Optimal. Leaf size=155 \[ -\frac {(a+b x)^5 (B d-A e)}{5 e (d+e x)^5 (b d-a e)}+\frac {4 b^3 B (b d-a e)}{e^6 (d+e x)}-\frac {3 b^2 B (b d-a e)^2}{e^6 (d+e x)^2}+\frac {4 b B (b d-a e)^3}{3 e^6 (d+e x)^3}-\frac {B (b d-a e)^4}{4 e^6 (d+e x)^4}+\frac {b^4 B \log (d+e x)}{e^6} \]
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Rubi [A] time = 0.15, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {27, 78, 43} \begin {gather*} -\frac {(a+b x)^5 (B d-A e)}{5 e (d+e x)^5 (b d-a e)}+\frac {4 b^3 B (b d-a e)}{e^6 (d+e x)}-\frac {3 b^2 B (b d-a e)^2}{e^6 (d+e x)^2}+\frac {4 b B (b d-a e)^3}{3 e^6 (d+e x)^3}-\frac {B (b d-a e)^4}{4 e^6 (d+e x)^4}+\frac {b^4 B \log (d+e x)}{e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 78
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)^6} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{(d+e x)^6} \, dx\\ &=-\frac {(B d-A e) (a+b x)^5}{5 e (b d-a e) (d+e x)^5}+\frac {B \int \frac {(a+b x)^4}{(d+e x)^5} \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^5}{5 e (b d-a e) (d+e x)^5}+\frac {B \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^5}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^4}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^3}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^2}+\frac {b^4}{e^4 (d+e x)}\right ) \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^5}{5 e (b d-a e) (d+e x)^5}-\frac {B (b d-a e)^4}{4 e^6 (d+e x)^4}+\frac {4 b B (b d-a e)^3}{3 e^6 (d+e x)^3}-\frac {3 b^2 B (b d-a e)^2}{e^6 (d+e x)^2}+\frac {4 b^3 B (b d-a e)}{e^6 (d+e x)}+\frac {b^4 B \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [B] time = 0.15, size = 332, normalized size = 2.14 \begin {gather*} \frac {-3 a^4 e^4 (4 A e+B (d+5 e x))-4 a^3 b e^3 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )-6 a^2 b^2 e^2 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-12 a b^3 e \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+b^4 \left (B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+60 b^4 B (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \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)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 526, normalized size = 3.39 \begin {gather*} \frac {137 \, B b^{4} d^{5} - 12 \, A a^{4} e^{5} - 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 6 \, {\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} - 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 60 \, {\left (5 \, B b^{4} d e^{4} - {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 60 \, {\left (15 \, 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} + 20 \, {\left (55 \, B b^{4} d^{3} e^{2} - 6 \, {\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} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 5 \, {\left (125 \, B b^{4} d^{4} e - 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} - 6 \, {\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} - 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x + 60 \, {\left (B b^{4} e^{5} x^{5} + 5 \, B b^{4} d e^{4} x^{4} + 10 \, B b^{4} d^{2} e^{3} x^{3} + 10 \, B b^{4} d^{3} e^{2} x^{2} + 5 \, B b^{4} d^{4} e x + B b^{4} d^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} 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 = 2.68 \begin {gather*} B b^{4} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (60 \, {\left (5 \, B b^{4} d e^{3} - 4 \, B a b^{3} e^{4} - A b^{4} e^{4}\right )} x^{4} + 60 \, {\left (15 \, B b^{4} d^{2} e^{2} - 8 \, B a b^{3} d e^{3} - 2 \, A b^{4} d e^{3} - 3 \, B a^{2} b^{2} e^{4} - 2 \, A a b^{3} e^{4}\right )} x^{3} + 20 \, {\left (55 \, B b^{4} d^{3} e - 24 \, B a b^{3} d^{2} e^{2} - 6 \, A b^{4} d^{2} e^{2} - 9 \, B a^{2} b^{2} d e^{3} - 6 \, A a b^{3} d e^{3} - 4 \, B a^{3} b e^{4} - 6 \, A a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (125 \, B b^{4} d^{4} - 48 \, B a b^{3} d^{3} e - 12 \, A b^{4} d^{3} e - 18 \, B a^{2} b^{2} d^{2} e^{2} - 12 \, A a b^{3} d^{2} e^{2} - 8 \, B a^{3} b d e^{3} - 12 \, A a^{2} b^{2} d e^{3} - 3 \, B a^{4} e^{4} - 12 \, A a^{3} b e^{4}\right )} x + {\left (137 \, B b^{4} d^{5} - 48 \, B a b^{3} d^{4} e - 12 \, A b^{4} d^{4} e - 18 \, B a^{2} b^{2} d^{3} e^{2} - 12 \, 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} - 3 \, B a^{4} d