Optimal. Leaf size=299 \[ \frac {e^{-\frac {a}{b n}} (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^4 n}+\frac {3 e^{-\frac {2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {3 e^{-\frac {3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {e^{-\frac {4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n} \]
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Rubi [A]
time = 0.33, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2446, 2436,
2337, 2209, 2437, 2347} \begin {gather*} \frac {3 g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {3 g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^4 n}+\frac {g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2446
Rubi steps
\begin {align*} \int \frac {(f+g x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx &=\int \left (\frac {(e f-d g)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {3 g (e f-d g)^2 (d+e x)}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {3 g^2 (e f-d g) (d+e x)^2}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {g^3 (d+e x)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=\frac {g^3 \int \frac {(d+e x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}+\frac {\left (3 g^2 (e f-d g)\right ) \int \frac {(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}+\frac {\left (3 g (e f-d g)^2\right ) \int \frac {d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}+\frac {(e f-d g)^3 \int \frac {1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}\\ &=\frac {g^3 \text {Subst}\left (\int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g^2 (e f-d g)\right ) \text {Subst}\left (\int \frac {x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g (e f-d g)^2\right ) \text {Subst}\left (\int \frac {x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}+\frac {(e f-d g)^3 \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}\\ &=\frac {\left (g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {4 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac {\left ((e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g)^3 (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^4 n}+\frac {3 e^{-\frac {2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {3 e^{-\frac {3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac {e^{-\frac {4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 266, normalized size = 0.89 \begin {gather*} \frac {e^{-\frac {4 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (e^{\frac {3 a}{b n}} (e f-d g)^3 \left (c (d+e x)^n\right )^{3/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )+g (d+e x) \left (3 e^{\frac {2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text {Ei}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-g (d+e x) \left (-3 e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {Ei}\left (\frac {3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-g (d+e x) \text {Ei}\left (\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )\right )\right )}{b e^4 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.12, size = 3160, normalized size = 10.57
method | result | size |
risch | \(\text {Expression too large to display}\) | \(3160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 303, normalized size = 1.01 \begin {gather*} \frac {{\left (g^{3} \operatorname {log\_integral}\left ({\left (x^{4} e^{4} + 4 \, d x^{3} e^{3} + 6 \, d^{2} x^{2} e^{2} + 4 \, d^{3} x e + d^{4}\right )} e^{\left (\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) - 3 \, {\left (d g^{3} - f g^{2} e\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + 3 \, {\left (d^{2} g^{3} - 2 \, d f g^{2} e + f^{2} g e^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left ({\left (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) - {\left (d^{3} g^{3} - 3 \, d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} - f^{3} e^{3}\right )} e^{\left (\frac {3 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \operatorname {log\_integral}\left ({\left (x e + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {4 \, {\left (b \log \left (c\right ) + a\right )}}{b n} - 4\right )}}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{3}}{a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.13, size = 582, normalized size = 1.95 \begin {gather*} -\frac {d^{3} g^{3} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} - 4\right )}}{b c^{\left (\frac {1}{n}\right )} n} + \frac {3 \, d^{2} f g^{2} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} - 3\right )}}{b c^{\left (\frac {1}{n}\right )} n} + \frac {3 \, d^{2} g^{3} {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {2 \, a}{b n} - 4\right )}}{b c^{\frac {2}{n}} n} - \frac {3 \, d f^{2} g {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} - 2\right )}}{b c^{\left (\frac {1}{n}\right )} n} - \frac {6 \, d f g^{2} {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {2 \, a}{b n} - 3\right )}}{b c^{\frac {2}{n}} n} - \frac {3 \, d g^{3} {\rm Ei}\left (\frac {3 \, \log \left (c\right )}{n} + \frac {3 \, a}{b n} + 3 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {3 \, a}{b n} - 4\right )}}{b c^{\frac {3}{n}} n} + \frac {f^{3} {\rm Ei}\left (\frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac {a}{b n} - 1\right )}}{b c^{\left (\frac {1}{n}\right )} n} + \frac {3 \, f^{2} g {\rm Ei}\left (\frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {2 \, a}{b n} - 2\right )}}{b c^{\frac {2}{n}} n} + \frac {3 \, f g^{2} {\rm Ei}\left (\frac {3 \, \log \left (c\right )}{n} + \frac {3 \, a}{b n} + 3 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {3 \, a}{b n} - 3\right )}}{b c^{\frac {3}{n}} n} + \frac {g^{3} {\rm Ei}\left (\frac {4 \, \log \left (c\right )}{n} + \frac {4 \, a}{b n} + 4 \, \log \left (x e + d\right )\right ) e^{\left (-\frac {4 \, a}{b n} - 4\right )}}{b c^{\frac {4}{n}} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^3}{a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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