \log \relax (c) + 36 \, b^{2} d^{2} x^{2} e \log \relax (c) - 48 \, b^{2} d^{3} p x + 36 \, a b d^{2} p e \log \left (b x^{2} + a\right ) + 24 \, b^{2} d^{3} x \log \relax (c) + 6 \, a b p x^{2} e^{3} + 48 \, a b d p x e^{2} - 6 \, a^{2} p e^{3} \log \left (b x^{2} + a\right )}{24 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.72, size = 1330, normalized size = 7.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 177, normalized size = 0.99 \[ \frac {1}{24} \, b p {\left (\frac {48 \, {\left (a b d^{3} - a^{2} d e^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} - \frac {3 \, b e^{3} x^{4} + 16 \, b d e^{2} x^{3} + 6 \, {\left (6 \, b d^{2} e - a e^{3}\right )} x^{2} + 48 \, {\left (b d^{3} - a d e^{2}\right )} x}{b^{2}} + \frac {6 \, {\left (6 \, a b d^{2} e - a^{2} e^{3}\right )} \log \left (b x^{2} + a\right )}{b^{3}}\right )} + \frac {1}{4} \, {\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 222, normalized size = 1.25 \[ \frac {e^3\,x^4\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{4}-2\,d^3\,p\,x-\frac {e^3\,p\,x^4}{8}+d^3\,x\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )+\frac {3\,d^2\,e\,x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{2}+d\,e^2\,x^3\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )-\frac {3\,d^2\,e\,p\,x^2}{2}-\frac {2\,d\,e^2\,p\,x^3}{3}+\frac {a\,e^3\,p\,x^2}{4\,b}+\frac {2\,\sqrt {a}\,d^3\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {a^2\,e^3\,p\,\ln \left (b\,x^2+a\right )}{4\,b^2}-\frac {2\,a^{3/2}\,d\,e^2\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{3/2}}+\frac {2\,a\,d\,e^2\,p\,x}{b}+\frac {3\,a\,d^2\,e\,p\,\ln \left (b\,x^2+a\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 47.87, size = 422, normalized size = 2.37 \[ \begin {cases} - \frac {i a^{\frac {3}{2}} d e^{2} p \log {\left (a + b x^{2} \right )}}{b^{2} \sqrt {\frac {1}{b}}} + \frac {2 i a^{\frac {3}{2}} d e^{2} p \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{b^{2} \sqrt {\frac {1}{b}}} + \frac {i \sqrt {a} d^{3} p \log {\left (a + b x^{2} \right )}}{b \sqrt {\frac {1}{b}}} - \frac {2 i \sqrt {a} d^{3} p \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + x \right )}}{b \sqrt {\frac {1}{b}}} - \frac {a^{2} e^{3} p \log {\left (a + b x^{2} \right )}}{4 b^{2}} + \frac {3 a d^{2} e p \log {\left (a + b x^{2} \right )}}{2 b} + \frac {2 a d e^{2} p x}{b} + \frac {a e^{3} p x^{2}}{4 b} + d^{3} p x \log {\left (a + b x^{2} \right )} - 2 d^{3} p x + d^{3} x \log {\relax (c )} + \frac {3 d^{2} e p x^{2} \log {\left (a + b x^{2} \right )}}{2} - \frac {3 d^{2} e p x^{2}}{2} + \frac {3 d^{2} e x^{2} \log {\relax (c )}}{2} + d e^{2} p x^{3} \log {\left (a + b x^{2} \right )} - \frac {2 d e^{2} p x^{3}}{3} + d e^{2} x^{3} \log {\relax (c )} + \frac {e^{3} p x^{4} \log {\left (a + b x^{2} \right )}}{4} - \frac {e^{3} p x^{4}}{8} + \frac {e^{3} x^{4} \log {\relax (c )}}{4} & \text {for}\: b \neq 0 \\\left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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