Optimal. Leaf size=172 \[ \frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{8 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{6 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \]
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Rubi [A] time = 0.180173, antiderivative size = 172, 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} \[ \frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{8 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{6 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (d+e x)^3 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{(b d-a e)^3 \left (a b+b^2 x\right )^5}{b^3}+\frac{3 e (b d-a e)^2 \left (a b+b^2 x\right )^6}{b^4}+\frac{3 e^2 (b d-a e) \left (a b+b^2 x\right )^7}{b^5}+\frac{e^3 \left (a b+b^2 x\right )^8}{b^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(b d-a e)^3 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac{3 e (b d-a e)^2 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac{3 e^2 (b d-a e) (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac{e^3 (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^4}\\ \end{align*}
Mathematica [A] time = 0.0872429, size = 253, normalized size = 1.47 \[ \frac{x \sqrt{(a+b x)^2} \left (84 a^3 b^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+36 a^2 b^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+126 a^4 b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+126 a^5 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+9 a b^4 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+b^5 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )\right )}{504 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.156, size = 322, normalized size = 1.9 \begin{align*}{\frac{x \left ( 56\,{e}^{3}{b}^{5}{x}^{8}+315\,{x}^{7}{e}^{3}a{b}^{4}+189\,{x}^{7}d{e}^{2}{b}^{5}+720\,{x}^{6}{e}^{3}{a}^{2}{b}^{3}+1080\,{x}^{6}d{e}^{2}a{b}^{4}+216\,{x}^{6}{d}^{2}e{b}^{5}+840\,{x}^{5}{e}^{3}{a}^{3}{b}^{2}+2520\,{x}^{5}d{e}^{2}{a}^{2}{b}^{3}+1260\,{x}^{5}{d}^{2}ea{b}^{4}+84\,{x}^{5}{d}^{3}{b}^{5}+504\,{a}^{4}b{e}^{3}{x}^{4}+3024\,{a}^{3}{b}^{2}d{e}^{2}{x}^{4}+3024\,{a}^{2}{b}^{3}{d}^{2}e{x}^{4}+504\,a{b}^{4}{d}^{3}{x}^{4}+126\,{x}^{3}{e}^{3}{a}^{5}+1890\,{x}^{3}d{e}^{2}{a}^{4}b+3780\,{x}^{3}{d}^{2}e{a}^{3}{b}^{2}+1260\,{x}^{3}{d}^{3}{a}^{2}{b}^{3}+504\,{x}^{2}d{e}^{2}{a}^{5}+2520\,{x}^{2}{d}^{2}e{a}^{4}b+1680\,{x}^{2}{d}^{3}{a}^{3}{b}^{2}+756\,x{d}^{2}e{a}^{5}+1260\,x{d}^{3}{a}^{4}b+504\,{d}^{3}{a}^{5} \right ) }{504\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{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 [B] time = 1.49215, size = 585, normalized size = 3.4 \begin{align*} \frac{1}{9} \, b^{5} e^{3} x^{9} + a^{5} d^{3} x + \frac{1}{8} \,{\left (3 \, b^{5} d e^{2} + 5 \, a b^{4} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, b^{5} d^{2} e + 15 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{3} + 15 \, a b^{4} d^{2} e + 30 \, a^{2} b^{3} d e^{2} + 10 \, a^{3} b^{2} e^{3}\right )} x^{6} +{\left (a b^{4} d^{3} + 6 \, a^{2} b^{3} d^{2} e + 6 \, a^{3} b^{2} d e^{2} + a^{4} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, a^{2} b^{3} d^{3} + 30 \, a^{3} b^{2} d^{2} e + 15 \, a^{4} b d e^{2} + a^{5} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} d^{3} + 15 \, a^{4} b d^{2} e + 3 \, a^{5} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{3} + 3 \, a^{5} d^{2} 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 )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19104, size = 595, normalized size = 3.46 \begin{align*} \frac{1}{9} \, b^{5} x^{9} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{8} \, b^{5} d x^{8} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, b^{5} d^{2} x^{7} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, b^{5} d^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{8} \, a b^{4} x^{8} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{7} \, a b^{4} d x^{7} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a b^{4} d^{2} x^{6} e \mathrm{sgn}\left (b x + a\right ) + a b^{4} d^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, a^{2} b^{3} x^{7} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{2} b^{3} d x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{3} d^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{3} b^{2} d x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{2} \, a^{3} b^{2} d^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{2} d^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} b x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{15}{4} \, a^{4} b d x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b d^{2} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b d^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a^{5} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{5} d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{5} d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{5} d^{3} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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