Optimal. Leaf size=200 \[ \frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^4 (a+b x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^4 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{2 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3}{5 e^4 (a+b x)} \]
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Rubi [A] time = 0.159973, antiderivative size = 200, 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 = {646, 43} \[ \frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^4 (a+b x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^4 (a+b x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{2 e^4 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3}{5 e^4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (d+e x)^4 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3 (d+e x)^4}{e^3}+\frac{3 b^4 (b d-a e)^2 (d+e x)^5}{e^3}-\frac{3 b^5 (b d-a e) (d+e x)^6}{e^3}+\frac{b^6 (d+e x)^7}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{(b d-a e)^3 (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}+\frac{b (b d-a e)^2 (d+e x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x)}-\frac{3 b^2 (b d-a e) (d+e x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}+\frac{b^3 (d+e x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.076116, size = 215, normalized size = 1.08 \[ \frac{x \sqrt{(a+b x)^2} \left (28 a^2 b x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+56 a^3 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+8 a b^2 x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+b^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )\right )}{280 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 264, normalized size = 1.3 \begin{align*}{\frac{x \left ( 35\,{b}^{3}{e}^{4}{x}^{7}+120\,{x}^{6}{b}^{2}a{e}^{4}+160\,{x}^{6}{b}^{3}d{e}^{3}+140\,{x}^{5}b{a}^{2}{e}^{4}+560\,{x}^{5}{b}^{2}ad{e}^{3}+280\,{x}^{5}{b}^{3}{d}^{2}{e}^{2}+56\,{x}^{4}{a}^{3}{e}^{4}+672\,{x}^{4}b{a}^{2}d{e}^{3}+1008\,{x}^{4}{b}^{2}a{d}^{2}{e}^{2}+224\,{x}^{4}{b}^{3}{d}^{3}e+280\,{x}^{3}{a}^{3}d{e}^{3}+1260\,{x}^{3}b{a}^{2}{d}^{2}{e}^{2}+840\,{x}^{3}{b}^{2}a{d}^{3}e+70\,{x}^{3}{b}^{3}{d}^{4}+560\,{a}^{3}{d}^{2}{e}^{2}{x}^{2}+1120\,{a}^{2}b{d}^{3}e{x}^{2}+280\,a{b}^{2}{d}^{4}{x}^{2}+560\,x{a}^{3}{d}^{3}e+420\,xb{a}^{2}{d}^{4}+280\,{a}^{3}{d}^{4} \right ) }{280\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57092, size = 471, normalized size = 2.36 \begin{align*} \frac{1}{8} \, b^{3} e^{4} x^{8} + a^{3} d^{4} x + \frac{1}{7} \,{\left (4 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, b^{3} d^{2} e^{2} + 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, b^{3} d^{3} e + 18 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{4} + 12 \, a b^{2} d^{3} e + 18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3}\right )} x^{4} +{\left (a b^{2} d^{4} + 4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17076, size = 482, normalized size = 2.41 \begin{align*} \frac{1}{8} \, b^{3} x^{8} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{4}{7} \, b^{3} d x^{7} e^{3} \mathrm{sgn}\left (b x + a\right ) + b^{3} d^{2} x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{4}{5} \, b^{3} d^{3} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, a b^{2} x^{7} e^{4} \mathrm{sgn}\left (b x + a\right ) + 2 \, a b^{2} d x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{18}{5} \, a b^{2} d^{2} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, a b^{2} d^{3} x^{4} e \mathrm{sgn}\left (b x + a\right ) + a b^{2} d^{4} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a^{2} b x^{6} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{12}{5} \, a^{2} b d x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{9}{2} \, a^{2} b d^{2} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{2} b d^{3} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d^{4} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, a^{3} x^{5} e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{3} d x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{3} d^{2} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{3} d^{3} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{3} d^{4} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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