Optimal. Leaf size=181 \[ \frac {(b d-a e)^3 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (1+2 p)}+\frac {3 e (b d-a e)^2 (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (1+p)}+\frac {3 e^2 (b d-a e) (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (3+2 p)}+\frac {e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (2+p)} \]
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Rubi [A]
time = 0.06, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {660, 45}
\begin {gather*} \frac {3 e^2 (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+3)}+\frac {3 e (a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+1)}+\frac {(a+b x) (b d-a e)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (2 p+1)}+\frac {e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^3 \, dx\\ &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (\frac {(b d-a e)^3 \left (a b+b^2 x\right )^{2 p}}{b^3}+\frac {3 e (b d-a e)^2 \left (a b+b^2 x\right )^{1+2 p}}{b^4}+\frac {3 e^2 (b d-a e) \left (a b+b^2 x\right )^{2+2 p}}{b^5}+\frac {e^3 \left (a b+b^2 x\right )^{3+2 p}}{b^6}\right ) \, dx\\ &=\frac {(b d-a e)^3 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (1+2 p)}+\frac {3 e (b d-a e)^2 (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (1+p)}+\frac {3 e^2 (b d-a e) (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^4 (3+2 p)}+\frac {e^3 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^4 (2+p)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 107, normalized size = 0.59 \begin {gather*} \frac {(a+b x) \left ((a+b x)^2\right )^p \left (\frac {2 (b d-a e)^3}{1+2 p}+\frac {3 e (b d-a e)^2 (a+b x)}{1+p}+\frac {6 e^2 (b d-a e) (a+b x)^2}{3+2 p}+\frac {e^3 (a+b x)^3}{2+p}\right )}{2 b^4} \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. \(404\) vs.
\(2(177)=354\).
time = 0.69, size = 405, normalized size = 2.24
method | result | size |
gosper | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} \left (-4 b^{3} e^{3} p^{3} x^{3}-12 b^{3} d \,e^{2} p^{3} x^{2}-12 b^{3} e^{3} p^{2} x^{3}+6 a \,b^{2} e^{3} p^{2} x^{2}-12 b^{3} d^{2} e \,p^{3} x -42 b^{3} d \,e^{2} p^{2} x^{2}-11 b^{3} e^{3} p \,x^{3}+12 a \,b^{2} d \,e^{2} p^{2} x +9 a \,b^{2} e^{3} p \,x^{2}-4 b^{3} d^{3} p^{3}-48 b^{3} d^{2} e \,p^{2} x -42 b^{3} d \,e^{2} p \,x^{2}-3 b^{3} x^{3} e^{3}-6 a^{2} b \,e^{3} p x +6 a \,b^{2} d^{2} e \,p^{2}+30 a \,b^{2} d \,e^{2} p x +3 a \,b^{2} e^{3} x^{2}-18 b^{3} d^{3} p^{2}-57 b^{3} d^{2} e p x -12 b^{3} d \,e^{2} x^{2}-6 a^{2} b d \,e^{2} p -3 a^{2} b \,e^{3} x +21 a \,b^{2} d^{2} e p +12 a \,b^{2} d \,e^{2} x -26 b^{3} d^{3} p -18 b^{3} d^{2} e x +3 e^{3} a^{3}-12 a^{2} b d \,e^{2}+18 a \,b^{2} d^{2} e -12 b^{3} d^{3}\right ) \left (b x +a \right )}{2 b^{4} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}\) | \(405\) |
