Optimal. Leaf size=84 \[ \frac {\left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^3 (1+n)}-\frac {2 c (e f-d g) (f+g x)^{2+n}}{g^3 (2+n)}+\frac {c e (f+g x)^{3+n}}{g^3 (3+n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {712}
\begin {gather*} \frac {(f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^3 (n+1)}-\frac {2 c (e f-d g) (f+g x)^{n+2}}{g^3 (n+2)}+\frac {c e (f+g x)^{n+3}}{g^3 (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx &=\int \left (\frac {\left (a g^2+c f (e f-2 d g)\right ) (f+g x)^n}{g^2}+\frac {2 c (-e f+d g) (f+g x)^{1+n}}{g^2}+\frac {c e (f+g x)^{2+n}}{g^2}\right ) \, dx\\ &=\frac {\left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^3 (1+n)}-\frac {2 c (e f-d g) (f+g x)^{2+n}}{g^3 (2+n)}+\frac {c e (f+g x)^{3+n}}{g^3 (3+n)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 92, normalized size = 1.10 \begin {gather*} \frac {(f+g x)^{1+n} \left (a g^2 \left (6+5 n+n^2\right )+2 c d g (3+n) (-f+g (1+n) x)+c e \left (2 f^2-2 f g (1+n) x+g^2 \left (2+3 n+n^2\right ) x^2\right )\right )}{g^3 (1+n) (2+n) (3+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 147, normalized size = 1.75
method | result | size |
gosper | \(\frac {\left (g x +f \right )^{1+n} \left (c e \,g^{2} n^{2} x^{2}+2 c d \,g^{2} n^{2} x +3 c e \,g^{2} n \,x^{2}+8 c d \,g^{2} n x -2 c e f g n x +2 c e \,x^{2} g^{2}+a \,g^{2} n^{2}-2 c d f g n +6 c d \,g^{2} x -2 c e f g x +5 a \,g^{2} n -6 c d f g +2 c e \,f^{2}+6 a \,g^{2}\right )}{g^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}\) | \(147\) |
norman | \(\frac {c e \,x^{3} {\mathrm e}^{n \ln \left (g x +f \right )}}{3+n}+\frac {f \left (a \,g^{2} n^{2}-2 c d f g n +5 a \,g^{2} n -6 c d f g +2 c e \,f^{2}+6 a \,g^{2}\right ) {\mathrm e}^{n \ln \left (g x +f \right )}}{g^{3} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (2 c d f g \,n^{2}+a \,g^{2} n^{2}+6 c d f g n -2 c e \,f^{2} n +5 a \,g^{2} n +6 a \,g^{2}\right ) x \,{\mathrm e}^{n \ln \left (g x +f \right )}}{g^{2} \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (2 d g n +e f n +6 d g \right ) c \,x^{2} {\mathrm e}^{n \ln \left (g x +f \right )}}{g \left (n^{2}+5 n +6\right )}\) | \(209\) |
risch | \(\frac {\left (c e \,g^{3} n^{2} x^{3}+2 c d \,g^{3} n^{2} x^{2}+c e f \,g^{2} n^{2} x^{2}+3 c e \,g^{3} n \,x^{3}+2 c d f \,g^{2} n^{2} x +8 c d \,g^{3} n \,x^{2}+c e f \,g^{2} n \,x^{2}+2 c e \,x^{3} g^{3}+a \,g^{3} n^{2} x +6 c d f \,g^{2} n x +6 c d \,g^{3} x^{2}-2 c e \,f^{2} g n x +a f \,g^{2} n^{2}+5 a \,g^{3} n x -2 c d \,f^{2} g n +5 a f \,g^{2} n +6 a \,g^{3} x -6 c d \,f^{2} g +2 c e \,f^{3}+6 a f \,g^{2}\right ) \left (g x +f \right )^{n}}{\left (2+n \right ) \left (3+n \right ) \left (1+n \right ) g^{3}}\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 135, normalized size = 1.61 \begin {gather*} \frac {2 \, {\left (g^{2} {\left (n + 1\right )} x^{2} + f g n x - f^{2}\right )} {\left (g x + f\right )}^{n} c d}{{\left (n^{2} + 3 \, n + 2\right )} g^{2}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} g^{3} x^{3} + {\left (n^{2} + n\right )} f g^{2} x^{2} - 2 \, f^{2} g n x + 2 \, f^{3}\right )} {\left (g x + f\right )}^{n} c e}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{3}} + \frac {{\left (g x + f\right )}^{n + 1} a}{g {\left (n + 1\right )}} \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. 224 vs.
