Optimal. Leaf size=58 \[ \frac{b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac{(a+b x)^5}{6 (d+e x)^6 (b d-a e)} \]
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Rubi [A] time = 0.0142256, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {27, 45, 37} \[ \frac{b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac{(a+b x)^5}{6 (d+e x)^6 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^7} \, dx\\ &=\frac{(a+b x)^5}{6 (b d-a e) (d+e x)^6}+\frac{b \int \frac{(a+b x)^4}{(d+e x)^6} \, dx}{6 (b d-a e)}\\ &=\frac{(a+b x)^5}{6 (b d-a e) (d+e x)^6}+\frac{b (a+b x)^5}{30 (b d-a e)^2 (d+e x)^5}\\ \end{align*}
Mathematica [B] time = 0.0451627, size = 144, normalized size = 2.48 \[ -\frac{3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+4 a^3 b e^3 (d+6 e x)+5 a^4 e^4+2 a b^3 e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 186, normalized size = 3.2 \begin{align*} -{\frac{4\,{b}^{3} \left ( ae-bd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{4\,b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18201, size = 319, normalized size = 5.5 \begin{align*} -\frac{15 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \,{\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \,{\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63437, size = 482, normalized size = 8.31 \begin{align*} -\frac{15 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \,{\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \,{\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 11.0187, size = 252, normalized size = 4.34 \begin{align*} - \frac{5 a^{4} e^{4} + 4 a^{3} b d e^{3} + 3 a^{2} b^{2} d^{2} e^{2} + 2 a b^{3} d^{3} e + b^{4} d^{4} + 15 b^{4} e^{4} x^{4} + x^{3} \left (40 a b^{3} e^{4} + 20 b^{4} d e^{3}\right ) + x^{2} \left (45 a^{2} b^{2} e^{4} + 30 a b^{3} d e^{3} + 15 b^{4} d^{2} e^{2}\right ) + x \left (24 a^{3} b e^{4} + 18 a^{2} b^{2} d e^{3} + 12 a b^{3} d^{2} e^{2} + 6 b^{4} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21412, size = 235, normalized size = 4.05 \begin{align*} -\frac{{\left (15 \, b^{4} x^{4} e^{4} + 20 \, b^{4} d x^{3} e^{3} + 15 \, b^{4} d^{2} x^{2} e^{2} + 6 \, b^{4} d^{3} x e + b^{4} d^{4} + 40 \, a b^{3} x^{3} e^{4} + 30 \, a b^{3} d x^{2} e^{3} + 12 \, a b^{3} d^{2} x e^{2} + 2 \, a b^{3} d^{3} e + 45 \, a^{2} b^{2} x^{2} e^{4} + 18 \, a^{2} b^{2} d x e^{3} + 3 \, a^{2} b^{2} d^{2} e^{2} + 24 \, a^{3} b x e^{4} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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