Optimal. Leaf size=142 \[ \frac {(b d-a e)^4 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 b (b d-a e)^3 (d+e x)^{2+m}}{e^5 (2+m)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 b^3 (b d-a e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {b^4 (d+e x)^{5+m}}{e^5 (5+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 142, 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 = {27, 45}
\begin {gather*} -\frac {4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}+\frac {(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac {b^4 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^m \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (d+e x)^m}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{1+m}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{2+m}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{3+m}}{e^4}+\frac {b^4 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 b (b d-a e)^3 (d+e x)^{2+m}}{e^5 (2+m)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 b^3 (b d-a e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {b^4 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 121, normalized size = 0.85 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {(b d-a e)^4}{1+m}-\frac {4 b (b d-a e)^3 (d+e x)}{2+m}+\frac {6 b^2 (b d-a e)^2 (d+e x)^2}{3+m}-\frac {4 b^3 (b d-a e) (d+e x)^3}{4+m}+\frac {b^4 (d+e x)^4}{5+m}\right )}{e^5} \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. \(734\) vs.
\(2(142)=284\).
time = 0.76, size = 735, normalized size = 5.18 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 394 vs.
\(2 (146) = 292\).
time = 0.30, size = 394, normalized size = 2.77 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a^{4} e^{\left (-1\right )}}{m + 1} + \frac {4 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a^{3} b e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {6 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} a^{2} b^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {4 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} a b^{3} e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} b^{4} e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \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. 771 vs.
\(2 (146) = 292\).
time = 2.94, size = 771, normalized size = 5.43 \begin {gather*} \frac {{\left (24 \, b^{4} d^{5} + {\left ({\left (b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}\right )} x^{5} + 4 \, {\left (a b^{3} m^{4} + 11 \, a b^{3} m^{3} + 41 \, a b^{3} m^{2} + 61 \, a b^{3} m + 30 \, a b^{3}\right )} x^{4} + 6 \, {\left (a^{2} b^{2} m^{4} + 12 \, a^{2} b^{2} m^{3} + 49 \, a^{2} b^{2} m^{2} + 78 \, a^{2} b^{2} m + 40 \, a^{2} b^{2}\right )} x^{3} + 4 \, {\left (a^{3} b m^{4} + 13 \, a^{3} b m^{3} + 59 \, a^{3} b m^{2} + 107 \, a^{3} b m + 60 \, a^{3} b\right )} x^{2} + {\left (a^{4} m^{4} + 14 \, a^{4} m^{3} + 71 \, a^{4} m^{2} + 154 \, a^{4} m + 120 \, a^{4}\right )} x\right )} e^{5} + {\left (a^{4} d m^{4} + 14 \, a^{4} d m^{3} + 71 \, a^{4} d m^{2} + 154 \, a^{4} d m + 120 \, a^{4} d + {\left (b^{4} d m^{4} + 6 \, b^{4} d m^{3} + 11 \, b^{4} d m^{2} + 6 \, b^{4} d m\right )} x^{4} + 4 \, {\left (a b^{3} d m^{4} + 8 \, a b^{3} d m^{3} + 17 \, a b^{3} d m^{2} + 10 \, a b^{3} d m\right )} x^{3} + 6 \, {\left (a^{2} b^{2} d m^{4} + 10 \, a^{2} b^{2} d m^{3} + 29 \, a^{2} b^{2} d m^{2} + 20 \, a^{2} b^{2} d m\right )} x^{2} + 4 \, {\left (a^{3} b d m^{4} + 12 \, a^{3} b d m^{3} + 47 \, a^{3} b d m^{2} + 60 \, a^{3} b d m\right )} x\right )} e^{4} - 4 \, {\left (a^{3} b d^{2} m^{3} + 12 \, a^{3} b d^{2} m^{2} + 47 \, a^{3} b d^{2} m + 60 \, a^{3} b d^{2} + {\left (b^{4} d^{2} m^{3} + 3 \, b^{4} d^{2} m^{2} + 2 \, b^{4} d^{2} m\right )} x^{3} + 3 \, {\left (a b^{3} d^{2} m^{3} + 6 \, a b^{3} d^{2} m^{2} + 5 \, a b^{3} d^{2} m\right )} x^{2} + 3 \, {\left (a^{2} b^{2} d^{2} m^{3} + 9 \, a^{2} b^{2} d^{2} m^{2} + 20 \, a^{2} b^{2} d^{2} m\right )} x\right )} e^{3} + 12 \, {\left (a^{2} b^{2} d^{3} m^{2} + 9 \, a^{2} b^{2} d^{3} m + 20 \, a^{2} b^{2} d^{3} + {\left (b^{4} d^{3} m^{2} + b^{4} d^{3} m\right )} x^{2} + 2 \, {\left (a b^{3} d^{3} m^{2} + 5 \, a b^{3} d^{3} m\right )} x\right )} e^{2} - 24 \, {\left (b^{4} d^{4} m x + a b^{3} d^{4} m + 5 \, a b^{3} d^{4}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \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. 8719 vs.
