Optimal. Leaf size=86 \[ \frac {c (b c-a d) x^n}{d^3 n}-\frac {(b c-a d) x^{2 n}}{2 d^2 n}+\frac {b x^{3 n}}{3 d n}-\frac {c^2 (b c-a d) \log \left (c+d x^n\right )}{d^4 n} \]
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Rubi [A]
time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {457, 78}
\begin {gather*} -\frac {c^2 (b c-a d) \log \left (c+d x^n\right )}{d^4 n}+\frac {c x^n (b c-a d)}{d^3 n}-\frac {x^{2 n} (b c-a d)}{2 d^2 n}+\frac {b x^{3 n}}{3 d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n} \left (a+b x^n\right )}{c+d x^n} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 (a+b x)}{c+d x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {c (b c-a d)}{d^3}+\frac {(-b c+a d) x}{d^2}+\frac {b x^2}{d}-\frac {c^2 (b c-a d)}{d^3 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {c (b c-a d) x^n}{d^3 n}-\frac {(b c-a d) x^{2 n}}{2 d^2 n}+\frac {b x^{3 n}}{3 d n}-\frac {c^2 (b c-a d) \log \left (c+d x^n\right )}{d^4 n}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 76, normalized size = 0.88 \begin {gather*} \frac {d x^n \left (3 a d \left (-2 c+d x^n\right )+b \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )\right )+6 c^2 (-b c+a d) \log \left (c+d x^n\right )}{6 d^4 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 91, normalized size = 1.06
method | result | size |
norman | \(\frac {b \,{\mathrm e}^{3 n \ln \left (x \right )}}{3 d n}+\frac {\left (a d -b c \right ) {\mathrm e}^{2 n \ln \left (x \right )}}{2 d^{2} n}-\frac {c \left (a d -b c \right ) {\mathrm e}^{n \ln \left (x \right )}}{d^{3} n}+\frac {c^{2} \left (a d -b c \right ) \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{d^{4} n}\) | \(91\) |
risch | \(\frac {b \,x^{3 n}}{3 d n}+\frac {x^{2 n} a}{2 d n}-\frac {x^{2 n} b c}{2 d^{2} n}-\frac {c \,x^{n} a}{d^{2} n}+\frac {c^{2} x^{n} b}{d^{3} n}+\frac {c^{2} \ln \left (x^{n}+\frac {c}{d}\right ) a}{d^{3} n}-\frac {c^{3} \ln \left (x^{n}+\frac {c}{d}\right ) b}{d^{4} n}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 112, normalized size = 1.30 \begin {gather*} -\frac {1}{6} \, 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 {\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 = 2.79, size = 82, normalized size = 0.95 \begin {gather*} \frac {2 \, b d^{3} x^{3 \, n} - 3 \, {\left (b c d^{2} - a d^{3}\right )} x^{2 \, n} + 6 \, {\left (b c^{2} d - a c d^{2}\right )} x^{n} - 6 \, {\left (b c^{3} - a c^{2} d\right )} \log \left (d x^{n} + c\right )}{6 \, d^{4} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.60, size = 139, normalized size = 1.62 \begin {gather*} \begin {cases} \frac {\left (a + b\right ) \log {\left (x \right )}}{c} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\left (a + b\right ) \log {\left (x \right )}}{c + d} & \text {for}\: n = 0 \\\frac {\frac {a x^{3 n}}{3 n} + \frac {b x^{4 n}}{4 n}}{c} & \text {for}\: d = 0 \\\frac {a c^{2} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{3} n} - \frac {a c x^{n}}{d^{2} n} + \frac {a x^{2 n}}{2 d n} - \frac {b c^{3} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{4} n} + \frac {b c^{2} x^{n}}{d^{3} n} - \frac {b c x^{2 n}}{2 d^{2} n} + \frac {b x^{3 n}}{3 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^{3\,n-1}\,\left (a+b\,x^n\right )}{c+d\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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