Optimal. Leaf size=76 \[ \frac {(b d-a e) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (1+2 p)}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 b^2 (1+p)} \]
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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {654, 623}
\begin {gather*} \frac {(a+b x) (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (2 p+1)}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b^2 (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 654
Rubi steps
\begin {align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 b^2 (1+p)}+\frac {\left (2 b^2 d-2 a b e\right ) \int \left (a^2+2 a b x+b^2 x^2\right )^p \, dx}{2 b^2}\\ &=\frac {(b d-a e) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (1+2 p)}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 b^2 (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 54, normalized size = 0.71 \begin {gather*} \frac {(a+b x) \left ((a+b x)^2\right )^p (-a e+2 b d (1+p)+b e (1+2 p) x)}{2 b^2 (1+p) (1+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 65, normalized size = 0.86
method | result | size |
gosper | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} \left (-2 b e p x -2 b d p -b e x +a e -2 b d \right ) \left (b x +a \right )}{2 b^{2} \left (2 p^{2}+3 p +1\right )}\) | \(65\) |
risch | \(-\frac {\left (-2 b^{2} e p \,x^{2}-2 a b e p x -2 b^{2} d p x -b^{2} e \,x^{2}-2 a b d p -2 b^{2} d x +a^{2} e -2 a b d \right ) \left (\left (b x +a \right )^{2}\right )^{p}}{2 b^{2} \left (1+p \right ) \left (1+2 p \right )}\) | \(85\) |
norman | \(\frac {\left (a e p +b d p +b d \right ) x \,{\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b \left (2 p^{2}+3 p +1\right )}+\frac {e \,x^{2} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 p +2}-\frac {a \left (-2 b d p +a e -2 b d \right ) {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{2} \left (2 p^{2}+3 p +1\right )}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 80, normalized size = 1.05 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} d}{b {\left (2 \, p + 1\right )}} + \frac {{\left (b^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )} e^{\left (2 \, p \log \left (b x + a\right ) + 1\right )}}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.00, size = 96, normalized size = 1.26 \begin {gather*} \frac {{\left (2 \, a b d p + 2 \, a b d + 2 \, {\left (b^{2} d p + b^{2} d\right )} x + {\left (2 \, a b p x + {\left (2 \, b^{2} p + b^{2}\right )} x^{2} - a^{2}\right )} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, {\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \left (d x + \frac {e x^{2}}{2}\right ) \left (a^{2}\right )^{p} & \text {for}\: b = 0 \\\frac {a e \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a e}{a b^{2} + b^{3} x} - \frac {b d}{a b^{2} + b^{3} x} + \frac {b e x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: p = -1 \\\int \frac {d + e x}{\sqrt {\left (a + b x\right )^{2}}}\, dx & \text {for}\: p = - \frac {1}{2} \\- \frac {a^{2} e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} + \frac {2 a b d p \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} + \frac {2 a b d \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} + \frac {2 a b e p x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} + \frac {2 b^{2} d p x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} + \frac {2 b^{2} d x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} + \frac {2 b^{2} e p x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} + \frac {b^{2} e x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{4 b^{2} p^{2} + 6 b^{2} p + 2 b^{2}} & \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. 228 vs.
\(2 (76) = 152\).
time = 0.95, size = 228, normalized size = 3.00 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} p x^{2} e + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} d p x + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b p x e + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} x^{2} e + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b d p + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} d x + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b d - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} e}{2 \, {\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 112, normalized size = 1.47 \begin {gather*} \left (\frac {x\,\left (2\,b^2\,d+2\,b^2\,d\,p+2\,a\,b\,e\,p\right )}{2\,b^2\,\left (2\,p^2+3\,p+1\right )}+\frac {a\,\left (2\,b\,d-a\,e+2\,b\,d\,p\right )}{2\,b^2\,\left (2\,p^2+3\,p+1\right )}+\frac {e\,x^2\,\left (2\,p+1\right )}{2\,\left (2\,p^2+3\,p+1\right )}\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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