\(2 (90) = 180\).
time = 4.93, size = 224, normalized size = 2.67 \begin {gather*} \frac {{\left (a f g^{2} n^{2} - 6 \, c d f^{2} g + 6 \, a f g^{2} + 2 \, {\left (c d g^{3} n^{2} + 4 \, c d g^{3} n + 3 \, c d g^{3}\right )} x^{2} - {\left (2 \, c d f^{2} g - 5 \, a f g^{2}\right )} n + {\left (6 \, a g^{3} + {\left (2 \, c d f g^{2} + a g^{3}\right )} n^{2} + {\left (6 \, c d f g^{2} + 5 \, a g^{3}\right )} n\right )} x - {\left (2 \, c f^{2} g n x - 2 \, c f^{3} - {\left (c g^{3} n^{2} + 3 \, c g^{3} n + 2 \, c g^{3}\right )} x^{3} - {\left (c f g^{2} n^{2} + c f g^{2} n\right )} x^{2}\right )} e\right )} {\left (g x + f\right )}^{n}}{g^{3} n^{3} + 6 \, g^{3} n^{2} + 11 \, g^{3} n + 6 \, g^{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. 1489 vs.
\(2 (75) = 150\).
time = 0.57, size = 1489, normalized size = 17.73 \begin {gather*} \begin {cases} f^{n} \left (a x + c d x^{2} + \frac {c e x^{3}}{3}\right ) & \text {for}\: g = 0 \\- \frac {a g^{2}}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} - \frac {2 c d f g}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} - \frac {4 c d g^{2} x}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} + \frac {2 c e f^{2} \log {\left (\frac {f}{g} + x \right )}}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} + \frac {3 c e f^{2}}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} + \frac {4 c e f g x \log {\left (\frac {f}{g} + x \right )}}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} + \frac {4 c e f g x}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} + \frac {2 c e g^{2} x^{2} \log {\left (\frac {f}{g} + x \right )}}{2 f^{2} g^{3} + 4 f g^{4} x + 2 g^{5} x^{2}} & \text {for}\: n = -3 \\- \frac {a g^{2}}{f g^{3} + g^{4} x} + \frac {2 c d f g \log {\left (\frac {f}{g} + x \right )}}{f g^{3} + g^{4} x} + \frac {2 c d f g}{f g^{3} + g^{4} x} + \frac {2 c d g^{2} x \log {\left (\frac {f}{g} + x \right )}}{f g^{3} + g^{4} x} - \frac {2 c e f^{2} \log {\left (\frac {f}{g} + x \right )}}{f g^{3} + g^{4} x} - \frac {2 c e f^{2}}{f g^{3} + g^{4} x} - \frac {2 c e f g x \log {\left (\frac {f}{g} + x \right )}}{f g^{3} + g^{4} x} + \frac {c e g^{2} x^{2}}{f g^{3} + g^{4} x} & \text {for}\: n = -2 \\\frac {a \log {\left (\frac {f}{g} + x \right )}}{g} - \frac {2 c d f \log {\left (\frac {f}{g} + x \right )}}{g^{2}} + \frac {2 c d x}{g} + \frac {c e f^{2} \log {\left (\frac {f}{g} + x \right )}}{g^{3}} - \frac {c e f x}{g^{2}} + \frac {c e x^{2}}{2 g} & \text {for}\: n = -1 \\\frac {a f g^{2} n^{2} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {5 a f g^{2} n \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {6 a f g^{2} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {a g^{3} n^{2} x \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {5 a g^{3} n x \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {6 a g^{3} x \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} - \frac {2 c d f^{2} g n \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} - \frac {6 c d f^{2} g \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {2 c d f g^{2} n^{2} x \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {6 c d f g^{2} n x \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {2 c d g^{3} n^{2} x^{2} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {8 c d g^{3} n x^{2} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {6 c d g^{3} x^{2} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {2 c e f^{3} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} - \frac {2 c e f^{2} g n x \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {c e f g^{2} n^{2} x^{2} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {c e f g^{2} n x^{2} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {c e g^{3} n^{2} x^{3} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {3 c e g^{3} n x^{3} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{3}} + \frac {2 c e g^{3} x^{3} \left (f + g x\right )^{n}}{g^{3} n^{3} + 6 g^{3} n^{2} + 11 g^{3} n + 6 g^{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. 373 vs.
