Optimal. Leaf size=78 \[ \frac {(b c-a d)^2 (a+b x)^{1+n}}{b^3 (1+n)}+\frac {2 d (b c-a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^2 (a+b x)^{3+n}}{b^3 (3+n)} \]
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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 (a+b x)^{n+1}}{b^3 (n+1)}+\frac {2 d (b c-a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d^2 (a+b x)^{n+3}}{b^3 (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^n (c+d x)^2 \, dx &=\int \left (\frac {(b c-a d)^2 (a+b x)^n}{b^2}+\frac {2 d (b c-a d) (a+b x)^{1+n}}{b^2}+\frac {d^2 (a+b x)^{2+n}}{b^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (a+b x)^{1+n}}{b^3 (1+n)}+\frac {2 d (b c-a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^2 (a+b x)^{3+n}}{b^3 (3+n)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 67, normalized size = 0.86 \begin {gather*} \frac {(a+b x)^{1+n} \left (\frac {(b c-a d)^2}{1+n}+\frac {2 d (b c-a d) (a+b x)}{2+n}+\frac {d^2 (a+b x)^2}{3+n}\right )}{b^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 = 0.08, size = 159, normalized size = 2.04
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{2} d^{2} n^{2} x^{2}+2 b^{2} c d \,n^{2} x +3 b^{2} d^{2} n \,x^{2}-2 a b \,d^{2} n x +b^{2} c^{2} n^{2}+8 b^{2} c d n x +2 b^{2} d^{2} x^{2}-2 a b c d n -2 a b \,d^{2} x +5 b^{2} c^{2} n +6 b^{2} c d x +2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}\right )}{b^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(159\) |
norman | \(\frac {d^{2} x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{3+n}+\frac {a \left (b^{2} c^{2} n^{2}-2 a b c d n +5 b^{2} c^{2} n +2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (a d n +2 b c n +6 b c \right ) d \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+5 n +6\right )}-\frac {\left (-2 a b c d \,n^{2}-b^{2} c^{2} n^{2}+2 a^{2} d^{2} n -6 a b c d n -5 b^{2} c^{2} n -6 b^{2} c^{2}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(226\) |
risch | \(\frac {\left (b^{3} d^{2} n^{2} x^{3}+a \,b^{2} d^{2} n^{2} x^{2}+2 b^{3} c d \,n^{2} x^{2}+3 b^{3} d^{2} n \,x^{3}+2 a \,b^{2} c d \,n^{2} x +a \,b^{2} d^{2} n \,x^{2}+b^{3} c^{2} n^{2} x +8 b^{3} c d n \,x^{2}+2 d^{2} x^{3} b^{3}-2 a^{2} b \,d^{2} n x +a \,b^{2} c^{2} n^{2}+6 a \,b^{2} c d n x +5 b^{3} c^{2} n x +6 b^{3} c d \,x^{2}-2 a^{2} b c d n +5 a \,b^{2} c^{2} n +6 b^{3} c^{2} x +2 a^{3} d^{2}-6 a^{2} b c d +6 a \,b^{2} c^{2}\right ) \left (b x +a \right )^{n}}{\left (2+n \right ) \left (3+n \right ) \left (1+n \right ) b^{3}}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 138, normalized size = 1.77 \begin {gather*} \frac {2 \, {\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{n + 1} c^{2}}{b {\left (n + 1\right )}} + \frac {{\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} d^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{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. 235 vs.
\(2 (78) = 156\).
