Optimal. Leaf size=219 \[ \frac{4 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac{4 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \]
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Rubi [A] time = 0.0896996, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {645} \[ \frac{4 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{9 b^5}+\frac{3 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{4 b^5}+\frac{4 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^4}{6 b^5}+\frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \]
Antiderivative was successfully verified.
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Rule 645
Rubi steps
\begin{align*} \int (d+e x)^4 \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 (\frac{(b d-a e)^4 \left (a b+b^2 x\right )^5}{b^4}+\frac{4 e (b d-a e)^3 \left (a b+b^2 x\right )^6}{b^5}+\frac{6 e^2 (b d-a e)^2 \left (a b+b^2 x\right )^7}{b^6}+\frac{4 e^3 (b d-a e) \left (a b+b^2 x\right )^8}{b^7}+\frac{e^4 \left (a b+b^2 x\right )^9}{b^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(b d-a e)^4 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^5}+\frac{4 e (b d-a e)^3 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac{3 e^2 (b d-a e)^2 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^5}+\frac{4 e^3 (b d-a e) (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac{e^4 (a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{10 b^5}\\ \end{align*}
Mathematica [A] time = 0.107724, size = 319, normalized size = 1.46 \[ \frac{x \sqrt{(a+b x)^2} \left (120 a^3 b^2 x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+45 a^2 b^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )+210 a^4 b x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+252 a^5 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+10 a b^4 x^4 \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )+b^5 x^5 \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )\right )}{1260 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.157, size = 414, normalized size = 1.9 \begin{align*}{\frac{x \left ( 126\,{e}^{4}{b}^{5}{x}^{9}+700\,{x}^{8}{e}^{4}a{b}^{4}+560\,{x}^{8}d{e}^{3}{b}^{5}+1575\,{x}^{7}{e}^{4}{a}^{2}{b}^{3}+3150\,{x}^{7}d{e}^{3}a{b}^{4}+945\,{x}^{7}{d}^{2}{e}^{2}{b}^{5}+1800\,{x}^{6}{e}^{4}{a}^{3}{b}^{2}+7200\,{x}^{6}d{e}^{3}{a}^{2}{b}^{3}+5400\,{x}^{6}{d}^{2}{e}^{2}a{b}^{4}+720\,{x}^{6}{d}^{3}e{b}^{5}+1050\,{x}^{5}{e}^{4}{a}^{4}b+8400\,{x}^{5}d{e}^{3}{a}^{3}{b}^{2}+12600\,{x}^{5}{d}^{2}{e}^{2}{a}^{2}{b}^{3}+4200\,{x}^{5}{d}^{3}ea{b}^{4}+210\,{x}^{5}{d}^{4}{b}^{5}+252\,{x}^{4}{e}^{4}{a}^{5}+5040\,{x}^{4}d{e}^{3}{a}^{4}b+15120\,{x}^{4}{d}^{2}{e}^{2}{a}^{3}{b}^{2}+10080\,{x}^{4}{d}^{3}e{a}^{2}{b}^{3}+1260\,{x}^{4}{d}^{4}a{b}^{4}+1260\,{x}^{3}d{e}^{3}{a}^{5}+9450\,{x}^{3}{d}^{2}{e}^{2}{a}^{4}b+12600\,{x}^{3}{d}^{3}e{a}^{3}{b}^{2}+3150\,{x}^{3}{d}^{4}{a}^{2}{b}^{3}+2520\,{x}^{2}{d}^{2}{e}^{2}{a}^{5}+8400\,{x}^{2}{d}^{3}e{a}^{4}b+4200\,{x}^{2}{d}^{4}{a}^{3}{b}^{2}+2520\,x{d}^{3}e{a}^{5}+3150\,x{d}^{4}{a}^{4}b+1260\,{d}^{4}{a}^{5} \right ) }{1260\, \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 = 1.60779, size = 760, normalized size = 3.47 \begin{align*} \frac{1}{10} \, b^{5} e^{4} x^{10} + a^{5} d^{4} x + \frac{1}{9} \,{\left (4 \, b^{5} d e^{3} + 5 \, a b^{4} e^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, b^{5} d^{2} e^{2} + 10 \, a b^{4} d e^{3} + 5 \, a^{2} b^{3} e^{4}\right )} x^{8} + \frac{2}{7} \,{\left (2 \, b^{5} d^{3} e + 15 \, a b^{4} d^{2} e^{2} + 20 \, a^{2} b^{3} d e^{3} + 5 \, a^{3} b^{2} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{4} + 20 \, a b^{4} d^{3} e + 60 \, a^{2} b^{3} d^{2} e^{2} + 40 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (5 \, a b^{4} d^{4} + 40 \, a^{2} b^{3} d^{3} e + 60 \, a^{3} b^{2} d^{2} e^{2} + 20 \, a^{4} b d e^{3} + a^{5} e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (5 \, a^{2} b^{3} d^{4} + 20 \, a^{3} b^{2} d^{3} e + 15 \, a^{4} b d^{2} e^{2} + 2 \, a^{5} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (5 \, a^{3} b^{2} d^{4} + 10 \, a^{4} b d^{3} e + 3 \, a^{5} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{4} + 4 \, a^{5} 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{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24433, size = 761, normalized size = 3.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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