Optimal. Leaf size=154 \[ -\frac {a (b c-a d)^3 (a+b x)^{1+n}}{b^5 (1+n)}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)^{2+n}}{b^5 (2+n)}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^{3+n}}{b^5 (3+n)}+\frac {d^2 (3 b c-4 a d) (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d^3 (a+b x)^{5+n}}{b^5 (5+n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 154, 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 = {78}
\begin {gather*} \frac {d^2 (3 b c-4 a d) (a+b x)^{n+4}}{b^5 (n+4)}-\frac {a (b c-a d)^3 (a+b x)^{n+1}}{b^5 (n+1)}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)^{n+2}}{b^5 (n+2)}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^{n+3}}{b^5 (n+3)}+\frac {d^3 (a+b x)^{n+5}}{b^5 (n+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int x (a+b x)^n (c+d x)^3 \, dx &=\int \left (\frac {a (-b c+a d)^3 (a+b x)^n}{b^4}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)^{1+n}}{b^4}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^{2+n}}{b^4}+\frac {d^2 (3 b c-4 a d) (a+b x)^{3+n}}{b^4}+\frac {d^3 (a+b x)^{4+n}}{b^4}\right ) \, dx\\ &=-\frac {a (b c-a d)^3 (a+b x)^{1+n}}{b^5 (1+n)}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)^{2+n}}{b^5 (2+n)}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^{3+n}}{b^5 (3+n)}+\frac {d^2 (3 b c-4 a d) (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d^3 (a+b x)^{5+n}}{b^5 (5+n)}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 132, normalized size = 0.86 \begin {gather*} \frac {(a+b x)^{1+n} \left (\frac {a (-b c+a d)^3}{1+n}+\frac {(b c-4 a d) (b c-a d)^2 (a+b x)}{2+n}+\frac {3 d (b c-2 a d) (b c-a d) (a+b x)^2}{3+n}+\frac {d^2 (3 b c-4 a d) (a+b x)^3}{4+n}+\frac {d^3 (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. \(586\) vs.
\(2(154)=308\).
time = 0.08, size = 587, normalized size = 3.81
method | result | size |
norman | \(\frac {d^{3} x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{5+n}+\frac {a^{2} \left (-b^{3} c^{3} n^{3}+6 a \,b^{2} c^{2} d \,n^{2}-12 b^{3} c^{3} n^{2}-18 a^{2} b c \,d^{2} n +54 a \,b^{2} c^{2} d n -47 b^{3} c^{3} n +24 a^{3} d^{3}-90 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -60 b^{3} c^{3}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {\left (3 a \,b^{2} c^{2} d \,n^{3}+b^{3} c^{3} n^{3}-9 a^{2} b c \,d^{2} n^{2}+27 a \,b^{2} c^{2} d \,n^{2}+12 b^{3} c^{3} n^{2}+12 a^{3} d^{3} n -45 a^{2} b c \,d^{2} n +60 a \,b^{2} c^{2} d n +47 b^{3} c^{3} n +60 b^{3} c^{3}\right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}+\frac {\left (a d n +3 b c n +15 b c \right ) d^{2} x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+9 n +20\right )}-\frac {\left (-3 a b c d \,n^{2}-3 b^{2} c^{2} n^{2}+4 a^{2} d^{2} n -15 a b c d n -27 b^{2} c^{2} n -60 b^{2} c^{2}\right ) d \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}-\frac {n a \left (-b^{3} c^{3} n^{3}+6 a \,b^{2} c^{2} d \,n^{2}-12 b^{3} c^{3} n^{2}-18 a^{2} b c \,d^{2} n +54 a \,b^{2} c^{2} d n -47 b^{3} c^{3} n +24 a^{3} d^{3}-90 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -60 b^{3} c^{3}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(587\) |
gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{4} d^{3} n^{4} x^{4}+3 b^{4} c \,d^{2} n^{4} x^{3}+10 b^{4} d^{3} n^{3} x^{4}-4 a \,b^{3} d^{3} n^{3} x^{3}+3 b^{4} c^{2} d \,n^{4} x^{2}+33 b^{4} c \,d^{2} n^{3} x^{3}+35 b^{4} d^{3} n^{2} x^{4}-9 a \,b^{3} c \,d^{2} n^{3} x^{2}-24 a \,b^{3} d^{3} n^{2} x^{3}+b^{4} c^{3} n^{4} x +36 b^{4} c^{2} d \,n^{3} x^{2}+123 b^{4} c \,d^{2} n^{2} x^{3}+50 b^{4} d^{3} n \,x^{4}+12 a^{2} b^{2} d^{3} n^{2} x^{2}-6 a \,b^{3} c^{2} d \,n^{3} x -72 a \,b^{3} c \,d^{2} n^{2} x^{2}-44 a \,b^{3} d^{3} n \,x^{3}+13 b^{4} c^{3} n^{3} x +147 b^{4} c^{2} d \,n^{2} x^{2}+183 b^{4} c \,d^{2} n \,x^{3}+24 d^{3} x^{4} b^{4}+18 a^{2} b^{2} c \,d^{2} n^{2} x +36 a^{2} b^{2} d^{3} n \,x^{2}-a \,b^{3} c^{3} n^{3}-60 a \,b^{3} c^{2} d \,n^{2} x -153 a \,b^{3} c \,d^{2} n \,x^{2}-24 a \,b^{3} d^{3} x^{3}+59 b^{4} c^{3} n^{2} x +234 b^{4} c^{2} d n \,x^{2}+90 b^{4} c \,d^{2} x^{3}-24 a^{3} b \,d^{3} n x +6 a^{2} b^{2} c^{2} d \,n^{2}+108 a^{2} b^{2} c \,d^{2} n x +24 a^{2} b^{2} d^{3} x^{2}-12 a \,b^{3} c^{3} n^{2}-174 a \,b^{3} c^{2} d n x -90 a \,b^{3} c \,d^{2} x^{2}+107 b^{4} c^{3} n x +120 b^{4} c^{2} d \,x^{2}-18 a^{3} b c \,d^{2} n -24 a^{3} b \,d^{3} x +54 a^{2} b^{2} c^{2} d n +90 a^{2} b^{2} c \,d^{2} x -47 a \,b^{3} c^{3} n -120 a \,b^{3} c^{2} d x +60 b^{4} c^{3} x +24 a^{4} d^{3}-90 a^{3} b c \,d^{2}+120 a^{2} b^{2} c^{2} d -60 a \,b^{3} c^{3}\right )}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(685\) |
risch | \(\frac {\left (b^{5} d^{3} n^{4} x^{5}+a \,b^{4} d^{3} n^{4} x^{4}+3 b^{5} c \,d^{2} n^{4} x^{4}+10 b^{5} d^{3} n^{3} x^{5}+3 a \,b^{4} c \,d^{2} n^{4} x^{3}+6 a \,b^{4} d^{3} n^{3} x^{4}+3 b^{5} c^{2} d \,n^{4} x^{3}+33 b^{5} c \,d^{2} n^{3} x^{4}+35 b^{5} d^{3} n^{2} x^{5}-4 a^{2} b^{3} d^{3} n^{3} x^{3}+3 a \,b^{4} c^{2} d \,n^{4} x^{2}+24 a \,b^{4} c \,d^{2} n^{3} x^{3}+11 a \,b^{4} d^{3} n^{2} x^{4}+b^{5} c^{3} n^{4} x^{2}+36 b^{5} c^{2} d \,n^{3} x^{3}+123 b^{5} c \,d^{2} n^{2} x^{4}+50 b^{5} d^{3} n \,x^{5}-9 a^{2} b^{3} c \,d^{2} n^{3} x^{2}-12 a^{2} b^{3} d^{3} n^{2} x^{3}+a \,b^{4} c^{3} n^{4} x +30 a \,b^{4} c^{2} d \,n^{3} x^{2}+51 a \,b^{4} c \,d^{2} n^{2} x^{3}+6 a \,b^{4} d^{3} n \,x^{4}+13 b^{5} c^{3} n^{3} x^{2}+147 b^{5} c^{2} d \,n^{2} x^{3}+183 b^{5} c \,d^{2} n \,x^{4}+24 d^{3} x^{5} b^{5}+12 a^{3} b^{2} d^{3} n^{2} x^{2}-6 a^{2} b^{3} c^{2} d \,n^{3} x -54 a^{2} b^{3} c \,d^{2} n^{2} x^{2}-8 a^{2} b^{3} d^{3} n \,x^{3}+12 a \,b^{4} c^{3} n^{3} x +87 a \,b^{4} c^{2} d \,n^{2} x^{2}+30 a \,b^{4} c \,d^{2} n \,x^{3}+59 b^{5} c^{3} n^{2} x^{2}+234 b^{5} c^{2} d n \,x^{3}+90 b^{5} c \,d^{2} x^{4}+18 a^{3} b^{2} c \,d^{2} n^{2} x +12 a^{3} b^{2} d^{3} n \,x^{2}-a^{2} b^{3} c^{3} n^{3}-54 a^{2} b^{3} c^{2} d \,n^{2} x -45 a^{2} b^{3} c \,d^{2} n \,x^{2}+47 a \,b^{4} c^{3} n^{2} x +60 a \,b^{4} c^{2} d n \,x^{2}+107 b^{5} c^{3} n \,x^{2}+120 b^{5} c^{2} d \,x^{3}-24 a^{4} b \,d^{3} n x +6 a^{3} b^{2} c^{2} d \,n^{2}+90 a^{3} b^{2} c \,d^{2} n x -12 a^{2} b^{3} c^{3} n^{2}-120 a^{2} b^{3} c^{2} d n x +60 a \,b^{4} c^{3} n x +60 b^{5} c^{3} x^{2}-18 a^{4} b c \,d^{2} n +54 a^{3} b^{2} c^{2} d n -47 a^{2} b^{3} c^{3} n +24 a^{5} d^{3}-90 a^{4} b c \,d^{2}+120 a^{3} b^{2} c^{2} d -60 a^{2} b^{3} c^{3}\right ) \left (b x +a \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{5}}\) | \(876\) |
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. 367 vs.
