Optimal. Leaf size=78 \[ \frac {(b c-a d)^2 (c+d x)^{1+n}}{d^3 (1+n)}-\frac {2 b (b c-a d) (c+d x)^{2+n}}{d^3 (2+n)}+\frac {b^2 (c+d x)^{3+n}}{d^3 (3+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} \frac {(b c-a d)^2 (c+d x)^{n+1}}{d^3 (n+1)}-\frac {2 b (b c-a d) (c+d x)^{n+2}}{d^3 (n+2)}+\frac {b^2 (c+d x)^{n+3}}{d^3 (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^2 (c+d x)^n \, dx &=\int \left (\frac {(-b c+a d)^2 (c+d x)^n}{d^2}-\frac {2 b (b c-a d) (c+d x)^{1+n}}{d^2}+\frac {b^2 (c+d x)^{2+n}}{d^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (c+d x)^{1+n}}{d^3 (1+n)}-\frac {2 b (b c-a d) (c+d x)^{2+n}}{d^3 (2+n)}+\frac {b^2 (c+d x)^{3+n}}{d^3 (3+n)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 67, normalized size = 0.86 \begin {gather*} \frac {(c+d x)^{1+n} \left (\frac {(b c-a d)^2}{1+n}-\frac {2 b (b c-a d) (c+d x)}{2+n}+\frac {b^2 (c+d x)^2}{3+n}\right )}{d^3} \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. \(158\) vs.
\(2(78)=156\).
time = 0.19, size = 159, normalized size = 2.04
method | result | size |
gosper | \(\frac {\left (d x +c \right )^{1+n} \left (b^{2} d^{2} n^{2} x^{2}+2 a b \,d^{2} n^{2} x +3 b^{2} d^{2} n \,x^{2}+a^{2} d^{2} n^{2}+8 a b \,d^{2} n x -2 b^{2} c d n x +2 b^{2} x^{2} d^{2}+5 a^{2} d^{2} n -2 a b c d n +6 a b \,d^{2} x -2 b^{2} c d x +6 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right )}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(159\) |
norman | \(\frac {b^{2} x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{3+n}+\frac {c \left (a^{2} d^{2} n^{2}+5 a^{2} d^{2} n -2 a b c d n +6 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (a^{2} d^{2} n^{2}+2 a b c d \,n^{2}+5 a^{2} d^{2} n +6 a b c d n -2 b^{2} c^{2} n +6 a^{2} d^{2}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (2 a d n +b c n +6 a d \right ) b \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+5 n +6\right )}\) | \(224\) |
risch | \(\frac {\left (b^{2} d^{3} n^{2} x^{3}+2 a b \,d^{3} n^{2} x^{2}+b^{2} c \,d^{2} n^{2} x^{2}+3 b^{2} d^{3} n \,x^{3}+a^{2} d^{3} n^{2} x +2 a b c \,d^{2} n^{2} x +8 a b \,d^{3} n \,x^{2}+b^{2} c \,d^{2} n \,x^{2}+2 b^{2} x^{3} d^{3}+a^{2} c \,d^{2} n^{2}+5 a^{2} d^{3} n x +6 a b c \,d^{2} n x +6 a b \,d^{3} x^{2}-2 b^{2} c^{2} d n x +5 a^{2} c \,d^{2} n +6 a^{2} d^{3} x -2 a b \,c^{2} d n +6 a^{2} c \,d^{2}-6 a b \,c^{2} d +2 b^{2} c^{3}\right ) \left (d x +c \right )^{n}}{\left (2+n \right ) \left (3+n \right ) \left (1+n \right ) d^{3}}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 138, normalized size = 1.77 \begin {gather*} \frac {2 \, {\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a b}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d x + c\right )}^{n + 1} a^{2}}{d {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} b^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} \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. 237 vs.
\(2 (78) = 156\).
time = 0.72, size = 237, normalized size = 3.04 \begin {gather*} \frac {{\left (a^{2} c d^{2} n^{2} + 2 \, b^{2} c^{3} - 6 \, a b c^{2} d + 6 \, a^{2} c d^{2} + {\left (b^{2} d^{3} n^{2} + 3 \, b^{2} d^{3} n + 2 \, b^{2} d^{3}\right )} x^{3} + {\left (6 \, a b d^{3} + {\left (b^{2} c d^{2} + 2 \, a b d^{3}\right )} n^{2} + {\left (b^{2} c d^{2} + 8 \, a b d^{3}\right )} n\right )} x^{2} - {\left (2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} n + {\left (6 \, a^{2} d^{3} + {\left (2 \, a b c d^{2} + a^{2} d^{3}\right )} n^{2} - {\left (2 \, b^{2} c^{2} d - 6 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \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. 1506 vs.