norman | \(\frac {\left (6 a \,b^{2} d^{2} e \,p^{3}+2 b^{3} d^{3} p^{3}-6 a^{2} b d \,e^{2} p^{2}+21 a \,b^{2} d^{2} e \,p^{2}+9 b^{3} d^{3} p^{2}+3 a^{3} e^{3} p -12 a^{2} b d \,e^{2} p +18 a \,b^{2} d^{2} e p +13 b^{3} d^{3} p +6 b^{3} d^{3}\right ) x \,{\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b^{3} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}+\frac {e^{2} \left (a e p +3 b d p +6 b d \right ) x^{3} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b \left (2 p^{2}+7 p +6\right )}+\frac {e^{3} x^{4} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{4+2 p}-\frac {a \left (-4 b^{3} d^{3} p^{3}+6 a \,b^{2} d^{2} e \,p^{2}-18 b^{3} d^{3} p^{2}-6 a^{2} b d \,e^{2} p +21 a \,b^{2} d^{2} e p -26 b^{3} d^{3} p +3 e^{3} a^{3}-12 a^{2} b d \,e^{2}+18 a \,b^{2} d^{2} e -12 b^{3} d^{3}\right ) {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{4} \left (4 p^{4}+20 p^{3}+35 p^{2}+25 p +6\right )}-\frac {3 \left (-2 a b d e \,p^{2}-2 b^{2} d^{2} p^{2}+a^{2} e^{2} p -4 a b d e p -7 b^{2} d^{2} p -6 b^{2} d^{2}\right ) e \,x^{2} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{2 b^{2} \left (2 p^{3}+9 p^{2}+13 p +6\right )}\) | \(498\) |
risch | \(-\frac {\left (-4 b^{4} e^{3} p^{3} x^{4}-4 a \,b^{3} e^{3} p^{3} x^{3}-12 b^{4} d \,e^{2} p^{3} x^{3}-12 b^{4} e^{3} p^{2} x^{4}-12 a \,b^{3} d \,e^{2} p^{3} x^{2}-6 a \,b^{3} e^{3} p^{2} x^{3}-12 b^{4} d^{2} e \,p^{3} x^{2}-42 b^{4} d \,e^{2} p^{2} x^{3}-11 b^{4} e^{3} p \,x^{4}+6 a^{2} b^{2} e^{3} p^{2} x^{2}-12 a \,b^{3} d^{2} e \,p^{3} x -30 a \,b^{3} d \,e^{2} p^{2} x^{2}-2 a \,b^{3} e^{3} p \,x^{3}-4 b^{4} d^{3} p^{3} x -48 b^{4} d^{2} e \,p^{2} x^{2}-42 b^{4} d \,e^{2} p \,x^{3}-3 e^{3} x^{4} b^{4}+12 a^{2} b^{2} d \,e^{2} p^{2} x +3 a^{2} b^{2} e^{3} p \,x^{2}-4 a \,b^{3} d^{3} p^{3}-42 a \,b^{3} d^{2} e \,p^{2} x -12 a \,b^{3} d \,e^{2} p \,x^{2}-18 b^{4} d^{3} p^{2} x -57 b^{4} d^{2} e p \,x^{2}-12 b^{4} d \,e^{2} x^{3}-6 a^{3} b \,e^{3} p x +6 a^{2} b^{2} d^{2} e \,p^{2}+24 a^{2} b^{2} d \,e^{2} p x -18 a \,b^{3} d^{3} p^{2}-36 a \,b^{3} d^{2} e p x -26 b^{4} d^{3} p x -18 b^{4} d^{2} e \,x^{2}-6 a^{3} b d \,e^{2} p +21 a^{2} b^{2} d^{2} e p -26 a \,b^{3} d^{3} p -12 b^{4} d^{3} x +3 e^{3} a^{4}-12 d \,e^{2} a^{3} b +18 d^{2} e \,a^{2} b^{2}-12 a \,b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{p}}{2 \left (3+2 p \right ) \left (2+p \right ) \left (1+p \right ) \left (1+2 p \right ) b^{4}}\) | \(558\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 278, normalized size = 1.54 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} d^{3}}{b {\left (2 \, p + 1\right )}} + \frac {3 \, {\left (b^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )} d^{2} e^{\left (2 \, p \log \left (b x + a\right ) + 1\right )}}{2 \, {\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac {3 \, {\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} + {\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )} d e^{\left (2 \, p \log \left (b x + a\right ) + 2\right )}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac {{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{4} + 2 \, {\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{3} - 3 \, {\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b p x - 3 \, a^{4}\right )} e^{\left (2 \, p \log \left (b x + a\right ) + 3\right )}}{2 \, {\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{4}} \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. 487 vs.
\(2 (179) = 358\).
