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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 6.90, size = 1256, normalized size = 16.10 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x \left (3 a^2+3 a b x+b^2 x^2\right ) c^n}{3},d\text {==}0\right \},\left \{\frac {b^2 c^2 \left (3+2 \text {Log}\left [\frac {c+d x}{d}\right ]\right )+2 b c d \left (-a+2 b x+2 b x \text {Log}\left [\frac {c+d x}{d}\right ]\right )+d^2 \left (-a^2-4 a b x+2 b^2 x^2 \text {Log}\left [\frac {c+d x}{d}\right ]\right )}{2 d^3 \left (c^2+2 c d x+d^2 x^2\right )},n\text {==}-3\right \},\left \{\frac {-2 b^2 c^2 \left (1+\text {Log}\left [\frac {c+d x}{d}\right ]\right )+2 b c d \left (a+a \text {Log}\left [\frac {c+d x}{d}\right ]-b x \text {Log}\left [\frac {c+d x}{d}\right ]\right )+d^2 \left (-a^2+2 a b x \text {Log}\left [\frac {c+d x}{d}\right ]+b^2 x^2\right )}{d^3 \left (c+d x\right )},n\text {==}-2\right \},\left \{\frac {b^2 c^2 \text {Log}\left [\frac {c+d x}{d}\right ]-b c d \left (2 a \text {Log}\left [\frac {c+d x}{d}\right ]+b x\right )+\frac {d^2 \left (2 a^2 \text {Log}\left [\frac {c+d x}{d}\right ]+4 a b x+b^2 x^2\right )}{2}}{d^3},n\text {==}-1\right \}\right \},\frac {6 a^2 c d^2 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {5 a^2 c d^2 n \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {a^2 c d^2 n^2 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {6 a^2 d^3 x \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {5 a^2 d^3 n x \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {a^2 d^3 n^2 x \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}-\frac {6 a b c^2 d \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}-\frac {2 a b c^2 d n \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {6 a b c d^2 n x \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {2 a b c d^2 n^2 x \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {6 a b d^3 x^2 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {8 a b d^3 n x^2 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {2 a b d^3 n^2 x^2 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {2 b^2 c^3 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}-\frac {2 b^2 c^2 d n x \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {b^2 c d^2 n x^2 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {b^2 c d^2 n^2 x^2 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {2 b^2 d^3 x^3 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {3 b^2 d^3 n x^3 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}+\frac {b^2 d^3 n^2 x^3 \left (c+d x\right )^n}{6 d^3+11 d^3 n+6 d^3 n^2+d^3 n^3}\right ] \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 = 0.29, 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 = 0.33, 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 [A]
time = 0.81, size = 1506, normalized size = 19.31
result too large to display
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 = 0.00, size = 427, normalized size = 5.47 \begin {gather*} \frac {a^{2} c d^{2} n^{2} \mathrm {e}^{n \ln \left (c+d x\right )}+5 a^{2} c d^{2} n \mathrm {e}^{n \ln \left (c+d x\right )}+6 a^{2} c d^{2} \mathrm {e}^{n \ln \left (c+d x\right )}+a^{2} d^{3} n^{2} x \mathrm {e}^{n \ln \left (c+d x\right )}+5 a^{2} d^{3} n x \mathrm {e}^{n \ln \left (c+d x\right )}+6 a^{2} d^{3} x \mathrm {e}^{n \ln \left (c+d x\right )}-2 a b c^{2} d n \mathrm {e}^{n \ln \left (c+d x\right )}-6 a b c^{2} d \mathrm {e}^{n \ln \left (c+d x\right )}+2 a b c d^{2} n^{2} x \mathrm {e}^{n \ln \left (c+d x\right )}+6 a b c d^{2} n x \mathrm {e}^{n \ln \left (c+d x\right )}+2 a b d^{3} n^{2} x^{2} \mathrm {e}^{n \ln \left (c+d x\right )}+8 a b d^{3} n x^{2} \mathrm {e}^{n \ln \left (c+d x\right )}+6 a b d^{3} x^{2} \mathrm {e}^{n \ln \left (c+d x\right )}+2 b^{2} c^{3} \mathrm {e}^{n \ln \left (c+d x\right )}-2 b^{2} c^{2} d n x \mathrm {e}^{n \ln \left (c+d x\right )}+b^{2} c d^{2} n^{2} x^{2} \mathrm {e}^{n \ln \left (c+d x\right )}+b^{2} c d^{2} n x^{2} \mathrm {e}^{n \ln \left (c+d x\right )}+b^{2} d^{3} n^{2} x^{3} \mathrm {e}^{n \ln \left (c+d x\right )}+3 b^{2} d^{3} n x^{3} \mathrm {e}^{n \ln \left (c+d x\right )}+2 b^{2} d^{3} x^{3} \mathrm {e}^{n \ln \left (c+d x\right )}}{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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