\(2 (66) = 132\).
time = 0.55, size = 1506, normalized size = 19.31 \begin {gather*} \begin {cases} c^{n} \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) & \text {for}\: d = 0 \\- \frac {a^{2} d^{2}}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} - \frac {2 a b c d}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} - \frac {4 a b d^{2} x}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} + \frac {2 b^{2} c^{2} \log {\left (\frac {c}{d} + x \right )}}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} + \frac {3 b^{2} c^{2}}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} + \frac {4 b^{2} c d x \log {\left (\frac {c}{d} + x \right )}}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} + \frac {4 b^{2} c d x}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} + \frac {2 b^{2} d^{2} x^{2} \log {\left (\frac {c}{d} + x \right )}}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} & \text {for}\: n = -3 \\- \frac {a^{2} d^{2}}{c d^{3} + d^{4} x} + \frac {2 a b c d \log {\left (\frac {c}{d} + x \right )}}{c d^{3} + d^{4} x} + \frac {2 a b c d}{c d^{3} + d^{4} x} + \frac {2 a b d^{2} x \log {\left (\frac {c}{d} + x \right )}}{c d^{3} + d^{4} x} - \frac {2 b^{2} c^{2} \log {\left (\frac {c}{d} + x \right )}}{c d^{3} + d^{4} x} - \frac {2 b^{2} c^{2}}{c d^{3} + d^{4} x} - \frac {2 b^{2} c d x \log {\left (\frac {c}{d} + x \right )}}{c d^{3} + d^{4} x} + \frac {b^{2} d^{2} x^{2}}{c d^{3} + d^{4} x} & \text {for}\: n = -2 \\\frac {a^{2} \log {\left (\frac {c}{d} + x \right )}}{d} - \frac {2 a b c \log {\left (\frac {c}{d} + x \right )}}{d^{2}} + \frac {2 a b x}{d} + \frac {b^{2} c^{2} \log {\left (\frac {c}{d} + x \right )}}{d^{3}} - \frac {b^{2} c x}{d^{2}} + \frac {b^{2} x^{2}}{2 d} & \text {for}\: n = -1 \\\frac {a^{2} c d^{2} n^{2} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {5 a^{2} c d^{2} n \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {6 a^{2} c d^{2} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {a^{2} d^{3} n^{2} x \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {5 a^{2} d^{3} n x \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {6 a^{2} d^{3} x \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} - \frac {2 a b c^{2} d n \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} - \frac {6 a b c^{2} d \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {2 a b c d^{2} n^{2} x \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {6 a b c d^{2} n x \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {2 a b d^{3} n^{2} x^{2} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {8 a b d^{3} n x^{2} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {6 a b d^{3} x^{2} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {2 b^{2} c^{3} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} - \frac {2 b^{2} c^{2} d n x \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {b^{2} c d^{2} n^{2} x^{2} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {b^{2} c d^{2} n x^{2} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {b^{2} d^{3} n^{2} x^{3} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {3 b^{2} d^{3} n x^{3} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} + \frac {2 b^{2} d^{3} x^{3} \left (c + d x\right )^{n}}{d^{3} n^{3} + 6 d^{3} n^{2} + 11 d^{3} n + 6 d^{3}} & \text {otherwise} \end {cases} \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. 385 vs.
\(2 (78) = 156\).
time = 1.61, size = 385, normalized size = 4.94 \begin {gather*} \frac {{\left (d x + c\right )}^{n} b^{2} d^{3} n^{2} x^{3} + {\left (d x + c\right )}^{n} b^{2} c d^{2} n^{2} x^{2} + 2 \, {\left (d x + c\right )}^{n} a b d^{3} n^{2} x^{2} + 3 \, {\left (d x + c\right )}^{n} b^{2} d^{3} n x^{3} + 2 \, {\left (d x + c\right )}^{n} a b c d^{2} n^{2} x + {\left (d x + c\right )}^{n} a^{2} d^{3} n^{2} x + {\left (d x + c\right )}^{n} b^{2} c d^{2} n x^{2} + 8 \, {\left (d x + c\right )}^{n} a b d^{3} n x^{2} + 2 \, {\left (d x + c\right )}^{n} b^{2} d^{3} x^{3} + {\left (d x + c\right )}^{n} a^{2} c d^{2} n^{2} - 2 \, {\left (d x + c\right )}^{n} b^{2} c^{2} d n x + 6 \, {\left (d x + c\right )}^{n} a b c d^{2} n x + 5 \, {\left (d x + c\right )}^{n} a^{2} d^{3} n x + 6 \, {\left (d x + c\right )}^{n} a b d^{3} x^{2} - 2 \, {\left (d x + c\right )}^{n} a b c^{2} d n + 5 \, {\left (d x + c\right )}^{n} a^{2} c d^{2} n + 6 \, {\left (d x + c\right )}^{n} a^{2} d^{3} x + 2 \, {\left (d x + c\right )}^{n} b^{2} c^{3} - 6 \, {\left (d x + c\right )}^{n} a b c^{2} d + 6 \, {\left (d x + c\right )}^{n} a^{2} c d^{2}}{d^{3} n^{3} + 6 \, d^{3} n^{2} + 11 \, d^{3} n + 6 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 226, normalized size = 2.90 \begin {gather*} {\left (c+d\,x\right )}^n\,\left (\frac {c\,\left (a^2\,d^2\,n^2+5\,a^2\,d^2\,n+6\,a^2\,d^2-2\,a\,b\,c\,d\,n-6\,a\,b\,c\,d+2\,b^2\,c^2\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b^2\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}+\frac {x\,\left (a^2\,d^3\,n^2+5\,a^2\,d^3\,n+6\,a^2\,d^3+2\,a\,b\,c\,d^2\,n^2+6\,a\,b\,c\,d^2\,n-2\,b^2\,c^2\,d\,n\right )}{d^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b\,x^2\,\left (n+1\right )\,\left (6\,a\,d+2\,a\,d\,n+b\,c\,n\right )}{d\,\left (n^3+6\,n^2+11\,n+6\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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