Optimal. Leaf size=158 \[ \frac {b x^{3 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 d^3 n}-\frac {b^2 x^{4 n} (b c-3 a d)}{4 d^2 n}-\frac {c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{d^6 n}+\frac {c x^n (b c-a d)^3}{d^5 n}-\frac {x^{2 n} (b c-a d)^3}{2 d^4 n}+\frac {b^3 x^{5 n}}{5 d n} \]
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Rubi [A] time = 0.15, antiderivative size = 158, 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 = {446, 88} \begin {gather*} \frac {b x^{3 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 d^3 n}-\frac {b^2 x^{4 n} (b c-3 a d)}{4 d^2 n}-\frac {c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{d^6 n}+\frac {c x^n (b c-a d)^3}{d^5 n}-\frac {x^{2 n} (b c-a d)^3}{2 d^4 n}+\frac {b^3 x^{5 n}}{5 d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 (a+b x)^3}{c+d x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {c (b c-a d)^3}{d^5}+\frac {(-b c+a d)^3 x}{d^4}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^2}{d^3}-\frac {b^2 (b c-3 a d) x^3}{d^2}+\frac {b^3 x^4}{d}-\frac {c^2 (b c-a d)^3}{d^5 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {c (b c-a d)^3 x^n}{d^5 n}-\frac {(b c-a d)^3 x^{2 n}}{2 d^4 n}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^{3 n}}{3 d^3 n}-\frac {b^2 (b c-3 a d) x^{4 n}}{4 d^2 n}+\frac {b^3 x^{5 n}}{5 d n}-\frac {c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{d^6 n}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 138, normalized size = 0.87 \begin {gather*} \frac {20 b d^3 x^{3 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )-15 b^2 d^4 x^{4 n} (b c-3 a d)-60 c^2 (b c-a d)^3 \log \left (c+d x^n\right )+30 d^2 x^{2 n} (a d-b c)^3+60 c d x^n (b c-a d)^3+12 b^3 d^5 x^{5 n}}{60 d^6 n} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 220, normalized size = 1.39 \begin {gather*} \frac {x^n \left (-60 a^3 c d^3+30 a^3 d^4 x^n+180 a^2 b c^2 d^2-90 a^2 b c d^3 x^n+60 a^2 b d^4 x^{2 n}-180 a b^2 c^3 d+90 a b^2 c^2 d^2 x^n-60 a b^2 c d^3 x^{2 n}+45 a b^2 d^4 x^{3 n}+60 b^3 c^4-30 b^3 c^3 d x^n+20 b^3 c^2 d^2 x^{2 n}-15 b^3 c d^3 x^{3 n}+12 b^3 d^4 x^{4 n}\right )}{60 d^5 n}-\frac {c^2 (b c-a d)^3 \log \left (c+d x^n\right )}{d^6 n} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 230, normalized size = 1.46 \begin {gather*} \frac {12 \, b^{3} d^{5} x^{5 \, n} - 15 \, {\left (b^{3} c d^{4} - 3 \, a b^{2} d^{5}\right )} x^{4 \, n} + 20 \, {\left (b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} x^{3 \, n} - 30 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2 \, n} + 60 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x^{n} - 60 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \log \left (d x^{n} + c\right )}{60 \, d^{6} n} \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*} \int \frac {{\left (b x^{n} + a\right )}^{3} x^{3 \, n - 1}}{d x^{n} + c}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 342, normalized size = 2.16 \begin {gather*} \frac {a^{3} c^{2} \ln \left (x^{n}+\frac {c}{d}\right )}{d^{3} n}-\frac {a^{3} c \,x^{n}}{d^{2} n}+\frac {a^{3} x^{2 n}}{2 d n}-\frac {3 a^{2} b \,c^{3} \ln \left (x^{n}+\frac {c}{d}\right )}{d^{4} n}+\frac {3 a^{2} b \,c^{2} x^{n}}{d^{3} n}-\frac {3 a^{2} b c \,x^{2 n}}{2 d^{2} n}+\frac {a^{2} b \,x^{3 n}}{d n}+\frac {3 a \,b^{2} c^{4} \ln \left (x^{n}+\frac {c}{d}\right )}{d^{5} n}-\frac {3 a \,b^{2} c^{3} x^{n}}{d^{4} n}+\frac {3 a \,b^{2} c^{2} x^{2 n}}{2 d^{3} n}-\frac {a \,b^{2} c \,x^{3 n}}{d^{2} n}+\frac {3 a \,b^{2} x^{4 n}}{4 d n}-\frac {b^{3} c^{5} \ln \left (x^{n}+\frac {c}{d}\right )}{d^{6} n}+\frac {b^{3} c^{4} x^{n}}{d^{5} n}-\frac {b^{3} c^{3} x^{2 n}}{2 d^{4} n}+\frac {b^{3} c^{2} x^{3 n}}{3 d^{3} n}-\frac {b^{3} c \,x^{4 n}}{4 d^{2} n}+\frac {b^{3} x^{5 n}}{5 d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 286, normalized size = 1.81 \begin {gather*} -\frac {1}{60} \, b^{3} {\left (\frac {60 \, c^{5} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{6} n} - \frac {12 \, d^{4} x^{5 \, n} - 15 \, c d^{3} x^{4 \, n} + 20 \, c^{2} d^{2} x^{3 \, n} - 30 \, c^{3} d x^{2 \, n} + 60 \, c^{4} x^{n}}{d^{5} n}\right )} + \frac {1}{4} \, a b^{2} {\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^{2} b {\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 {1}{2} \, a^{3} {\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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{3\,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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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