Optimal. Leaf size=263 \[ \frac{5 e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^2}{b^6}+\frac{10 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \]
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Rubi [A] time = 0.376533, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{5 e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^2}{b^6}+\frac{10 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^5 \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 (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^5 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^5 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(b d-a e)^5 (a+b x)^6}{b^5}+\frac{5 e (b d-a e)^4 (a+b x)^7}{b^5}+\frac{10 e^2 (b d-a e)^3 (a+b x)^8}{b^5}+\frac{10 e^3 (b d-a e)^2 (a+b x)^9}{b^5}+\frac{5 e^4 (b d-a e) (a+b x)^{10}}{b^5}+\frac{e^5 (a+b x)^{11}}{b^5}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^5 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac{5 e (b d-a e)^4 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{8 b^6}+\frac{10 e^2 (b d-a e)^3 (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac{e^3 (b d-a e)^2 (a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{b^6}+\frac{5 e^4 (b d-a e) (a+b x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{11 b^6}+\frac{e^5 (a+b x)^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{12 b^6}\\ \end{align*}
Mathematica [A] time = 0.148601, size = 448, normalized size = 1.7 \[ \frac{x \sqrt{(a+b x)^2} \left (495 a^4 b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+220 a^3 b^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )+66 a^2 b^4 x^4 \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )+792 a^5 b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+924 a^6 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+12 a b^5 x^5 \left (3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1980 d^4 e x+462 d^5+1386 d e^4 x^4+252 e^5 x^5\right )+b^6 x^6 \left (6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+3465 d^4 e x+792 d^5+2520 d e^4 x^4+462 e^5 x^5\right )\right )}{5544 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 598, normalized size = 2.3 \begin{align*}{\frac{x \left ( 462\,{e}^{5}{b}^{6}{x}^{11}+3024\,{x}^{10}{e}^{5}{b}^{5}a+2520\,{x}^{10}d{e}^{4}{b}^{6}+8316\,{x}^{9}{e}^{5}{a}^{2}{b}^{4}+16632\,{x}^{9}d{e}^{4}{b}^{5}a+5544\,{x}^{9}{d}^{2}{e}^{3}{b}^{6}+12320\,{x}^{8}{e}^{5}{a}^{3}{b}^{3}+46200\,{x}^{8}d{e}^{4}{a}^{2}{b}^{4}+36960\,{x}^{8}{d}^{2}{e}^{3}{b}^{5}a+6160\,{x}^{8}{d}^{3}{e}^{2}{b}^{6}+10395\,{x}^{7}{e}^{5}{a}^{4}{b}^{2}+69300\,{x}^{7}d{e}^{4}{a}^{3}{b}^{3}+103950\,{x}^{7}{d}^{2}{e}^{3}{a}^{2}{b}^{4}+41580\,{x}^{7}{d}^{3}{e}^{2}{b}^{5}a+3465\,{x}^{7}{d}^{4}e{b}^{6}+4752\,{x}^{6}{e}^{5}{a}^{5}b+59400\,{x}^{6}d{e}^{4}{a}^{4}{b}^{2}+158400\,{x}^{6}{d}^{2}{e}^{3}{a}^{3}{b}^{3}+118800\,{x}^{6}{d}^{3}{e}^{2}{a}^{2}{b}^{4}+23760\,{x}^{6}{d}^{4}e{b}^{5}a+792\,{x}^{6}{d}^{5}{b}^{6}+924\,{x}^{5}{e}^{5}{a}^{6}+27720\,{x}^{5}d{e}^{4}{a}^{5}b+138600\,{x}^{5}{d}^{2}{e}^{3}{a}^{4}{b}^{2}+184800\,{x}^{5}{d}^{3}{e}^{2}{a}^{3}{b}^{3}+69300\,{x}^{5}{d}^{4}e{a}^{2}{b}^{4}+5544\,{x}^{5}{d}^{5}{b}^{5}a+5544\,{a}^{6}d{e}^{4}{x}^{4}+66528\,{a}^{5}b{d}^{2}{e}^{3}{x}^{4}+166320\,{a}^{4}{b}^{2}{d}^{3}{e}^{2}{x}^{4}+110880\,{a}^{3}{b}^{3}{d}^{4}e{x}^{4}+16632\,{a}^{2}{b}^{4}{d}^{5}{x}^{4}+13860\,{x}^{3}{d}^{2}{e}^{3}{a}^{6}+83160\,{x}^{3}{d}^{3}{e}^{2}{a}^{5}b+103950\,{x}^{3}{d}^{4}e{a}^{4}{b}^{2}+27720\,{x}^{3}{d}^{5}{a}^{3}{b}^{3}+18480\,{x}^{2}{d}^{3}{e}^{2}{a}^{6}+55440\,{x}^{2}{d}^{4}e{a}^{5}b+27720\,{x}^{2}{d}^{5}{a}^{4}{b}^{2}+13860\,x{d}^{4}e{a}^{6}+16632\,x{d}^{5}{a}^{5}b+5544\,{d}^{5}{a}^{6} \right ) }{5544\, \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 = 1.58699, size = 1085, normalized size = 4.13 \begin{align*} \frac{1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac{1}{11} \,{\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac{5}{8} \,{\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} 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 (a + b x\right ) \left (d + e x\right )^{5} \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 = 1.16611, size = 1094, normalized size = 4.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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