\(2 (90) = 180\).
time = 3.81, size = 373, normalized size = 4.44 \begin {gather*} \frac {{\left (g x + f\right )}^{n} c g^{3} n^{2} x^{3} e + 2 \, {\left (g x + f\right )}^{n} c d g^{3} n^{2} x^{2} + {\left (g x + f\right )}^{n} c f g^{2} n^{2} x^{2} e + 3 \, {\left (g x + f\right )}^{n} c g^{3} n x^{3} e + 2 \, {\left (g x + f\right )}^{n} c d f g^{2} n^{2} x + 8 \, {\left (g x + f\right )}^{n} c d g^{3} n x^{2} + {\left (g x + f\right )}^{n} c f g^{2} n x^{2} e + 2 \, {\left (g x + f\right )}^{n} c g^{3} x^{3} e + 6 \, {\left (g x + f\right )}^{n} c d f g^{2} n x + {\left (g x + f\right )}^{n} a g^{3} n^{2} x + 6 \, {\left (g x + f\right )}^{n} c d g^{3} x^{2} - 2 \, {\left (g x + f\right )}^{n} c f^{2} g n x e - 2 \, {\left (g x + f\right )}^{n} c d f^{2} g n + {\left (g x + f\right )}^{n} a f g^{2} n^{2} + 5 \, {\left (g x + f\right )}^{n} a g^{3} n x - 6 \, {\left (g x + f\right )}^{n} c d f^{2} g + 5 \, {\left (g x + f\right )}^{n} a f g^{2} n + 6 \, {\left (g x + f\right )}^{n} a g^{3} x + 2 \, {\left (g x + f\right )}^{n} c f^{3} e + 6 \, {\left (g x + f\right )}^{n} a f g^{2}}{g^{3} n^{3} + 6 \, g^{3} n^{2} + 11 \, g^{3} n + 6 \, g^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.07, size = 211, normalized size = 2.51 \begin {gather*} {\left (f+g\,x\right )}^n\,\left (\frac {f\,\left (2\,c\,e\,f^2-2\,c\,d\,f\,g\,n-6\,c\,d\,f\,g+a\,g^2\,n^2+5\,a\,g^2\,n+6\,a\,g^2\right )}{g^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {x\,\left (-2\,c\,e\,f^2\,g\,n+2\,c\,d\,f\,g^2\,n^2+6\,c\,d\,f\,g^2\,n+a\,g^3\,n^2+5\,a\,g^3\,n+6\,a\,g^3\right )}{g^3\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {c\,e\,x^3\,\left (n^2+3\,n+2\right )}{n^3+6\,n^2+11\,n+6}+\frac {c\,x^2\,\left (n+1\right )\,\left (6\,d\,g+2\,d\,g\,n+e\,f\,n\right )}{g\,\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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