Optimal. Leaf size=266 \[ \frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac{10 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]
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Rubi [A] time = 0.302778, antiderivative size = 266, 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} \[ \frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac{10 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int (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 \left (a b+b^2 x\right )^5 (d+e x)^5 \, 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)^5 \left (a b+b^2 x\right )^5}{b^5}+\frac{5 e (b d-a e)^4 \left (a b+b^2 x\right )^6}{b^6}+\frac{10 e^2 (b d-a e)^3 \left (a b+b^2 x\right )^7}{b^7}+\frac{10 e^3 (b d-a e)^2 \left (a b+b^2 x\right )^8}{b^8}+\frac{5 e^4 (b d-a e) \left (a b+b^2 x\right )^9}{b^9}+\frac{e^5 \left (a b+b^2 x\right )^{10}}{b^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(b d-a e)^5 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^6}+\frac{5 e (b d-a e)^4 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac{5 e^2 (b d-a e)^3 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^6}+\frac{10 e^3 (b d-a e)^2 (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac{e^4 (b d-a e) (a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^6}+\frac{e^5 (a+b x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{11 b^6}\\ \end{align*}
Mathematica [A] time = 0.132222, size = 385, normalized size = 1.45 \[ \frac{x \sqrt{(a+b x)^2} \left (165 a^3 b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+55 a^2 b^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )+330 a^4 b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+462 a^5 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+11 a b^4 x^4 \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )+b^5 x^5 \left (3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1980 d^4 e x+462 d^5+1386 d e^4 x^4+252 e^5 x^5\right )\right )}{2772 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.157, size = 506, normalized size = 1.9 \begin{align*}{\frac{x \left ( 252\,{b}^{5}{e}^{5}{x}^{10}+1386\,{x}^{9}a{b}^{4}{e}^{5}+1386\,{x}^{9}{b}^{5}d{e}^{4}+3080\,{x}^{8}{a}^{2}{b}^{3}{e}^{5}+7700\,{x}^{8}a{b}^{4}d{e}^{4}+3080\,{x}^{8}{b}^{5}{d}^{2}{e}^{3}+3465\,{x}^{7}{a}^{3}{b}^{2}{e}^{5}+17325\,{x}^{7}{a}^{2}{b}^{3}d{e}^{4}+17325\,{x}^{7}a{b}^{4}{d}^{2}{e}^{3}+3465\,{x}^{7}{b}^{5}{d}^{3}{e}^{2}+1980\,{x}^{6}{a}^{4}b{e}^{5}+19800\,{x}^{6}{a}^{3}{b}^{2}d{e}^{4}+39600\,{x}^{6}{a}^{2}{b}^{3}{d}^{2}{e}^{3}+19800\,{x}^{6}a{b}^{4}{d}^{3}{e}^{2}+1980\,{x}^{6}{b}^{5}{d}^{4}e+462\,{x}^{5}{a}^{5}{e}^{5}+11550\,{x}^{5}{a}^{4}bd{e}^{4}+46200\,{x}^{5}{a}^{3}{b}^{2}{d}^{2}{e}^{3}+46200\,{x}^{5}{a}^{2}{b}^{3}{d}^{3}{e}^{2}+11550\,{x}^{5}a{b}^{4}{d}^{4}e+462\,{x}^{5}{b}^{5}{d}^{5}+2772\,{a}^{5}d{e}^{4}{x}^{4}+27720\,{a}^{4}b{d}^{2}{e}^{3}{x}^{4}+55440\,{a}^{3}{b}^{2}{d}^{3}{e}^{2}{x}^{4}+27720\,{a}^{2}{b}^{3}{d}^{4}e{x}^{4}+2772\,a{b}^{4}{d}^{5}{x}^{4}+6930\,{x}^{3}{a}^{5}{d}^{2}{e}^{3}+34650\,{x}^{3}{a}^{4}b{d}^{3}{e}^{2}+34650\,{x}^{3}{a}^{3}{b}^{2}{d}^{4}e+6930\,{x}^{3}{a}^{2}{b}^{3}{d}^{5}+9240\,{x}^{2}{a}^{5}{d}^{3}{e}^{2}+23100\,{x}^{2}{a}^{4}b{d}^{4}e+9240\,{x}^{2}{a}^{3}{b}^{2}{d}^{5}+6930\,x{a}^{5}{d}^{4}e+6930\,x{a}^{4}b{d}^{5}+2772\,{a}^{5}{d}^{5} \right ) }{2772\, \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 = 1.85984, size = 883, normalized size = 3.32 \begin{align*} \frac{1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac{1}{2} \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac{5}{4} \,{\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac{5}{7} \,{\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} +{\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac{5}{2} \,{\left (a^{4} b d^{5} + a^{5} 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 (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 = 1.30252, size = 926, normalized size = 3.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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