e^{4} - 12 \, A a^{3} b d e^{4} - 12 \, A a^{4} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 651, normalized size = 4.20 \begin {gather*} -\frac {A \,a^{4}}{5 \left (e x +d \right )^{5} e}+\frac {4 A \,a^{3} b d}{5 \left (e x +d \right )^{5} e^{2}}-\frac {6 A \,a^{2} b^{2} d^{2}}{5 \left (e x +d \right )^{5} e^{3}}+\frac {4 A a \,b^{3} d^{3}}{5 \left (e x +d \right )^{5} e^{4}}-\frac {A \,b^{4} d^{4}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {B \,a^{4} d}{5 \left (e x +d \right )^{5} e^{2}}-\frac {4 B \,a^{3} b \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}+\frac {6 B \,a^{2} b^{2} d^{3}}{5 \left (e x +d \right )^{5} e^{4}}-\frac {4 B a \,b^{3} d^{4}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {B \,b^{4} d^{5}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {A \,a^{3} b}{\left (e x +d \right )^{4} e^{2}}+\frac {3 A \,a^{2} b^{2} d}{\left (e x +d \right )^{4} e^{3}}-\frac {3 A a \,b^{3} d^{2}}{\left (e x +d \right )^{4} e^{4}}+\frac {A \,b^{4} d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {B \,a^{4}}{4 \left (e x +d \right )^{4} e^{2}}+\frac {2 B \,a^{3} b d}{\left (e x +d \right )^{4} e^{3}}-\frac {9 B \,a^{2} b^{2} d^{2}}{2 \left (e x +d \right )^{4} e^{4}}+\frac {4 B a \,b^{3} d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {5 B \,b^{4} d^{4}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {2 A \,a^{2} b^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {4 A a \,b^{3} d}{\left (e x +d \right )^{3} e^{4}}-\frac {2 A \,b^{4} d^{2}}{\left (e x +d \right )^{3} e^{5}}-\frac {4 B \,a^{3} b}{3 \left (e x +d \right )^{3} e^{3}}+\frac {6 B \,a^{2} b^{2} d}{\left (e x +d \right )^{3} e^{4}}-\frac {8 B a \,b^{3} d^{2}}{\left (e x +d \right )^{3} e^{5}}+\frac {10 B \,b^{4} d^{3}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {2 A a \,b^{3}}{\left (e x +d \right )^{2} e^{4}}+\frac {2 A \,b^{4} d}{\left (e x +d \right )^{2} e^{5}}-\frac {3 B \,a^{2} b^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {8 B a \,b^{3} d}{\left (e x +d \right )^{2} e^{5}}-\frac {5 B \,b^{4} d^{2}}{\left (e x +d \right )^{2} e^{6}}-\frac {A \,b^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 B a \,b^{3}}{\left (e x +d \right ) e^{5}}+\frac {5 B \,b^{4} d}{\left (e x +d \right ) e^{6}}+\frac {B \,b^{4} \ln \left (e x +d \right )}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 459, normalized size = 2.96 \begin {gather*} \frac {137 \, B b^{4} d^{5} - 12 \, A a^{4} e^{5} - 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 6 \, {\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} - 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 60 \, {\left (5 \, B b^{4} d e^{4} - {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 60 \, {\left (15 \, 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} + 20 \, {\left (55 \, B b^{4} d^{3} e^{2} - 6 \, {\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} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 5 \, {\left (125 \, B b^{4} d^{4} e - 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} - 6 \, {\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} - 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac {B b^{4} \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 = 0.20, size = 465, normalized size = 3.00 \begin {gather*} \frac {B\,b^4\,\ln \left (d+e\,x\right )}{e^6}-\frac {\frac {3\,B\,a^4\,d\,e^4+12\,A\,a^4\,e^5+8\,B\,a^3\,b\,d^2\,e^3+12\,A\,a^3\,b\,d\,e^4+18\,B\,a^2\,b^2\,d^3\,e^2+12\,A\,a^2\,b^2\,d^2\,e^3+48\,B\,a\,b^3\,d^4\,e+12\,A\,a\,b^3\,d^3\,e^2-137\,B\,b^4\,d^5+12\,A\,b^4\,d^4\,e}{60\,e^6}+\frac {x\,\left (3\,B\,a^4\,e^4+8\,B\,a^3\,b\,d\,e^3+12\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3+48\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2-125\,B\,b^4\,d^4+12\,A\,b^4\,d^3\,e\right )}{12\,e^5}+\frac {x^2\,\left (4\,B\,a^3\,b\,e^3+9\,B\,a^2\,b^2\,d\,e^2+6\,A\,a^2\,b^2\,e^3+24\,B\,a\,b^3\,d^2\,e+6\,A\,a\,b^3\,d\,e^2-55\,B\,b^4\,d^3+6\,A\,b^4\,d^2\,e\right )}{3\,e^4}+\frac {x^3\,\left (3\,B\,a^2\,b^2\,e^2+8\,B\,a\,b^3\,d\,e+2\,A\,a\,b^3\,e^2-15\,B\,b^4\,d^2+2\,A\,b^4\,d\,e\right )}{e^3}+\frac {b^3\,x^4\,\left (A\,b\,e+4\,B\,a\,e-5\,B\,b\,d\right )}{e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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