Optimal. Leaf size=185 \[ -\frac {a \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^6 (1+n)}+\frac {\left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}-\frac {2 a d \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {2 d \left (b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+n)} \]
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Rubi [A]
time = 0.07, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {786}
\begin {gather*} -\frac {a \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^6 (n+1)}+\frac {\left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac {2 a d \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {2 d \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac {5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^2 (a+b x)^{n+6}}{b^6 (n+6)} \end {gather*}
Antiderivative was successfully verified.
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Rule 786
Rubi steps
\begin {align*} \int x (a+b x)^n \left (c+d x^2\right )^2 \, dx &=\int \left (-\frac {a \left (b^2 c+a^2 d\right )^2 (a+b x)^n}{b^5}+\frac {\left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{1+n}}{b^5}-\frac {2 a d \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^5}+\frac {2 d \left (b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^5}-\frac {5 a d^2 (a+b x)^{4+n}}{b^5}+\frac {d^2 (a+b x)^{5+n}}{b^5}\right ) \, dx\\ &=-\frac {a \left (b^2 c+a^2 d\right )^2 (a+b x)^{1+n}}{b^6 (1+n)}+\frac {\left (b^2 c+a^2 d\right ) \left (b^2 c+5 a^2 d\right ) (a+b x)^{2+n}}{b^6 (2+n)}-\frac {2 a d \left (3 b^2 c+5 a^2 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {2 d \left (b^2 c+5 a^2 d\right ) (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^2 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^2 (a+b x)^{6+n}}{b^6 (6+n)}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 279, normalized size = 1.51 \begin {gather*} \frac {(a+b x)^{1+n} \left (-120 a^5 d^2+120 a^4 b d^2 (1+n) x-12 a^3 b^2 d \left (c \left (30+11 n+n^2\right )+5 d \left (2+3 n+n^2\right ) x^2\right )+4 a^2 b^3 d (1+n) x \left (3 c \left (30+11 n+n^2\right )+5 d \left (6+5 n+n^2\right ) x^2\right )+b^5 \left (15+23 n+9 n^2+n^3\right ) x \left (c^2 \left (24+10 n+n^2\right )+2 c d \left (12+8 n+n^2\right ) x^2+d^2 \left (8+6 n+n^2\right ) x^4\right )-a b^4 \left (c^2 \left (360+342 n+119 n^2+18 n^3+n^4\right )+6 c d \left (60+112 n+65 n^2+14 n^3+n^4\right ) x^2+5 d^2 \left (24+50 n+35 n^2+10 n^3+n^4\right ) x^4\right )\right )}{b^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+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. \(600\) vs.
