Optimal. Leaf size=123 \[ -\frac{8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac{8 b \sqrt{d+e x} (b d-a e)^3}{e^5}-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5} \]
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Rubi [A] time = 0.0406202, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {27, 43} \[ -\frac{8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac{8 b \sqrt{d+e x} (b d-a e)^3}{e^5}-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{3/2}}-\frac{4 b (b d-a e)^3}{e^4 \sqrt{d+e x}}+\frac{6 b^2 (b d-a e)^2 \sqrt{d+e x}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{3/2}}{e^4}+\frac{b^4 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}-\frac{8 b (b d-a e)^3 \sqrt{d+e x}}{e^5}+\frac{4 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{5/2}}{5 e^5}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5}\\ \end{align*}
Mathematica [A] time = 0.069786, size = 101, normalized size = 0.82 \[ \frac{2 \left (70 b^2 (d+e x)^2 (b d-a e)^2-28 b^3 (d+e x)^3 (b d-a e)-140 b (d+e x) (b d-a e)^3-35 (b d-a e)^4+5 b^4 (d+e x)^4\right )}{35 e^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 186, normalized size = 1.5 \begin{align*} -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-56\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+112\,{x}^{2}a{b}^{3}d{e}^{3}-32\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-280\,x{a}^{3}b{e}^{4}+560\,x{a}^{2}{b}^{2}d{e}^{3}-448\,xa{b}^{3}{d}^{2}{e}^{2}+128\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}-560\,{a}^{3}bd{e}^{3}+1120\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-896\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{35\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04781, size = 255, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{4} - 28 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 70 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 140 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} \sqrt{e x + d}}{e^{4}} - \frac{35 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{35 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50403, size = 412, normalized size = 3.35 \begin{align*} \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \,{\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 40.107, size = 168, normalized size = 1.37 \begin{align*} \frac{2 b^{4} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (8 a b^{3} e - 8 b^{4} d\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (12 a^{2} b^{2} e^{2} - 24 a b^{3} d e + 12 b^{4} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (8 a^{3} b e^{3} - 24 a^{2} b^{2} d e^{2} + 24 a b^{3} d^{2} e - 8 b^{4} d^{3}\right )}{e^{5}} - \frac{2 \left (a e - b d\right )^{4}}{e^{5} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18195, size = 320, normalized size = 2.6 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} e^{30} - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d e^{30} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{2} e^{30} - 140 \, \sqrt{x e + d} b^{4} d^{3} e^{30} + 28 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} e^{31} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d e^{31} + 420 \, \sqrt{x e + d} a b^{3} d^{2} e^{31} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} e^{32} - 420 \, \sqrt{x e + d} a^{2} b^{2} d e^{32} + 140 \, \sqrt{x e + d} a^{3} b e^{33}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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