\(2 (124) = 248\).
time = 1.99, size = 8719, normalized size = 61.40 \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. 1529 vs.
\(2 (146) = 292\).
time = 0.95, size = 1529, normalized size = 10.77 \begin {gather*} \frac {{\left (x e + d\right )}^{m} b^{4} m^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} b^{4} d m^{4} x^{4} e^{4} + 4 \, {\left (x e + d\right )}^{m} a b^{3} m^{4} x^{4} e^{5} + 10 \, {\left (x e + d\right )}^{m} b^{4} m^{3} x^{5} e^{5} + 4 \, {\left (x e + d\right )}^{m} a b^{3} d m^{4} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} b^{4} d m^{3} x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} b^{4} d^{2} m^{3} x^{3} e^{3} + 6 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m^{4} x^{3} e^{5} + 44 \, {\left (x e + d\right )}^{m} a b^{3} m^{3} x^{4} e^{5} + 35 \, {\left (x e + d\right )}^{m} b^{4} m^{2} x^{5} e^{5} + 6 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m^{4} x^{2} e^{4} + 32 \, {\left (x e + d\right )}^{m} a b^{3} d m^{3} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} b^{4} d m^{2} x^{4} e^{4} - 12 \, {\left (x e + d\right )}^{m} a b^{3} d^{2} m^{3} x^{2} e^{3} - 12 \, {\left (x e + d\right )}^{m} b^{4} d^{2} m^{2} x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} b^{4} d^{3} m^{2} x^{2} e^{2} + 4 \, {\left (x e + d\right )}^{m} a^{3} b m^{4} x^{2} e^{5} + 72 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m^{3} x^{3} e^{5} + 164 \, {\left (x e + d\right )}^{m} a b^{3} m^{2} x^{4} e^{5} + 50 \, {\left (x e + d\right )}^{m} b^{4} m x^{5} e^{5} + 4 \, {\left (x e + d\right )}^{m} a^{3} b d m^{4} x e^{4} + 60 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m^{3} x^{2} e^{4} + 68 \, {\left (x e + d\right )}^{m} a b^{3} d m^{2} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} b^{4} d m x^{4} e^{4} - 12 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{2} m^{3} x e^{3} - 72 \, {\left (x e + d\right )}^{m} a b^{3} d^{2} m^{2} x^{2} e^{3} - 8 \, {\left (x e + d\right )}^{m} b^{4} d^{2} m x^{3} e^{3} + 24 \, {\left (x e + d\right )}^{m} a b^{3} d^{3} m^{2} x e^{2} + 12 \, {\left (x e + d\right )}^{m} b^{4} d^{3} m x^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} b^{4} d^{4} m x e + {\left (x e + d\right )}^{m} a^{4} m^{4} x e^{5} + 52 \, {\left (x e + d\right )}^{m} a^{3} b m^{3} x^{2} e^{5} + 294 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m^{2} x^{3} e^{5} + 244 \, {\left (x e + d\right )}^{m} a b^{3} m x^{4} e^{5} + 24 \, {\left (x e + d\right )}^{m} b^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} a^{4} d m^{4} e^{4} + 48 \, {\left (x e + d\right )}^{m} a^{3} b d m^{3} x e^{4} + 174 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m^{2} x^{2} e^{4} + 40 \, {\left (x e + d\right )}^{m} a b^{3} d m x^{3} e^{4} - 4 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} m^{3} e^{3} - 108 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{2} m^{2} x e^{3} - 60 \, {\left (x e + d\right )}^{m} a b^{3} d^{2} m x^{2} e^{3} + 12 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{3} m^{2} e^{2} + 120 \, {\left (x e + d\right )}^{m} a b^{3} d^{3} m x e^{2} - 24 \, {\left (x e + d\right )}^{m} a b^{3} d^{4} m e + 24 \, {\left (x e + d\right )}^{m} b^{4} d^{5} + 14 \, {\left (x e + d\right )}^{m} a^{4} m^{3} x e^{5} + 236 \, {\left (x e + d\right )}^{m} a^{3} b m^{2} x^{2} e^{5} + 468 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m x^{3} e^{5} + 120 \, {\left (x e + d\right )}^{m} a b^{3} x^{4} e^{5} + 14 \, {\left (x e + d\right )}^{m} a^{4} d m^{3} e^{4} + 188 \, {\left (x e + d\right )}^{m} a^{3} b d m^{2} x e^{4} + 120 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m x^{2} e^{4} - 48 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} m^{2} e^{3} - 240 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{2} m x e^{3} + 108 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{3} m e^{2} - 120 \, {\left (x e + d\right )}^{m} a b^{3} d^{4} e + 71 \, {\left (x e + d\right )}^{m} a^{4} m^{2} x e^{5} + 428 \, {\left (x e + d\right )}^{m} a^{3} b m x^{2} e^{5} + 240 \, {\left (x e + d\right )}^{m} a^{2} b^{2} x^{3} e^{5} + 71 \, {\left (x e + d\right )}^{m} a^{4} d m^{2} e^{4} + 240 \, {\left (x e + d\right )}^{m} a^{3} b d m x e^{4} - 188 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} m e^{3} + 240 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{3} e^{2} + 154 \, {\left (x e + d\right )}^{m} a^{4} m x e^{5} + 240 \, {\left (x e + d\right )}^{m} a^{3} b x^{2} e^{5} + 154 \, {\left (x e + d\right )}^{m} a^{4} d m e^{4} - 240 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} e^{3} + 120 \, {\left (x e + d\right )}^{m} a^{4} x e^{5} + 120 \, {\left (x e + d\right )}^{m} a^{4} d e^{4}}{m^{5} e^{5} + 15 \, m^{4} e^{5} + 85 \, m^{3} e^{5} + 225 \, m^{2} e^{5} + 274 \, m e^{5} + 120 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 831, normalized size = 5.85 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (a^4\,d\,e^4\,m^4+14\,a^4\,d\,e^4\,m^3+71\,a^4\,d\,e^4\,m^2+154\,a^4\,d\,e^4\,m+120\,a^4\,d\,e^4-4\,a^3\,b\,d^2\,e^3\,m^3-48\,a^3\,b\,d^2\,e^3\,m^2-188\,a^3\,b\,d^2\,e^3\,m-240\,a^3\,b\,d^2\,e^3+12\,a^2\,b^2\,d^3\,e^2\,m^2+108\,a^2\,b^2\,d^3\,e^2\,m+240\,a^2\,b^2\,d^3\,e^2-24\,a\,b^3\,d^4\,e\,m-120\,a\,b^3\,d^4\,e+24\,b^4\,d^5\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^4\,e^5\,m^4+14\,a^4\,e^5\,m^3+71\,a^4\,e^5\,m^2+154\,a^4\,e^5\,m+120\,a^4\,e^5+4\,a^3\,b\,d\,e^4\,m^4+48\,a^3\,b\,d\,e^4\,m^3+188\,a^3\,b\,d\,e^4\,m^2+240\,a^3\,b\,d\,e^4\,m-12\,a^2\,b^2\,d^2\,e^3\,m^3-108\,a^2\,b^2\,d^2\,e^3\,m^2-240\,a^2\,b^2\,d^2\,e^3\,m+24\,a\,b^3\,d^3\,e^2\,m^2+120\,a\,b^3\,d^3\,e^2\,m-24\,b^4\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b^4\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,b^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (3\,a^2\,e^2\,m^2+27\,a^2\,e^2\,m+60\,a^2\,e^2+2\,a\,b\,d\,e\,m^2+10\,a\,b\,d\,e\,m-2\,b^2\,d^2\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b^3\,x^4\,{\left (d+e\,x\right )}^m\,\left (20\,a\,e+4\,a\,e\,m+b\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,b\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (2\,a^3\,e^3\,m^3+24\,a^3\,e^3\,m^2+94\,a^3\,e^3\,m+120\,a^3\,e^3+3\,a^2\,b\,d\,e^2\,m^3+27\,a^2\,b\,d\,e^2\,m^2+60\,a^2\,b\,d\,e^2\,m-6\,a\,b^2\,d^2\,e\,m^2-30\,a\,b^2\,d^2\,e\,m+6\,b^3\,d^3\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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