\(2(185)=370\).
time = 0.08, size = 601, normalized size = 3.25
method | result | size |
norman | \(\frac {d^{2} x^{6} {\mathrm e}^{n \ln \left (b x +a \right )}}{6+n}+\frac {n a \left (b^{4} c^{2} n^{4}+18 b^{4} c^{2} n^{3}+12 a^{2} b^{2} c d \,n^{2}+119 b^{4} c^{2} n^{2}+132 a^{2} b^{2} c d n +342 b^{4} c^{2} n +120 a^{4} d^{2}+360 a^{2} b^{2} c d +360 b^{4} c^{2}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}+\frac {n a \,d^{2} x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+11 n +30\right )}-\frac {a^{2} \left (b^{4} c^{2} n^{4}+18 b^{4} c^{2} n^{3}+12 a^{2} b^{2} c d \,n^{2}+119 b^{4} c^{2} n^{2}+132 a^{2} b^{2} c d n +342 b^{4} c^{2} n +120 a^{4} d^{2}+360 a^{2} b^{2} c d +360 b^{4} c^{2}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}-\frac {\left (-b^{4} c^{2} n^{4}+6 a^{2} b^{2} c d \,n^{3}-18 b^{4} c^{2} n^{3}+66 a^{2} b^{2} c d \,n^{2}-119 b^{4} c^{2} n^{2}+60 a^{4} d^{2} n +180 a^{2} b^{2} c d n -342 b^{4} c^{2} n -360 b^{4} c^{2}\right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+20 n^{4}+155 n^{3}+580 n^{2}+1044 n +720\right )}-\frac {d \left (-2 b^{2} c \,n^{2}+5 a^{2} d n -22 b^{2} c n -60 b^{2} c \right ) x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+15 n^{2}+74 n +120\right )}+\frac {2 \left (b^{2} c \,n^{2}+11 b^{2} c n +10 a^{2} d +30 b^{2} c \right ) a d n \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+18 n^{3}+119 n^{2}+342 n +360\right )}\) | \(601\) |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{5} d^{2} n^{5} x^{5}-15 b^{5} d^{2} n^{4} x^{5}+5 a \,b^{4} d^{2} n^{4} x^{4}-2 b^{5} c d \,n^{5} x^{3}-85 b^{5} d^{2} n^{3} x^{5}+50 a \,b^{4} d^{2} n^{3} x^{4}-34 b^{5} c d \,n^{4} x^{3}-225 b^{5} d^{2} n^{2} x^{5}-20 a^{2} b^{3} d^{2} n^{3} x^{3}+6 a \,b^{4} c d \,n^{4} x^{2}+175 a \,b^{4} d^{2} n^{2} x^{4}-b^{5} c^{2} n^{5} x -214 b^{5} c d \,n^{3} x^{3}-274 b^{5} d^{2} n \,x^{5}-120 a^{2} b^{3} d^{2} n^{2} x^{3}+84 a \,b^{4} c d \,n^{3} x^{2}+250 a \,b^{4} d^{2} n \,x^{4}-19 b^{5} c^{2} n^{4} x -614 b^{5} c d \,n^{2} x^{3}-120 d^{2} x^{5} b^{5}+60 a^{3} b^{2} d^{2} n^{2} x^{2}-12 a^{2} b^{3} c d \,n^{3} x -220 a^{2} b^{3} d^{2} n \,x^{3}+a \,b^{4} c^{2} n^{4}+390 a \,b^{4} c d \,n^{2} x^{2}+120 a \,d^{2} x^{4} b^{4}-137 b^{5} c^{2} n^{3} x -792 b^{5} c d n \,x^{3}+180 a^{3} b^{2} d^{2} n \,x^{2}-144 a^{2} b^{3} c d \,n^{2} x -120 a^{2} b^{3} d^{2} x^{3}+18 a \,b^{4} c^{2} n^{3}+672 a \,b^{4} c d n \,x^{2}-461 b^{5} c^{2} n^{2} x -360 b^{5} c d \,x^{3}-120 a^{4} b \,d^{2} n x +12 