Optimal. Leaf size=126 \[ -\frac {a \left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {6 a^2 d (a+b x)^{3+n}}{b^5 (3+n)}-\frac {4 a d (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d (a+b x)^{5+n}}{b^5 (5+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1634}
\begin {gather*} -\frac {a \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^5 (n+1)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {6 a^2 d (a+b x)^{n+3}}{b^5 (n+3)}-\frac {4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d (a+b x)^{n+5}}{b^5 (n+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 1634
Rubi steps
\begin {align*} \int x (a+b x)^n \left (c+d x^3\right ) \, dx &=\int \left (\frac {a \left (-b^3 c+a^3 d\right ) (a+b x)^n}{b^4}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{1+n}}{b^4}+\frac {6 a^2 d (a+b x)^{2+n}}{b^4}-\frac {4 a d (a+b x)^{3+n}}{b^4}+\frac {d (a+b x)^{4+n}}{b^4}\right ) \, dx\\ &=-\frac {a \left (b^3 c-a^3 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {6 a^2 d (a+b x)^{3+n}}{b^5 (3+n)}-\frac {4 a d (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d (a+b x)^{5+n}}{b^5 (5+n)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 104, normalized size = 0.83 \begin {gather*} \frac {(a+b x)^{1+n} \left (\frac {a \left (-b^3 c+a^3 d\right )}{1+n}+\frac {\left (b^3 c-4 a^3 d\right ) (a+b x)}{2+n}+\frac {6 a^2 d (a+b x)^2}{3+n}-\frac {4 a d (a+b x)^3}{4+n}+\frac {d (a+b x)^4}{5+n}\right )}{b^5} \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. \(282\) vs.
\(2(126)=252\).
time = 0.24, size = 283, normalized size = 2.25
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{4} d \,n^{4} x^{4}+10 b^{4} d \,n^{3} x^{4}-4 a \,b^{3} d \,n^{3} x^{3}+35 b^{4} d \,n^{2} x^{4}-24 a \,b^{3} d \,n^{2} x^{3}+b^{4} c \,n^{4} x +50 b^{4} d n \,x^{4}+12 a^{2} b^{2} d \,n^{2} x^{2}-44 a \,b^{3} d n \,x^{3}+13 b^{4} c \,n^{3} x +24 d \,x^{4} b^{4}+36 a^{2} b^{2} d n \,x^{2}-a \,b^{3} c \,n^{3}-24 a d \,x^{3} b^{3}+59 b^{4} c \,n^{2} x -24 a^{3} b d n x +24 d \,a^{2} x^{2} b^{2}-12 a \,b^{3} c \,n^{2}+107 b^{4} c n x -24 a^{3} b d x -47 a \,b^{3} c n +60 b^{4} c x +24 a^{4} d -60 a \,b^{3} c \right )}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(283\) |
norman | \(\frac {d \,x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{5+n}+\frac {a^{2} \left (-b^{3} c \,n^{3}-12 b^{3} c \,n^{2}-47 b^{3} c n +24 a^{3} d -60 b^{3} c \right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {\left (b^{3} c \,n^{3}+12 b^{3} c \,n^{2}+12 a^{3} d n +47 b^{3} c n +60 b^{3} c \right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}+\frac {d a n \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+9 n +20\right )}-\frac {n a \left (-b^{3} c \,n^{3}-12 b^{3} c \,n^{2}-47 b^{3} c n +24 a^{3} d -60 b^{3} c \right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}-\frac {4 n \,a^{2} d \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}\) | \(320\) |
risch | \(\frac {\left (b^{5} d \,n^{4} x^{5}+a \,b^{4} d \,n^{4} x^{4}+10 b^{5} d \,n^{3} x^{5}+6 a \,b^{4} d \,n^{3} x^{4}+35 b^{5} d \,n^{2} x^{5}-4 a^{2} b^{3} d \,n^{3} x^{3}+11 a \,b^{4} d \,n^{2} x^{4}+b^{5} c \,n^{4} x^{2}+50 b^{5} d n \,x^{5}-12 a^{2} b^{3} d \,n^{2} x^{3}+a \,b^{4} c \,n^{4} x +6 a \,b^{4} d n \,x^{4}+13 b^{5} c \,n^{3} x^{2}+24 d \,x^{5} b^{5}+12 a^{3} b^{2} d \,n^{2} x^{2}-8 a^{2} b^{3} d n \,x^{3}+12 a \,b^{4} c \,n^{3} x +59 b^{5} c \,n^{2} x^{2}+12 a^{3} b^{2} d n \,x^{2}-a^{2} b^{3} c \,n^{3}+47 a \,b^{4} c \,n^{2} x +107 b^{5} c n \,x^{2}-24 a^{4} b d n x -12 a^{2} b^{3} c \,n^{2}+60 a \,b^{4} c n x +60 b^{5} c \,x^{2}-47 a^{2} b^{3} c n +24 a^{5} d -60 a^{2} b^{3} c \right ) \left (b x +a \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{5}}\) | \(365\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 184, normalized size = 1.46 \begin {gather*} \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \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. 348 vs.
