Optimal. Leaf size=62 \[ \frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45}
\begin {gather*} \frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{-1+3 n} \left (a+b x^n\right )^8 \, dx &=\frac {\text {Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2 (a+b x)^8}{b^2}-\frac {2 a (a+b x)^9}{b^2}+\frac {(a+b x)^{10}}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 113, normalized size = 1.82 \begin {gather*} \frac {x^{3 n} \left (165 a^8+990 a^7 b x^n+2772 a^6 b^2 x^{2 n}+4620 a^5 b^3 x^{3 n}+4950 a^4 b^4 x^{4 n}+3465 a^3 b^5 x^{5 n}+1540 a^2 b^6 x^{6 n}+396 a b^7 x^{7 n}+45 b^8 x^{8 n}\right )}{495 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs.
\(2(56)=112\).
time = 0.23, size = 136, normalized size = 2.19
method | result | size |
risch | \(\frac {b^{8} x^{11 n}}{11 n}+\frac {4 a \,b^{7} x^{10 n}}{5 n}+\frac {28 a^{2} b^{6} x^{9 n}}{9 n}+\frac {7 a^{3} b^{5} x^{8 n}}{n}+\frac {10 a^{4} b^{4} x^{7 n}}{n}+\frac {28 a^{5} b^{3} x^{6 n}}{3 n}+\frac {28 a^{6} b^{2} x^{5 n}}{5 n}+\frac {2 a^{7} b \,x^{4 n}}{n}+\frac {a^{8} x^{3 n}}{3 n}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs.
\(2 (56) = 112\).
time = 0.29, size = 135, normalized size = 2.18 \begin {gather*} \frac {b^{8} x^{11 \, n}}{11 \, n} + \frac {4 \, a b^{7} x^{10 \, n}}{5 \, n} + \frac {28 \, a^{2} b^{6} x^{9 \, n}}{9 \, n} + \frac {7 \, a^{3} b^{5} x^{8 \, n}}{n} + \frac {10 \, a^{4} b^{4} x^{7 \, n}}{n} + \frac {28 \, a^{5} b^{3} x^{6 \, n}}{3 \, n} + \frac {28 \, a^{6} b^{2} x^{5 \, n}}{5 \, n} + \frac {2 \, a^{7} b x^{4 \, n}}{n} + \frac {a^{8} x^{3 \, n}}{3 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs.
\(2 (56) = 112\).
time = 0.38, size = 113, normalized size = 1.82 \begin {gather*} \frac {45 \, b^{8} x^{11 \, n} + 396 \, a b^{7} x^{10 \, n} + 1540 \, a^{2} b^{6} x^{9 \, n} + 3465 \, a^{3} b^{5} x^{8 \, n} + 4950 \, a^{4} b^{4} x^{7 \, n} + 4620 \, a^{5} b^{3} x^{6 \, n} + 2772 \, a^{6} b^{2} x^{5 \, n} + 990 \, a^{7} b x^{4 \, n} + 165 \, a^{8} x^{3 \, n}}{495 \, 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. 141 vs.
\(2 (49) = 98\).
time = 30.89, size = 141, normalized size = 2.27 \begin {gather*} \begin {cases} \frac {a^{8} x^{3 n}}{3 n} + \frac {2 a^{7} b x^{4 n}}{n} + \frac {28 a^{6} b^{2} x^{5 n}}{5 n} + \frac {28 a^{5} b^{3} x^{6 n}}{3 n} + \frac {10 a^{4} b^{4} x^{7 n}}{n} + \frac {7 a^{3} b^{5} x^{8 n}}{n} + \frac {28 a^{2} b^{6} x^{9 n}}{9 n} + \frac {4 a b^{7} x^{10 n}}{5 n} + \frac {b^{8} x^{11 n}}{11 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \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 [B]
time = 1.45, size = 135, normalized size = 2.18 \begin {gather*} \frac {a^8\,x^{3\,n}}{3\,n}+\frac {b^8\,x^{11\,n}}{11\,n}+\frac {28\,a^6\,b^2\,x^{5\,n}}{5\,n}+\frac {28\,a^5\,b^3\,x^{6\,n}}{3\,n}+\frac {10\,a^4\,b^4\,x^{7\,n}}{n}+\frac {7\,a^3\,b^5\,x^{8\,n}}{n}+\frac {28\,a^2\,b^6\,x^{9\,n}}{9\,n}+\frac {2\,a^7\,b\,x^{4\,n}}{n}+\frac {4\,a\,b^7\,x^{10\,n}}{5\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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