a^{3} b^{2} c d \,n^{2}+120 a^{3} b^{2} d^{2} x^{2}-492 a^{2} b^{3} c d n x +119 a \,b^{4} c^{2} n^{2}+360 a \,b^{4} c d \,x^{2}-702 b^{5} c^{2} n x -120 a^{4} b \,d^{2} x +132 a^{3} b^{2} c d n -360 a^{2} b^{3} c d x +342 a \,b^{4} c^{2} n -360 b^{5} c^{2} x +120 a^{5} d^{2}+360 a^{3} b^{2} c d +360 a \,b^{4} c^{2}\right )}{b^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(677\) |
risch | \(-\frac {\left (-b^{6} d^{2} n^{5} x^{6}-a \,b^{5} d^{2} n^{5} x^{5}-15 b^{6} d^{2} n^{4} x^{6}-10 a \,b^{5} d^{2} n^{4} x^{5}-2 b^{6} c d \,n^{5} x^{4}-85 b^{6} d^{2} n^{3} x^{6}+5 a^{2} b^{4} d^{2} n^{4} x^{4}-2 a \,b^{5} c d \,n^{5} x^{3}-35 a \,b^{5} d^{2} n^{3} x^{5}-34 b^{6} c d \,n^{4} x^{4}-225 b^{6} d^{2} n^{2} x^{6}+30 a^{2} b^{4} d^{2} n^{3} x^{4}-28 a \,b^{5} c d \,n^{4} x^{3}-50 a \,b^{5} d^{2} n^{2} x^{5}-b^{6} c^{2} n^{5} x^{2}-214 b^{6} c d \,n^{3} x^{4}-274 b^{6} d^{2} n \,x^{6}-20 a^{3} b^{3} d^{2} n^{3} x^{3}+6 a^{2} b^{4} c d \,n^{4} x^{2}+55 a^{2} b^{4} d^{2} n^{2} x^{4}-a \,b^{5} c^{2} n^{5} x -130 a \,b^{5} c d \,n^{3} x^{3}-24 a \,b^{5} d^{2} n \,x^{5}-19 b^{6} c^{2} n^{4} x^{2}-614 b^{6} c d \,n^{2} x^{4}-120 d^{2} x^{6} b^{6}-60 a^{3} b^{3} d^{2} n^{2} x^{3}+72 a^{2} b^{4} c d \,n^{3} x^{2}+30 a^{2} b^{4} d^{2} n \,x^{4}-18 a \,b^{5} c^{2} n^{4} x -224 a \,b^{5} c d \,n^{2} x^{3}-137 b^{6} c^{2} n^{3} x^{2}-792 b^{6} c d n \,x^{4}+60 a^{4} b^{2} d^{2} n^{2} x^{2}-12 a^{3} b^{3} c d \,n^{3} x -40 a^{3} b^{3} d^{2} n \,x^{3}+a^{2} b^{4} c^{2} n^{4}+246 a^{2} b^{4} c d \,n^{2} x^{2}-119 a \,b^{5} c^{2} n^{3} x -120 a \,b^{5} c d n \,x^{3}-461 b^{6} c^{2} n^{2} x^{2}-360 b^{6} c d \,x^{4}+60 a^{4} b^{2} d^{2} n \,x^{2}-132 a^{3} b^{3} c d \,n^{2} x +18 a^{2} b^{4} c^{2} n^{3}+180 a^{2} b^{4} c d n \,x^{2}-342 a \,b^{5} c^{2} n^{2} x -702 b^{6} c^{2} n \,x^{2}-120 a^{5} b \,d^{2} n x +12 a^{4} b^{2} c d \,n^{2}-360 a^{3} b^{3} c d n x +119 a^{2} b^{4} c^{2} n^{2}-360 a \,b^{5} c^{2} n x -360 b^{6} c^{2} x^{2}+132 a^{4} b^{2} c d n +342 a^{2} b^{4} c^{2} n +120 a^{6} d^{2}+360 a^{4} b^{2} c d +360 a^{2} b^{4} c^{2}\right ) \left (b x +a \right )^{n}}{\left (5+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{6}}\) | \(845\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 335, normalized size = 1.81 \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^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\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}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \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. 757 vs.
\(2 (185) = 370\).