\(2 (126) = 252\).
time = 0.44, size = 348, normalized size = 2.76 \begin {gather*} -\frac {{\left (a^{2} b^{3} c n^{3} + 12 \, a^{2} b^{3} c n^{2} + 47 \, a^{2} b^{3} c n + 60 \, a^{2} b^{3} c - 24 \, a^{5} d - {\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} - {\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} + 4 \, {\left (a^{2} b^{3} d n^{3} + 3 \, a^{2} b^{3} d n^{2} + 2 \, a^{2} b^{3} d n\right )} x^{3} - {\left (b^{5} c n^{4} + 13 \, b^{5} c n^{3} + 60 \, b^{5} c + {\left (59 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n^{2} + {\left (107 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n\right )} x^{2} - {\left (a b^{4} c n^{4} + 12 \, a b^{4} c n^{3} + 47 \, a b^{4} c n^{2} + 12 \, {\left (5 \, a b^{4} c - 2 \, a^{4} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \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. 3704 vs.
\(2 (112) = 224\).
time = 1.30, size = 3704, normalized size = 29.40 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 577 vs.
\(2 (126) = 252\).
time = 4.88, size = 577, normalized size = 4.58 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{5} d n^{4} x^{5} + {\left (b x + a\right )}^{n} a b^{4} d n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d n^{3} x^{5} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d n^{2} x^{5} + {\left (b x + a\right )}^{n} b^{5} c n^{4} x^{2} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{3} x^{3} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d n x^{5} + {\left (b x + a\right )}^{n} a b^{4} c n^{4} x + 13 \, {\left (b x + a\right )}^{n} b^{5} c n^{3} x^{2} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{2} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d x^{5} + 12 \, {\left (b x + a\right )}^{n} a b^{4} c n^{3} x + 59 \, {\left (b x + a\right )}^{n} b^{5} c n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n^{2} x^{2} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n x^{3} - {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{3} + 47 \, {\left (b x + a\right )}^{n} a b^{4} c n^{2} x + 107 \, {\left (b x + a\right )}^{n} b^{5} c n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n x^{2} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{2} + 60 \, {\left (b x + a\right )}^{n} a b^{4} c n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d n x + 60 \, {\left (b x + a\right )}^{n} b^{5} c x^{2} - 47 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n - 60 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c + 24 \, {\left (b x + a\right )}^{n} a^{5} d}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.95, size = 363, normalized size = 2.88 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {d\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}-\frac {a^2\,\left (-24\,d\,a^3+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {x^2\,\left (n+1\right )\,\left (12\,d\,a^3\,n+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x\,\left (-24\,d\,a^3+c\,b^3\,n^3+12\,c\,b^3\,n^2+47\,c\,b^3\,n+60\,c\,b^3\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,d\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {4\,a^2\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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