time = 0.48, size = 235, normalized size = 3.01 \begin {gather*} \frac {{\left (a b^{2} c^{2} n^{2} + 6 \, a b^{2} c^{2} - 6 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (b^{3} d^{2} n^{2} + 3 \, b^{3} d^{2} n + 2 \, b^{3} d^{2}\right )} x^{3} + {\left (6 \, b^{3} c d + {\left (2 \, b^{3} c d + a b^{2} d^{2}\right )} n^{2} + {\left (8 \, b^{3} c d + a b^{2} d^{2}\right )} n\right )} x^{2} + {\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d\right )} n + {\left (6 \, b^{3} c^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d\right )} n^{2} + {\left (5 \, b^{3} c^{2} + 6 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{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 = 0.61, size = 1506, normalized size = 19.31 \begin {gather*} \begin {cases} a^{n} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {for}\: b = 0 \\\frac {2 a^{2} d^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {3 a^{2} d^{2}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} - \frac {2 a b c d}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b d^{2} x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b d^{2} x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} - \frac {b^{2} c^{2}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} - \frac {4 b^{2} c d x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {2 b^{2} d^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} & \text {for}\: n = -3 \\- \frac {2 a^{2} d^{2} \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {2 a^{2} d^{2}}{a b^{3} + b^{4} x} + \frac {2 a b c d \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {2 a b c d}{a b^{3} + b^{4} x} - \frac {2 a b d^{2} x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {b^{2} c^{2}}{a b^{3} + b^{4} x} + \frac {2 b^{2} c d x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {b^{2} d^{2} x^{2}}{a b^{3} + b^{4} x} & \text {for}\: n = -2 \\\frac {a^{2} d^{2} \log {\left (\frac {a}{b} + x \right )}}{b^{3}} - \frac {2 a c d \log {\left (\frac {a}{b} + x \right )}}{b^{2}} - \frac {a d^{2} x}{b^{2}} + \frac {c^{2} \log {\left (\frac {a}{b} + x \right )}}{b} + \frac {2 c d x}{b} + \frac {d^{2} x^{2}}{2 b} & \text {for}\: n = -1 \\\frac {2 a^{3} d^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {2 a^{2} b c d n \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {6 a^{2} b c d \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {2 a^{2} b d^{2} n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} c^{2} n^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {5 a b^{2} c^{2} n \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {6 a b^{2} c^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {2 a b^{2} c d n^{2} x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {6 a b^{2} c d n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} d^{2} n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} d^{2} n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} c^{2} n^{2} x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {5 b^{3} c^{2} n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {6 b^{3} c^{2} x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {2 b^{3} c d n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {8 b^{3} c d n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {6 b^{3} c d x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} d^{2} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {3 b^{3} d^{2} n x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {2 b^{3} d^{2} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{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 = 1.28, size = 385, normalized size = 4.94 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{3} d^{2} n^{2} x^{3} + 2 \, {\left (b x + a\right )}^{n} b^{3} c d n^{2} x^{2} + {\left (b x + a\right )}^{n} a b^{2} d^{2} n^{2} x^{2} + 3 \, {\left (b x + a\right )}^{n} b^{3} d^{2} n x^{3} + {\left (b x + a\right )}^{n} b^{3} c^{2} n^{2} x + 2 \, {\left (b x + a\right )}^{n} a b^{2} c d n^{2} x + 8 \, {\left (b x + a\right )}^{n} b^{3} c d n x^{2} + {\left (b x + a\right )}^{n} a b^{2} d^{2} n x^{2} + 2 \, {\left (b x + a\right )}^{n} b^{3} d^{2} x^{3} + {\left (b x + a\right )}^{n} a b^{2} c^{2} n^{2} + 5 \, {\left (b x + a\right )}^{n} b^{3} c^{2} n x + 6 \, {\left (b x + a\right )}^{n} a b^{2} c d n x - 2 \, {\left (b x + a\right )}^{n} a^{2} b d^{2} n x + 6 \, {\left (b x + a\right )}^{n} b^{3} c d x^{2} + 5 \, {\left (b x + a\right )}^{n} a b^{2} c^{2} n - 2 \, {\left (b x + a\right )}^{n} a^{2} b c d n + 6 \, {\left (b x + a\right )}^{n} b^{3} c^{2} x + 6 \, {\left (b x + a\right )}^{n} a b^{2} c^{2} - 6 \, {\left (b x + a\right )}^{n} a^{2} b c d + 2 \, {\left (b x + a\right )}^{n} a^{3} d^{2}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 226, normalized size = 2.90 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {a\,\left (2\,a^2\,d^2-2\,a\,b\,c\,d\,n-6\,a\,b\,c\,d+b^2\,c^2\,n^2+5\,b^2\,c^2\,n+6\,b^2\,c^2\right )}{b^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {d^2\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}+\frac {x\,\left (-2\,a^2\,b\,d^2\,n+2\,a\,b^2\,c\,d\,n^2+6\,a\,b^2\,c\,d\,n+b^3\,c^2\,n^2+5\,b^3\,c^2\,n+6\,b^3\,c^2\right )}{b^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {d\,x^2\,\left (n+1\right )\,\left (6\,b\,c+a\,d\,n+2\,b\,c\,n\right )}{b\,\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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