time = 2.03, size = 757, normalized size = 4.09 \begin {gather*} -\frac {{\left (a^{2} b^{4} c^{2} n^{4} + 18 \, a^{2} b^{4} c^{2} n^{3} + 360 \, a^{2} b^{4} c^{2} + 360 \, a^{4} b^{2} c d + 120 \, a^{6} d^{2} - {\left (b^{6} d^{2} n^{5} + 15 \, b^{6} d^{2} n^{4} + 85 \, b^{6} d^{2} n^{3} + 225 \, b^{6} d^{2} n^{2} + 274 \, b^{6} d^{2} n + 120 \, b^{6} d^{2}\right )} x^{6} - {\left (a b^{5} d^{2} n^{5} + 10 \, a b^{5} d^{2} n^{4} + 35 \, a b^{5} d^{2} n^{3} + 50 \, a b^{5} d^{2} n^{2} + 24 \, a b^{5} d^{2} n\right )} x^{5} - {\left (2 \, b^{6} c d n^{5} + 360 \, b^{6} c d + {\left (34 \, b^{6} c d - 5 \, a^{2} b^{4} d^{2}\right )} n^{4} + 2 \, {\left (107 \, b^{6} c d - 15 \, a^{2} b^{4} d^{2}\right )} n^{3} + {\left (614 \, b^{6} c d - 55 \, a^{2} b^{4} d^{2}\right )} n^{2} + 6 \, {\left (132 \, b^{6} c d - 5 \, a^{2} b^{4} d^{2}\right )} n\right )} x^{4} - 2 \, {\left (a b^{5} c d n^{5} + 14 \, a b^{5} c d n^{4} + 5 \, {\left (13 \, a b^{5} c d + 2 \, a^{3} b^{3} d^{2}\right )} n^{3} + 2 \, {\left (56 \, a b^{5} c d + 15 \, a^{3} b^{3} d^{2}\right )} n^{2} + 20 \, {\left (3 \, a b^{5} c d + a^{3} b^{3} d^{2}\right )} n\right )} x^{3} + {\left (119 \, a^{2} b^{4} c^{2} + 12 \, a^{4} b^{2} c d\right )} n^{2} - {\left (b^{6} c^{2} n^{5} + 360 \, b^{6} c^{2} + {\left (19 \, b^{6} c^{2} - 6 \, a^{2} b^{4} c d\right )} n^{4} + {\left (137 \, b^{6} c^{2} - 72 \, a^{2} b^{4} c d\right )} n^{3} + {\left (461 \, b^{6} c^{2} - 246 \, a^{2} b^{4} c d - 60 \, a^{4} b^{2} d^{2}\right )} n^{2} + 6 \, {\left (117 \, b^{6} c^{2} - 30 \, a^{2} b^{4} c d - 10 \, a^{4} b^{2} d^{2}\right )} n\right )} x^{2} + 6 \, {\left (57 \, a^{2} b^{4} c^{2} + 22 \, a^{4} b^{2} c d\right )} n - {\left (a b^{5} c^{2} n^{5} + 18 \, a b^{5} c^{2} n^{4} + {\left (119 \, a b^{5} c^{2} + 12 \, a^{3} b^{3} c d\right )} n^{3} + 6 \, {\left (57 \, a b^{5} c^{2} + 22 \, a^{3} b^{3} c d\right )} n^{2} + 120 \, {\left (3 \, a b^{5} c^{2} + 3 \, a^{3} b^{3} c d + a^{5} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} \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. 8940 vs.
\(2 (170) = 340\).
time = 2.67, size = 8940, normalized size = 48.32 \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. 1266 vs.
\(2 (185) = 370\).
time = 2.76, size = 1266, normalized size = 6.84 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{6} d^{2} n^{5} x^{6} + {\left (b x + a\right )}^{n} a b^{5} d^{2} n^{5} x^{5} + 15 \, {\left (b x + a\right )}^{n} b^{6} d^{2} n^{4} x^{6} + 2 \, {\left (b x + a\right )}^{n} b^{6} c d n^{5} x^{4} + 10 \, {\left (b x + a\right )}^{n} a b^{5} d^{2} n^{4} x^{5} + 85 \, {\left (b x + a\right )}^{n} b^{6} d^{2} n^{3} x^{6} + 2 \, {\left (b x + a\right )}^{n} a b^{5} c d n^{5} x^{3} + 34 \, {\left (b x + a\right )}^{n} b^{6} c d n^{4} x^{4} - 5 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d^{2} n^{4} x^{4} + 35 \, {\left (b x + a\right )}^{n} a b^{5} d^{2} n^{3} x^{5} + 225 \, {\left (b x + a\right )}^{n} b^{6} d^{2} n^{2} x^{6} + {\left (b x + a\right )}^{n} b^{6} c^{2} n^{5} x^{2} + 28 \, {\left (b x + a\right )}^{n} a b^{5} c d n^{4} x^{3} + 214 \, {\left (b x + a\right )}^{n} b^{6} c d n^{3} x^{4} - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d^{2} n^{3} x^{4} + 50 \, {\left (b x + a\right )}^{n} a b^{5} d^{2} n^{2} x^{5} + 274 \, {\left (b x + a\right )}^{n} b^{6} d^{2} n x^{6} + {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{5} x + 19 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{4} x^{2} - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c d n^{4} x^{2} + 130 \, {\left (b x + a\right )}^{n} a b^{5} c d n^{3} x^{3} + 20 \, {\left (b x + a\right )}^{n} a^{3} b^{3} d^{2} n^{3} x^{3} + 614 \, {\left (b x + a\right )}^{n} b^{6} c d n^{2} x^{4} - 55 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d^{2} n^{2} x^{4} + 24 \, {\left (b x + a\right )}^{n} a b^{5} d^{2} n x^{5} + 120 \, {\left (b x + a\right )}^{n} b^{6} d^{2} x^{6} + 18 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{4} x + 137 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{3} x^{2} - 72 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c d n^{3} x^{2} + 224 \, {\left (b x + a\right )}^{n} a b^{5} c d n^{2} x^{3} + 60 \, {\left (b x + a\right )}^{n} a^{3} b^{3} d^{2} n^{2} x^{3} + 792 \, {\left (b x + a\right )}^{n} b^{6} c d n x^{4} - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} d^{2} n x^{4} - {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{4} + 119 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{3} x + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c d n^{3} x + 461 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{2} x^{2} - 246 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c d n^{2} x^{2} - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} d^{2} n^{2} x^{2} + 120 \, {\left (b x + a\right )}^{n} a b^{5} c d n x^{3} + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{3} d^{2} n x^{3} + 360 \, {\left (b x + a\right )}^{n} b^{6} c d x^{4} - 18 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{3} + 342 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{2} x + 132 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c d n^{2} x + 702 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n x^{2} - 180 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c d n x^{2} - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} d^{2} n x^{2} - 119 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{2} - 12 \, {\left (b x + a\right )}^{n} a^{4} b^{2} c d n^{2} + 360 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n x + 360 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c d n x + 120 \, {\left (b x + a\right )}^{n} a^{5} b d^{2} n x + 360 \, {\left (b x + a\right )}^{n} b^{6} c^{2} x^{2} - 342 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n - 132 \, {\left (b x + a\right )}^{n} a^{4} b^{2} c d n - 360 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} - 360 \, {\left (b x + a\right )}^{n} a^{4} b^{2} c d - 120 \, {\left (b x + a\right )}^{n} a^{6} d^{2}}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.05, size = 723, normalized size = 3.91 \begin {gather*} \frac {d^2\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}-\frac {a^2\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+132\,a^2\,b^2\,c\,d\,n+360\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-60\,a^4\,d^2\,n-6\,a^2\,b^2\,c\,d\,n^3-66\,a^2\,b^2\,c\,d\,n^2-180\,a^2\,b^2\,c\,d\,n+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {d\,x^4\,{\left (a+b\,x\right )}^n\,\left (-5\,d\,a^2\,n+2\,c\,b^2\,n^2+22\,c\,b^2\,n+60\,c\,b^2\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{b^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,n\,x\,{\left (a+b\,x\right )}^n\,\left (120\,a^4\,d^2+12\,a^2\,b^2\,c\,d\,n^2+132\,a^2\,b^2\,c\,d\,n+360\,a^2\,b^2\,c\,d+b^4\,c^2\,n^4+18\,b^4\,c^2\,n^3+119\,b^4\,c^2\,n^2+342\,b^4\,c^2\,n+360\,b^4\,c^2\right )}{b^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,d^2\,n\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {2\,a\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (10\,d\,a^2+c\,b^2\,n^2+11\,c\,b^2\,n+30\,c\,b^2\right )}{b^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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