\(2 (154) = 308\).
time = 0.31, size = 367, normalized size = 2.38 \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^{3}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {3 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \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^{3}}{{\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. 766 vs.
\(2 (154) = 308\).
time = 0.45, size = 766, normalized size = 4.97 \begin {gather*} -\frac {{\left (a^{2} b^{3} c^{3} n^{3} + 60 \, a^{2} b^{3} c^{3} - 120 \, a^{3} b^{2} c^{2} d + 90 \, a^{4} b c d^{2} - 24 \, a^{5} d^{3} - {\left (b^{5} d^{3} n^{4} + 10 \, b^{5} d^{3} n^{3} + 35 \, b^{5} d^{3} n^{2} + 50 \, b^{5} d^{3} n + 24 \, b^{5} d^{3}\right )} x^{5} - {\left (90 \, b^{5} c d^{2} + {\left (3 \, b^{5} c d^{2} + a b^{4} d^{3}\right )} n^{4} + 3 \, {\left (11 \, b^{5} c d^{2} + 2 \, a b^{4} d^{3}\right )} n^{3} + {\left (123 \, b^{5} c d^{2} + 11 \, a b^{4} d^{3}\right )} n^{2} + 3 \, {\left (61 \, b^{5} c d^{2} + 2 \, a b^{4} d^{3}\right )} n\right )} x^{4} - {\left (120 \, b^{5} c^{2} d + 3 \, {\left (b^{5} c^{2} d + a b^{4} c d^{2}\right )} n^{4} + 4 \, {\left (9 \, b^{5} c^{2} d + 6 \, a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )} n^{3} + 3 \, {\left (49 \, b^{5} c^{2} d + 17 \, a b^{4} c d^{2} - 4 \, a^{2} b^{3} d^{3}\right )} n^{2} + 2 \, {\left (117 \, b^{5} c^{2} d + 15 \, a b^{4} c d^{2} - 4 \, a^{2} b^{3} d^{3}\right )} n\right )} x^{3} + 6 \, {\left (2 \, a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d\right )} n^{2} - {\left (60 \, b^{5} c^{3} + {\left (b^{5} c^{3} + 3 \, a b^{4} c^{2} d\right )} n^{4} + {\left (13 \, b^{5} c^{3} + 30 \, a b^{4} c^{2} d - 9 \, a^{2} b^{3} c d^{2}\right )} n^{3} + {\left (59 \, b^{5} c^{3} + 87 \, a b^{4} c^{2} d - 54 \, a^{2} b^{3} c d^{2} + 12 \, a^{3} b^{2} d^{3}\right )} n^{2} + {\left (107 \, b^{5} c^{3} + 60 \, a b^{4} c^{2} d - 45 \, a^{2} b^{3} c d^{2} + 12 \, a^{3} b^{2} d^{3}\right )} n\right )} x^{2} + {\left (47 \, a^{2} b^{3} c^{3} - 54 \, a^{3} b^{2} c^{2} d + 18 \, a^{4} b c d^{2}\right )} n - {\left (a b^{4} c^{3} n^{4} + 6 \, {\left (2 \, a b^{4} c^{3} - a^{2} b^{3} c^{2} d\right )} n^{3} + {\left (47 \, a b^{4} c^{3} - 54 \, a^{2} b^{3} c^{2} d + 18 \, a^{3} b^{2} c d^{2}\right )} n^{2} + 6 \, {\left (10 \, a b^{4} c^{3} - 20 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - 4 \, a^{4} b d^{3}\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. 7803 vs.
\(2 (138) = 276\).
time = 2.22, size = 7803, normalized size = 50.67 \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. 1305 vs.
