Optimal. Leaf size=189 \[ -\frac {b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac {2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac {2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac {(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac {b^4 B x^2}{2 e^4} \]
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Rubi [A] time = 0.22, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 77} \begin {gather*} -\frac {b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac {2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac {2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac {(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac {b^4 B x^2}{2 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{(d+e x)^4} \, dx\\ &=\int \left (\frac {b^3 (-4 b B d+A b e+4 a B e)}{e^5}+\frac {b^4 B x}{e^4}+\frac {(-b d+a e)^4 (-B d+A e)}{e^5 (d+e x)^4}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e)}{e^5 (d+e x)^3}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e)}{e^5 (d+e x)^2}-\frac {2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e)}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac {b^3 (4 b B d-A b e-4 a B e) x}{e^5}+\frac {b^4 B x^2}{2 e^4}+\frac {(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}-\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e)}{2 e^6 (d+e x)^2}+\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{e^6 (d+e x)}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 351, normalized size = 1.86 \begin {gather*} \frac {-a^4 e^4 (2 A e+B (d+3 e x))-4 a^3 b e^3 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (B d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )\right )+4 a b^3 e \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )+12 b^2 (d+e x)^3 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)+b^4 \left (2 A e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+B \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )\right )}{6 e^6 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 650, normalized size = 3.44 \begin {gather*} \frac {3 \, B b^{4} e^{5} x^{5} + 47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 3 \, {\left (5 \, B b^{4} d e^{4} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} - 9 \, {\left (7 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4}\right )} x^{3} - 3 \, {\left (3 \, B b^{4} d^{3} e^{2} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} - 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \, {\left (27 \, B b^{4} d^{4} e - 18 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x + 12 \, {\left (5 \, B b^{4} d^{5} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + {\left (5 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 3 \, {\left (5 \, B b^{4} d^{3} e^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4}\right )} x^{2} + 3 \, {\left (5 \, B b^{4} d^{4} e - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 415, normalized size = 2.20 \begin {gather*} 2 \, {\left (5 \, B b^{4} d^{2} - 8 \, B a b^{3} d e - 2 \, A b^{4} d e + 3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B b^{4} x^{2} e^{4} - 8 \, B b^{4} d x e^{3} + 8 \, B a b^{3} x e^{4} + 2 \, A b^{4} x e^{4}\right )} e^{\left (-8\right )} + \frac {{\left (47 \, B b^{4} d^{5} - 104 \, B a b^{3} d^{4} e - 26 \, A b^{4} d^{4} e + 66 \, B a^{2} b^{2} d^{3} e^{2} + 44 \, A a b^{3} d^{3} e^{2} - 8 \, B a^{3} b d^{2} e^{3} - 12 \, A a^{2} b^{2} d^{2} e^{3} - B a^{4} d e^{4} - 4 \, A a^{3} b d e^{4} - 2 \, A a^{4} e^{5} + 12 \, {\left (5 \, B b^{4} d^{3} e^{2} - 12 \, B a b^{3} d^{2} e^{3} - 3 \, A b^{4} d^{2} e^{3} + 9 \, B a^{2} b^{2} d e^{4} + 6 \, A a b^{3} d e^{4} - 2 \, B a^{3} b e^{5} - 3 \, A a^{2} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B b^{4} d^{4} e - 80 \, B a b^{3} d^{3} e^{2} - 20 \, A b^{4} d^{3} e^{2} + 54 \, B a^{2} b^{2} d^{2} e^{3} + 36 \, A a b^{3} d^{2} e^{3} - 8 \, B a^{3} b d e^{4} - 12 \, A a^{2} b^{2} d e^{4} - B a^{4} e^{5} - 4 \, A a^{3} b e^{5}\right )} x\right )} e^{\left (-6\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 626, normalized size = 3.31 \begin {gather*} -\frac {A \,a^{4}}{3 \left (e x +d \right )^{3} e}+\frac {4 A \,a^{3} b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 A \,a^{2} b^{2} d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {4 A a \,b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {A \,b^{4} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,a^{4} d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {4 B \,a^{3} b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 B \,a^{2} b^{2} d^{3}}{\left (e x +d \right )^{3} e^{4}}-\frac {4 B a \,b^{3} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,b^{4} d^{5}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {2 A \,a^{3} b}{\left (e x +d \right )^{2} e^{2}}+\frac {6 A \,a^{2} b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {6 A a \,b^{3} d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {2 A \,b^{4} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {B \,a^{4}}{2 \left (e x +d \right )^{2} e^{2}}+\frac {4 B \,a^{3} b d}{\left (e x +d \right )^{2} e^{3}}-\frac {9 B \,a^{2} b^{2} d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {8 B a \,b^{3} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {5 B \,b^{4} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {B \,b^{4} x^{2}}{2 e^{4}}-\frac {6 A \,a^{2} b^{2}}{\left (e x +d \right ) e^{3}}+\frac {12 A a \,b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {4 A a \,b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {6 A \,b^{4} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,b^{4} d \ln \left (e x +d \right )}{e^{5}}+\frac {A \,b^{4} x}{e^{4}}-\frac {4 B \,a^{3} b}{\left (e x +d \right ) e^{3}}+\frac {18 B \,a^{2} b^{2} d}{\left (e x +d \right ) e^{4}}+\frac {6 B \,a^{2} b^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {24 B a \,b^{3} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {16 B a \,b^{3} d \ln \left (e x +d \right )}{e^{5}}+\frac {4 B a \,b^{3} x}{e^{4}}+\frac {10 B \,b^{4} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {10 B \,b^{4} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,b^{4} d x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 431, normalized size = 2.28 \begin {gather*} \frac {47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 12 \, {\left (5 \, B b^{4} d^{3} e^{2} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B b^{4} d^{4} e - 20 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac {B b^{4} e x^{2} - 2 \, {\left (4 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x}{2 \, e^{5}} + \frac {2 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 451, normalized size = 2.39 \begin {gather*} x\,\left (\frac {A\,b^4+4\,B\,a\,b^3}{e^4}-\frac {4\,B\,b^4\,d}{e^5}\right )-\frac {\frac {B\,a^4\,d\,e^4+2\,A\,a^4\,e^5+8\,B\,a^3\,b\,d^2\,e^3+4\,A\,a^3\,b\,d\,e^4-66\,B\,a^2\,b^2\,d^3\,e^2+12\,A\,a^2\,b^2\,d^2\,e^3+104\,B\,a\,b^3\,d^4\,e-44\,A\,a\,b^3\,d^3\,e^2-47\,B\,b^4\,d^5+26\,A\,b^4\,d^4\,e}{6\,e}+x\,\left (\frac {B\,a^4\,e^4}{2}+4\,B\,a^3\,b\,d\,e^3+2\,A\,a^3\,b\,e^4-27\,B\,a^2\,b^2\,d^2\,e^2+6\,A\,a^2\,b^2\,d\,e^3+40\,B\,a\,b^3\,d^3\,e-18\,A\,a\,b^3\,d^2\,e^2-\frac {35\,B\,b^4\,d^4}{2}+10\,A\,b^4\,d^3\,e\right )+x^2\,\left (4\,B\,a^3\,b\,e^4-18\,B\,a^2\,b^2\,d\,e^3+6\,A\,a^2\,b^2\,e^4+24\,B\,a\,b^3\,d^2\,e^2-12\,A\,a\,b^3\,d\,e^3-10\,B\,b^4\,d^3\,e+6\,A\,b^4\,d^2\,e^2\right )}{d^3\,e^5+3\,d^2\,e^6\,x+3\,d\,e^7\,x^2+e^8\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (6\,B\,a^2\,b^2\,e^2-16\,B\,a\,b^3\,d\,e+4\,A\,a\,b^3\,e^2+10\,B\,b^4\,d^2-4\,A\,b^4\,d\,e\right )}{e^6}+\frac {B\,b^4\,x^2}{2\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 21.58, size = 486, normalized size = 2.57 \begin {gather*} \frac {B b^{4} x^{2}}{2 e^{4}} + \frac {2 b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right ) \log {\left (d + e x \right )}}{e^{6}} + x \left (\frac {A b^{4}}{e^{4}} + \frac {4 B a b^{3}}{e^{4}} - \frac {4 B b^{4} d}{e^{5}}\right ) + \frac {- 2 A a^{4} e^{5} - 4 A a^{3} b d e^{4} - 12 A a^{2} b^{2} d^{2} e^{3} + 44 A a b^{3} d^{3} e^{2} - 26 A b^{4} d^{4} e - B a^{4} d e^{4} - 8 B a^{3} b d^{2} e^{3} + 66 B a^{2} b^{2} d^{3} e^{2} - 104 B a b^{3} d^{4} e + 47 B b^{4} d^{5} + x^{2} \left (- 36 A a^{2} b^{2} e^{5} + 72 A a b^{3} d e^{4} - 36 A b^{4} d^{2} e^{3} - 24 B a^{3} b e^{5} + 108 B a^{2} b^{2} d e^{4} - 144 B a b^{3} d^{2} e^{3} + 60 B b^{4} d^{3} e^{2}\right ) + x \left (- 12 A a^{3} b e^{5} - 36 A a^{2} b^{2} d e^{4} + 108 A a b^{3} d^{2} e^{3} - 60 A b^{4} d^{3} e^{2} - 3 B a^{4} e^{5} - 24 B a^{3} b d e^{4} + 162 B a^{2} b^{2} d^{2} e^{3} - 240 B a b^{3} d^{3} e^{2} + 105 B b^{4} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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