Optimal. Leaf size=266 \[ \frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac{10 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]
[Out]
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Rubi [A] time = 0.684912, 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 \[ \frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac{10 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 45.5094, size = 255, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{11 e} + \frac{\left (5 a + 5 b x\right ) \left (d + e x\right )^{6} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{110 e^{2}} + \frac{2 \left (d + e x\right )^{6} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 e^{3}} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{6} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{396 e^{4}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{462 e^{5}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2772 e^{6} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.321646, size = 385, normalized size = 1.45 \[ \frac{x \sqrt{(a+b x)^2} \left (462 a^5 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+330 a^4 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+165 a^3 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+55 a^2 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+11 a b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )\right )}{2772 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.012, size = 506, normalized size = 1.9 \[{\frac{x \left ( 252\,{b}^{5}{e}^{5}{x}^{10}+1386\,{x}^{9}a{b}^{4}{e}^{5}+1386\,{x}^{9}{b}^{5}d{e}^{4}+3080\,{x}^{8}{a}^{2}{b}^{3}{e}^{5}+7700\,{x}^{8}a{b}^{4}d{e}^{4}+3080\,{x}^{8}{b}^{5}{d}^{2}{e}^{3}+3465\,{x}^{7}{a}^{3}{b}^{2}{e}^{5}+17325\,{x}^{7}{a}^{2}{b}^{3}d{e}^{4}+17325\,{x}^{7}a{b}^{4}{d}^{2}{e}^{3}+3465\,{x}^{7}{b}^{5}{d}^{3}{e}^{2}+1980\,{x}^{6}{a}^{4}b{e}^{5}+19800\,{x}^{6}{a}^{3}{b}^{2}d{e}^{4}+39600\,{x}^{6}{a}^{2}{b}^{3}{d}^{2}{e}^{3}+19800\,{x}^{6}a{b}^{4}{d}^{3}{e}^{2}+1980\,{x}^{6}{b}^{5}{d}^{4}e+462\,{x}^{5}{a}^{5}{e}^{5}+11550\,{x}^{5}{a}^{4}bd{e}^{4}+46200\,{x}^{5}{a}^{3}{b}^{2}{d}^{2}{e}^{3}+46200\,{x}^{5}{a}^{2}{b}^{3}{d}^{3}{e}^{2}+11550\,{x}^{5}a{b}^{4}{d}^{4}e+462\,{x}^{5}{b}^{5}{d}^{5}+2772\,{a}^{5}d{e}^{4}{x}^{4}+27720\,{a}^{4}b{d}^{2}{e}^{3}{x}^{4}+55440\,{a}^{3}{b}^{2}{d}^{3}{e}^{2}{x}^{4}+27720\,{a}^{2}{b}^{3}{d}^{4}e{x}^{4}+2772\,a{b}^{4}{d}^{5}{x}^{4}+6930\,{x}^{3}{a}^{5}{d}^{2}{e}^{3}+34650\,{x}^{3}{a}^{4}b{d}^{3}{e}^{2}+34650\,{x}^{3}{a}^{3}{b}^{2}{d}^{4}e+6930\,{x}^{3}{a}^{2}{b}^{3}{d}^{5}+9240\,{x}^{2}{a}^{5}{d}^{3}{e}^{2}+23100\,{x}^{2}{a}^{4}b{d}^{4}e+9240\,{x}^{2}{a}^{3}{b}^{2}{d}^{5}+6930\,x{a}^{5}{d}^{4}e+6930\,x{a}^{4}b{d}^{5}+2772\,{a}^{5}{d}^{5} \right ) }{2772\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208475, size = 576, normalized size = 2.17 \[ \frac{1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac{1}{2} \,{\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac{5}{4} \,{\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac{5}{7} \,{\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} +{\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac{5}{2} \,{\left (a^{4} b d^{5} + a^{5} d^{4} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{5} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.224969, size = 926, normalized size = 3.48 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^5,x, algorithm="giac")
[Out]