\(2 (154) = 308\).
time = 0.96, size = 1305, normalized size = 8.47 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{5} d^{3} n^{4} x^{5} + 3 \, {\left (b x + a\right )}^{n} b^{5} c d^{2} n^{4} x^{4} + {\left (b x + a\right )}^{n} a b^{4} d^{3} n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d^{3} n^{3} x^{5} + 3 \, {\left (b x + a\right )}^{n} b^{5} c^{2} d n^{4} x^{3} + 3 \, {\left (b x + a\right )}^{n} a b^{4} c d^{2} n^{4} x^{3} + 33 \, {\left (b x + a\right )}^{n} b^{5} c d^{2} n^{3} x^{4} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d^{3} n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d^{3} n^{2} x^{5} + {\left (b x + a\right )}^{n} b^{5} c^{3} n^{4} x^{2} + 3 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} d n^{4} x^{2} + 36 \, {\left (b x + a\right )}^{n} b^{5} c^{2} d n^{3} x^{3} + 24 \, {\left (b x + a\right )}^{n} a b^{4} c d^{2} n^{3} x^{3} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{3} n^{3} x^{3} + 123 \, {\left (b x + a\right )}^{n} b^{5} c d^{2} n^{2} x^{4} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d^{3} n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d^{3} n x^{5} + {\left (b x + a\right )}^{n} a b^{4} c^{3} n^{4} x + 13 \, {\left (b x + a\right )}^{n} b^{5} c^{3} n^{3} x^{2} + 30 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} d n^{3} x^{2} - 9 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d^{2} n^{3} x^{2} + 147 \, {\left (b x + a\right )}^{n} b^{5} c^{2} d n^{2} x^{3} + 51 \, {\left (b x + a\right )}^{n} a b^{4} c d^{2} n^{2} x^{3} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{3} n^{2} x^{3} + 183 \, {\left (b x + a\right )}^{n} b^{5} c d^{2} n x^{4} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d^{3} n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d^{3} x^{5} + 12 \, {\left (b x + a\right )}^{n} a b^{4} c^{3} n^{3} x - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} d n^{3} x + 59 \, {\left (b x + a\right )}^{n} b^{5} c^{3} n^{2} x^{2} + 87 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} d n^{2} x^{2} - 54 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d^{2} n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d^{3} n^{2} x^{2} + 234 \, {\left (b x + a\right )}^{n} b^{5} c^{2} d n x^{3} + 30 \, {\left (b x + a\right )}^{n} a b^{4} c d^{2} n x^{3} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d^{3} n x^{3} + 90 \, {\left (b x + a\right )}^{n} b^{5} c d^{2} x^{4} - {\left (b x + a\right )}^{n} a^{2} b^{3} c^{3} n^{3} + 47 \, {\left (b x + a\right )}^{n} a b^{4} c^{3} n^{2} x - 54 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} d n^{2} x + 18 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c d^{2} n^{2} x + 107 \, {\left (b x + a\right )}^{n} b^{5} c^{3} n x^{2} + 60 \, {\left (b x + a\right )}^{n} a b^{4} c^{2} d n x^{2} - 45 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c d^{2} n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d^{3} n x^{2} + 120 \, {\left (b x + a\right )}^{n} b^{5} c^{2} d x^{3} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{3} n^{2} + 6 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} d n^{2} + 60 \, {\left (b x + a\right )}^{n} a b^{4} c^{3} n x - 120 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{2} d n x + 90 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c d^{2} n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d^{3} n x + 60 \, {\left (b x + a\right )}^{n} b^{5} c^{3} x^{2} - 47 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{3} n + 54 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} d n - 18 \, {\left (b x + a\right )}^{n} a^{4} b c d^{2} n - 60 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c^{3} + 120 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c^{2} d - 90 \, {\left (b x + a\right )}^{n} a^{4} b c d^{2} + 24 \, {\left (b x + a\right )}^{n} a^{5} d^{3}}{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 = 1.36, size = 663, normalized size = 4.31 \begin {gather*} \frac {d^3\,x^5\,{\left (a+b\,x\right )}^n\,\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 (a+b\,x\right )}^n\,\left (-24\,a^3\,d^3+18\,a^2\,b\,c\,d^2\,n+90\,a^2\,b\,c\,d^2-6\,a\,b^2\,c^2\,d\,n^2-54\,a\,b^2\,c^2\,d\,n-120\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^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 (a+b\,x\right )}^n\,\left (12\,a^3\,d^3\,n-9\,a^2\,b\,c\,d^2\,n^2-45\,a^2\,b\,c\,d^2\,n+3\,a\,b^2\,c^2\,d\,n^3+27\,a\,b^2\,c^2\,d\,n^2+60\,a\,b^2\,c^2\,d\,n+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d^2\,x^4\,{\left (a+b\,x\right )}^n\,\left (15\,b\,c+a\,d\,n+3\,b\,c\,n\right )\,\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 {d\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-4\,a^2\,d^2\,n+3\,a\,b\,c\,d\,n^2+15\,a\,b\,c\,d\,n+3\,b^2\,c^2\,n^2+27\,b^2\,c^2\,n+60\,b^2\,c^2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (-24\,a^3\,d^3+18\,a^2\,b\,c\,d^2\,n+90\,a^2\,b\,c\,d^2-6\,a\,b^2\,c^2\,d\,n^2-54\,a\,b^2\,c^2\,d\,n-120\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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