Optimal. Leaf size=130 \[ -\frac {(b c-a d)^3 x^n}{d^4 n}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^{2 n}}{2 d^3 n}-\frac {b^2 (b c-3 a d) x^{3 n}}{3 d^2 n}+\frac {b^3 x^{4 n}}{4 d n}+\frac {c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n} \]
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Rubi [A]
time = 0.09, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {457, 78}
\begin {gather*} \frac {b x^{2 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{2 d^3 n}-\frac {b^2 x^{3 n} (b c-3 a d)}{3 d^2 n}+\frac {c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n}-\frac {x^n (b c-a d)^3}{d^4 n}+\frac {b^3 x^{4 n}}{4 d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx &=\frac {\text {Subst}\left (\int \frac {x (a+b x)^3}{c+d x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {(-b c+a d)^3}{d^4}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x}{d^3}-\frac {b^2 (b c-3 a d) x^2}{d^2}+\frac {b^3 x^3}{d}+\frac {c (b c-a d)^3}{d^4 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {(b c-a d)^3 x^n}{d^4 n}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^{2 n}}{2 d^3 n}-\frac {b^2 (b c-3 a d) x^{3 n}}{3 d^2 n}+\frac {b^3 x^{4 n}}{4 d n}+\frac {c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 134, normalized size = 1.03 \begin {gather*} \frac {d x^n \left (12 a^3 d^3+18 a^2 b d^2 \left (-2 c+d x^n\right )+6 a b^2 d \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )+b^3 \left (-12 c^3+6 c^2 d x^n-4 c d^2 x^{2 n}+3 d^3 x^{3 n}\right )\right )+12 c (b c-a d)^3 \log \left (c+d x^n\right )}{12 d^5 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 188, normalized size = 1.45
method | result | size |
norman | \(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) {\mathrm e}^{n \ln \left (x \right )}}{d^{4} n}+\frac {b^{3} {\mathrm e}^{4 n \ln \left (x \right )}}{4 d n}+\frac {b \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) {\mathrm e}^{2 n \ln \left (x \right )}}{2 d^{3} n}+\frac {b^{2} \left (3 a d -b c \right ) {\mathrm e}^{3 n \ln \left (x \right )}}{3 d^{2} n}-\frac {c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{d^{5} n}\) | \(188\) |
risch | \(\frac {b^{3} x^{4 n}}{4 d n}+\frac {b^{2} x^{3 n} a}{d n}-\frac {b^{3} x^{3 n} c}{3 d^{2} n}+\frac {3 b \,x^{2 n} a^{2}}{2 d n}-\frac {3 b^{2} x^{2 n} a c}{2 d^{2} n}+\frac {b^{3} x^{2 n} c^{2}}{2 d^{3} n}+\frac {x^{n} a^{3}}{d n}-\frac {3 x^{n} a^{2} b c}{d^{2} n}+\frac {3 x^{n} a \,b^{2} c^{2}}{d^{3} n}-\frac {x^{n} b^{3} c^{3}}{d^{4} n}-\frac {c \ln \left (x^{n}+\frac {c}{d}\right ) a^{3}}{d^{2} n}+\frac {3 c^{2} \ln \left (x^{n}+\frac {c}{d}\right ) a^{2} b}{d^{3} n}-\frac {3 c^{3} \ln \left (x^{n}+\frac {c}{d}\right ) a \,b^{2}}{d^{4} n}+\frac {c^{4} \ln \left (x^{n}+\frac {c}{d}\right ) b^{3}}{d^{5} n}\) | \(264\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 231, normalized size = 1.78 \begin {gather*} a^{3} {\left (\frac {x^{n}}{d n} - \frac {c \log \left (\frac {d x^{n} + c}{d}\right )}{d^{2} n}\right )} + \frac {1}{12} \, b^{3} {\left (\frac {12 \, c^{4} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{5} n} + \frac {3 \, d^{3} x^{4 \, n} - 4 \, c d^{2} x^{3 \, n} + 6 \, c^{2} d x^{2 \, n} - 12 \, c^{3} x^{n}}{d^{4} n}\right )} - \frac {1}{2} \, a b^{2} {\left (\frac {6 \, c^{3} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{4} n} - \frac {2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {2 \, c^{2} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{3} n} + \frac {d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.41, size = 177, normalized size = 1.36 \begin {gather*} \frac {3 \, b^{3} d^{4} x^{4 \, n} - 4 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} x^{3 \, n} + 6 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x^{2 \, n} - 12 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x^{n} + 12 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \log \left (d x^{n} + c\right )}{12 \, d^{5} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs.
\(2 (114) = 228\).
time = 70.04, size = 320, normalized size = 2.46 \begin {gather*} \begin {cases} \frac {\left (a + b\right )^{3} \log {\left (x \right )}}{c} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\left (a + b\right )^{3} \log {\left (x \right )}}{c + d} & \text {for}\: n = 0 \\\frac {\frac {a^{3} x^{2 n}}{2 n} + \frac {a^{2} b x^{3 n}}{n} + \frac {3 a b^{2} x^{4 n}}{4 n} + \frac {b^{3} x^{5 n}}{5 n}}{c} & \text {for}\: d = 0 \\- \frac {a^{3} c \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{2} n} + \frac {a^{3} x^{n}}{d n} + \frac {3 a^{2} b c^{2} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{3} n} - \frac {3 a^{2} b c x^{n}}{d^{2} n} + \frac {3 a^{2} b x^{2 n}}{2 d n} - \frac {3 a b^{2} c^{3} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{4} n} + \frac {3 a b^{2} c^{2} x^{n}}{d^{3} n} - \frac {3 a b^{2} c x^{2 n}}{2 d^{2} n} + \frac {a b^{2} x^{3 n}}{d n} + \frac {b^{3} c^{4} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{5} n} - \frac {b^{3} c^{3} x^{n}}{d^{4} n} + \frac {b^{3} c^{2} x^{2 n}}{2 d^{3} n} - \frac {b^{3} c x^{3 n}}{3 d^{2} n} + \frac {b^{3} x^{4 n}}{4 d n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {x^{2\,n-1}\,{\left (a+b\,x^n\right )}^3}{c+d\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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