time = 2.26, size = 487, normalized size = 2.69 \begin {gather*} \frac {{\left (4 \, a b^{3} d^{3} p^{3} + 18 \, a b^{3} d^{3} p^{2} + 26 \, a b^{3} d^{3} p + 12 \, a b^{3} d^{3} + 2 \, {\left (2 \, b^{4} d^{3} p^{3} + 9 \, b^{4} d^{3} p^{2} + 13 \, b^{4} d^{3} p + 6 \, b^{4} d^{3}\right )} x + {\left (6 \, a^{3} b p x + {\left (4 \, b^{4} p^{3} + 12 \, b^{4} p^{2} + 11 \, b^{4} p + 3 \, b^{4}\right )} x^{4} - 3 \, a^{4} + 2 \, {\left (2 \, a b^{3} p^{3} + 3 \, a b^{3} p^{2} + a b^{3} p\right )} x^{3} - 3 \, {\left (2 \, a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{2}\right )} e^{3} + 6 \, {\left (a^{3} b d p + 2 \, a^{3} b d + {\left (2 \, b^{4} d p^{3} + 7 \, b^{4} d p^{2} + 7 \, b^{4} d p + 2 \, b^{4} d\right )} x^{3} + {\left (2 \, a b^{3} d p^{3} + 5 \, a b^{3} d p^{2} + 2 \, a b^{3} d p\right )} x^{2} - 2 \, {\left (a^{2} b^{2} d p^{2} + 2 \, a^{2} b^{2} d p\right )} x\right )} e^{2} - 3 \, {\left (2 \, a^{2} b^{2} d^{2} p^{2} + 7 \, a^{2} b^{2} d^{2} p + 6 \, a^{2} b^{2} d^{2} - {\left (4 \, b^{4} d^{2} p^{3} + 16 \, b^{4} d^{2} p^{2} + 19 \, b^{4} d^{2} p + 6 \, b^{4} d^{2}\right )} x^{2} - 2 \, {\left (2 \, a b^{3} d^{2} p^{3} + 7 \, a b^{3} d^{2} p^{2} + 6 \, a b^{3} d^{2} p\right )} x\right )} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, {\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\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*} \text {Too large to display} \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. 1267 vs.
\(2 (179) = 358\).
time = 0.83, size = 1267, normalized size = 7.00 \begin {gather*} \frac {4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} p^{3} x^{4} e^{3} + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d p^{3} x^{3} e^{2} + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{2} p^{3} x^{2} e + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{3} p^{3} x + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} p^{3} x^{3} e^{3} + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} p^{2} x^{4} e^{3} + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d p^{3} x^{2} e^{2} + 42 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d p^{2} x^{3} e^{2} + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{2} p^{3} x e + 48 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{2} p^{2} x^{2} e + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{3} p^{3} + 18 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{3} p^{2} x + 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} p^{2} x^{3} e^{3} + 11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} p x^{4} e^{3} + 30 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d p^{2} x^{2} e^{2} + 42 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d p x^{3} e^{2} + 42 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{2} p^{2} x e + 57 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{2} p x^{2} e + 18 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{3} p^{2} + 26 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{3} p x - 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} p^{2} x^{2} e^{3} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} p x^{3} e^{3} + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} x^{4} e^{3} - 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d p^{2} x e^{2} + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d p x^{2} e^{2} + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d x^{3} e^{2} - 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d^{2} p^{2} e + 36 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{2} p x e + 18 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{2} x^{2} e + 26 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{3} p + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{4} d^{3} x - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} p x^{2} e^{3} - 24 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d p x e^{2} - 21 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d^{2} p e + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{3} d^{3} + 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} b p x e^{3} + 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} b d p e^{2} - 18 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b^{2} d^{2} e + 12 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} b d e^{2} - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{4} e^{3}}{2 \, {\left (4 \, b^{4} p^{4} + 20 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 25 \, b^{4} p + 6 \, b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 484, normalized size = 2.67 \begin {gather*} {\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {a\,\left (-3\,a^3\,e^3+6\,a^2\,b\,d\,e^2\,p+12\,a^2\,b\,d\,e^2-6\,a\,b^2\,d^2\,e\,p^2-21\,a\,b^2\,d^2\,e\,p-18\,a\,b^2\,d^2\,e+4\,b^3\,d^3\,p^3+18\,b^3\,d^3\,p^2+26\,b^3\,d^3\,p+12\,b^3\,d^3\right )}{2\,b^4\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {e^3\,x^4\,\left (4\,p^3+12\,p^2+11\,p+3\right )}{2\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {x\,\left (6\,a^3\,b\,e^3\,p-12\,a^2\,b^2\,d\,e^2\,p^2-24\,a^2\,b^2\,d\,e^2\,p+12\,a\,b^3\,d^2\,e\,p^3+42\,a\,b^3\,d^2\,e\,p^2+36\,a\,b^3\,d^2\,e\,p+4\,b^4\,d^3\,p^3+18\,b^4\,d^3\,p^2+26\,b^4\,d^3\,p+12\,b^4\,d^3\right )}{2\,b^4\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {3\,e\,x^2\,\left (2\,p+1\right )\,\left (-a^2\,e^2\,p+2\,a\,b\,d\,e\,p^2+4\,a\,b\,d\,e\,p+2\,b^2\,d^2\,p^2+7\,b^2\,d^2\,p+6\,b^2\,d^2\right )}{2\,b^2\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}+\frac {e^2\,x^3\,\left (2\,p^2+3\,p+1\right )\,\left (6\,b\,d+a\,e\,p+3\,b\,d\,p\right )}{b\,\left (4\,p^4+20\,p^3+35\,p^2+25\,p+6\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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