Optimal. Leaf size=229 \[ -3 d p x-\frac {3}{4} e p x^2-\frac {\sqrt {3} \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3}}+\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e} \]
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Rubi [A]
time = 0.22, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2513, 1901,
1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac {\sqrt {3} \sqrt [3]{a} p \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (\sqrt [3]{a} e+2 \sqrt [3]{b} d\right )}{2 b^{2/3}}+\frac {\sqrt [3]{a} p \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}-3 d p x-\frac {3}{4} e p x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1874
Rule 1885
Rule 1901
Rule 2513
Rubi steps
\begin {align*} \int (d+e x) \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {(3 b p) \int \frac {x^2 (d+e x)^2}{a+b x^3} \, dx}{2 e}\\ &=\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {(3 b p) \int \left (\frac {2 d e}{b}+\frac {e^2 x}{b}-\frac {2 a d e+a e^2 x-b d^2 x^2}{b \left (a+b x^3\right )}\right ) \, dx}{2 e}\\ &=-3 d p x-\frac {3}{4} e p x^2+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac {(3 p) \int \frac {2 a d e+a e^2 x-b d^2 x^2}{a+b x^3} \, dx}{2 e}\\ &=-3 d p x-\frac {3}{4} e p x^2+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac {(3 p) \int \frac {2 a d e+a e^2 x}{a+b x^3} \, dx}{2 e}-\frac {\left (3 b d^2 p\right ) \int \frac {x^2}{a+b x^3} \, dx}{2 e}\\ &=-3 d p x-\frac {3}{4} e p x^2-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac {p \int \frac {\sqrt [3]{a} \left (4 a \sqrt [3]{b} d e+a^{4/3} e^2\right )+\sqrt [3]{b} \left (-2 a \sqrt [3]{b} d e+a^{4/3} e^2\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{2/3} \sqrt [3]{b} e}+\frac {1}{2} \left (\sqrt [3]{a} \left (2 d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx\\ &=-3 d p x-\frac {3}{4} e p x^2+\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}-\frac {\left (\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 b^{2/3}}+\frac {1}{4} \left (3 a^{2/3} \left (2 d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx\\ &=-3 d p x-\frac {3}{4} e p x^2+\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}+\frac {\left (3 \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{2 b^{2/3}}\\ &=-3 d p x-\frac {3}{4} e p x^2-\frac {\sqrt {3} \sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3}}+\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3}}-\frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3}}-\frac {d^2 p \log \left (a+b x^3\right )}{2 e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 204, normalized size = 0.89 \begin {gather*} -3 d p x-\frac {3}{4} e p x^2+\frac {\sqrt {3} \sqrt [3]{a} d p \tan ^{-1}\left (\frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}\right )}{\sqrt [3]{b}}+\frac {3}{4} e p x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a}\right )+\frac {\sqrt [3]{a} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+d x \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{2} e x^2 \log \left (c \left (a+b x^3\right )^p\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.56, size = 335, normalized size = 1.46
method | result | size |
risch | \(\left (\frac {1}{2} e \,x^{2}+d x \right ) \ln \left (\left (x^{3} b +a \right )^{p}\right )+\frac {i \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) x^{2} e \pi }{4}-\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{4}-\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{3}}{4}+\frac {i \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} x^{2} e \pi }{4}+\frac {i \pi d \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} x}{2}-\frac {i \pi d \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) x}{2}-\frac {i \pi d \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{3} x}{2}+\frac {i \pi d \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) x}{2}+\frac {\ln \left (c \right ) e \,x^{2}}{2}-\frac {3 e p \,x^{2}}{4}+\ln \left (c \right ) d x -3 d p x +\frac {a p \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (e \textit {\_R} +2 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{2 b}\) | \(335\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 192, normalized size = 0.84 \begin {gather*} -\frac {1}{4} \, b p {\left (\frac {3 \, {\left (x^{2} e + 4 \, d x\right )}}{b} - \frac {2 \, \sqrt {3} {\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} e + 2 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} e - 2 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (a \left (\frac {a}{b}\right )^{\frac {1}{3}} e - 2 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.21, size = 2251, normalized size = 9.83 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.90, size = 112, normalized size = 0.49 \begin {gather*} 3 a d p \operatorname {RootSum} {\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log {\left (3 t a + x \right )} \right )\right )} + \frac {3 a e p \operatorname {RootSum} {\left (27 t^{3} a b^{2} + 1, \left ( t \mapsto t \log {\left (9 t^{2} a b + x \right )} \right )\right )}}{2} - 3 d p x + d x \log {\left (c \left (a + b x^{3}\right )^{p} \right )} - \frac {3 e p x^{2}}{4} + \frac {e x^{2} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.12, size = 220, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, p x^{2} e \log \left (b x^{3} + a\right ) - \frac {3}{4} \, p x^{2} e + d p x \log \left (b x^{3} + a\right ) + \frac {1}{2} \, x^{2} e \log \left (c\right ) - 3 \, d p x + d x \log \left (c\right ) - \frac {{\left (a p \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + 2 \, a d p\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{2 \, a} + \frac {{\left (2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b d p - \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} p e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{2 \, b^{2}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d p + \left (-a b^{2}\right )^{\frac {2}{3}} p e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 210, normalized size = 0.92 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\,\left (\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\,a\,b^2\,9+9\,a\,b^2\,d\,p\,x\right )+\frac {9\,a^2\,b\,d\,e\,p^2}{2}+\frac {9\,a^2\,b\,e^2\,p^2\,x}{4}\right )\,\mathrm {root}\left (8\,b^2\,c^3+12\,a\,b\,c\,d\,e\,p^2-8\,a\,b\,d^3\,p^3+a^2\,e^3\,p^3,c,k\right )\right )+\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,\left (\frac {e\,x^2}{2}+d\,x\right )-\frac {3\,e\,p\,x^2}{4}-3\,d\,p\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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