Optimal. Leaf size=99 \[ \frac {c^2 x^{n+1} (3 a d+b c)}{n+1}+\frac {d^2 x^{3 n+1} (a d+3 b c)}{3 n+1}+\frac {3 c d x^{2 n+1} (a d+b c)}{2 n+1}+a c^3 x+\frac {b d^3 x^{4 n+1}}{4 n+1} \]
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Rubi [A] time = 0.07, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {373} \[ \frac {c^2 x^{n+1} (3 a d+b c)}{n+1}+\frac {d^2 x^{3 n+1} (a d+3 b c)}{3 n+1}+\frac {3 c d x^{2 n+1} (a d+b c)}{2 n+1}+a c^3 x+\frac {b d^3 x^{4 n+1}}{4 n+1} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin {align*} \int \left (a+b x^n\right ) \left (c+d x^n\right )^3 \, dx &=\int \left (a c^3+c^2 (b c+3 a d) x^n+3 c d (b c+a d) x^{2 n}+d^2 (3 b c+a d) x^{3 n}+b d^3 x^{4 n}\right ) \, dx\\ &=a c^3 x+\frac {c^2 (b c+3 a d) x^{1+n}}{1+n}+\frac {3 c d (b c+a d) x^{1+2 n}}{1+2 n}+\frac {d^2 (3 b c+a d) x^{1+3 n}}{1+3 n}+\frac {b d^3 x^{1+4 n}}{1+4 n}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 90, normalized size = 0.91 \[ \frac {b x \left (c+d x^n\right )^4-x \left (c^3+\frac {3 c^2 d x^n}{n+1}+\frac {3 c d^2 x^{2 n}}{2 n+1}+\frac {d^3 x^{3 n}}{3 n+1}\right ) (b c-a d (4 n+1))}{4 d n+d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 319, normalized size = 3.22 \[ \frac {{\left (6 \, b d^{3} n^{3} + 11 \, b d^{3} n^{2} + 6 \, b d^{3} n + b d^{3}\right )} x x^{4 \, n} + {\left (3 \, b c d^{2} + a d^{3} + 8 \, {\left (3 \, b c d^{2} + a d^{3}\right )} n^{3} + 14 \, {\left (3 \, b c d^{2} + a d^{3}\right )} n^{2} + 7 \, {\left (3 \, b c d^{2} + a d^{3}\right )} n\right )} x x^{3 \, n} + 3 \, {\left (b c^{2} d + a c d^{2} + 12 \, {\left (b c^{2} d + a c d^{2}\right )} n^{3} + 19 \, {\left (b c^{2} d + a c d^{2}\right )} n^{2} + 8 \, {\left (b c^{2} d + a c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (b c^{3} + 3 \, a c^{2} d + 24 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} n^{3} + 26 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} n^{2} + 9 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} n\right )} x x^{n} + {\left (24 \, a c^{3} n^{4} + 50 \, a c^{3} n^{3} + 35 \, a c^{3} n^{2} + 10 \, a c^{3} n + a c^{3}\right )} x}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 450, normalized size = 4.55 \[ \frac {24 \, a c^{3} n^{4} x + 6 \, b d^{3} n^{3} x x^{4 \, n} + 24 \, b c d^{2} n^{3} x x^{3 \, n} + 8 \, a d^{3} n^{3} x x^{3 \, n} + 36 \, b c^{2} d n^{3} x x^{2 \, n} + 36 \, a c d^{2} n^{3} x x^{2 \, n} + 24 \, b c^{3} n^{3} x x^{n} + 72 \, a c^{2} d n^{3} x x^{n} + 50 \, a c^{3} n^{3} x + 11 \, b d^{3} n^{2} x x^{4 \, n} + 42 \, b c d^{2} n^{2} x x^{3 \, n} + 14 \, a d^{3} n^{2} x x^{3 \, n} + 57 \, b c^{2} d n^{2} x x^{2 \, n} + 57 \, a c d^{2} n^{2} x x^{2 \, n} + 26 \, b c^{3} n^{2} x x^{n} + 78 \, a c^{2} d n^{2} x x^{n} + 35 \, a c^{3} n^{2} x + 6 \, b d^{3} n x x^{4 \, n} + 21 \, b c d^{2} n x x^{3 \, n} + 7 \, a d^{3} n x x^{3 \, n} + 24 \, b c^{2} d n x x^{2 \, n} + 24 \, a c d^{2} n x x^{2 \, n} + 9 \, b c^{3} n x x^{n} + 27 \, a c^{2} d n x x^{n} + 10 \, a c^{3} n x + b d^{3} x x^{4 \, n} + 3 \, b c d^{2} x x^{3 \, n} + a d^{3} x x^{3 \, n} + 3 \, b c^{2} d x x^{2 \, n} + 3 \, a c d^{2} x x^{2 \, n} + b c^{3} x x^{n} + 3 \, a c^{2} d x x^{n} + a c^{3} x}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 104, normalized size = 1.05 \[ \frac {b \,d^{3} x \,{\mathrm e}^{4 n \ln \relax (x )}}{4 n +1}+a \,c^{3} x +\frac {\left (3 a d +b c \right ) c^{2} x \,{\mathrm e}^{n \ln \relax (x )}}{n +1}+\frac {3 \left (a d +b c \right ) c d x \,{\mathrm e}^{2 n \ln \relax (x )}}{2 n +1}+\frac {\left (a d +3 b c \right ) d^{2} x \,{\mathrm e}^{3 n \ln \relax (x )}}{3 n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 140, normalized size = 1.41 \[ a c^{3} x + \frac {b d^{3} x^{4 \, n + 1}}{4 \, n + 1} + \frac {3 \, b c d^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {a d^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, b c^{2} d x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a c d^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {b c^{3} x^{n + 1}}{n + 1} + \frac {3 \, a c^{2} d x^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 99, normalized size = 1.00 \[ a\,c^3\,x+\frac {x\,x^n\,\left (b\,c^3+3\,a\,d\,c^2\right )}{n+1}+\frac {x\,x^{3\,n}\,\left (a\,d^3+3\,b\,c\,d^2\right )}{3\,n+1}+\frac {b\,d^3\,x\,x^{4\,n}}{4\,n+1}+\frac {3\,c\,d\,x\,x^{2\,n}\,\left (a\,d+b\,c\right )}{2\,n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.35, size = 1540, normalized size = 15.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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