Optimal. Leaf size=172 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{3 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{8 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^4} \]
[Out]
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Rubi [A] time = 0.539171, antiderivative size = 172, 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 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{3 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{8 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 44.5392, size = 141, normalized size = 0.82 \[ \frac{\left (d + e x\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{10 b} - \frac{\left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{30 b^{2}} + \frac{\left (d + e x\right ) \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{120 b^{3}} - \frac{\left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{840 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.200314, size = 294, normalized size = 1.71 \[ \frac{x \sqrt{(a+b x)^2} \left (210 a^6 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+252 a^5 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+210 a^4 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+120 a^3 b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+45 a^2 b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+10 a b^5 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+b^6 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right )}{840 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.014, size = 380, normalized size = 2.2 \[{\frac{x \left ( 84\,{e}^{3}{b}^{6}{x}^{9}+560\,{x}^{8}{e}^{3}a{b}^{5}+280\,{x}^{8}d{e}^{2}{b}^{6}+1575\,{x}^{7}{e}^{3}{a}^{2}{b}^{4}+1890\,{x}^{7}d{e}^{2}a{b}^{5}+315\,{x}^{7}{d}^{2}e{b}^{6}+2400\,{x}^{6}{a}^{3}{b}^{3}{e}^{3}+5400\,{x}^{6}{a}^{2}{b}^{4}d{e}^{2}+2160\,{x}^{6}a{b}^{5}{d}^{2}e+120\,{x}^{6}{d}^{3}{b}^{6}+2100\,{x}^{5}{e}^{3}{b}^{2}{a}^{4}+8400\,{x}^{5}d{e}^{2}{a}^{3}{b}^{3}+6300\,{x}^{5}{d}^{2}e{a}^{2}{b}^{4}+840\,{x}^{5}{d}^{3}a{b}^{5}+1008\,{x}^{4}{e}^{3}{a}^{5}b+7560\,{x}^{4}d{e}^{2}{b}^{2}{a}^{4}+10080\,{x}^{4}{d}^{2}e{a}^{3}{b}^{3}+2520\,{x}^{4}{d}^{3}{a}^{2}{b}^{4}+210\,{x}^{3}{e}^{3}{a}^{6}+3780\,{x}^{3}d{e}^{2}{a}^{5}b+9450\,{x}^{3}{d}^{2}e{b}^{2}{a}^{4}+4200\,{x}^{3}{d}^{3}{a}^{3}{b}^{3}+840\,{a}^{6}d{e}^{2}{x}^{2}+5040\,{a}^{5}b{d}^{2}e{x}^{2}+4200\,{a}^{4}{b}^{2}{d}^{3}{x}^{2}+1260\,x{d}^{2}e{a}^{6}+2520\,x{d}^{3}{a}^{5}b+840\,{d}^{3}{a}^{6} \right ) }{840\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^3*(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)*(b*x + a)*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280034, size = 441, normalized size = 2.56 \[ \frac{1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac{1}{3} \,{\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac{3}{8} \,{\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (2 \, a^{5} b d^{3} + a^{6} d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.285974, size = 706, normalized size = 4.1 \[ \frac{1}{10} \, b^{6} x^{10} e^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, b^{6} d x^{9} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{8} \, b^{6} d^{2} x^{8} e{\rm sign}\left (b x + a\right ) + \frac{1}{7} \, b^{6} d^{3} x^{7}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, a b^{5} x^{9} e^{3}{\rm sign}\left (b x + a\right ) + \frac{9}{4} \, a b^{5} d x^{8} e^{2}{\rm sign}\left (b x + a\right ) + \frac{18}{7} \, a b^{5} d^{2} x^{7} e{\rm sign}\left (b x + a\right ) + a b^{5} d^{3} x^{6}{\rm sign}\left (b x + a\right ) + \frac{15}{8} \, a^{2} b^{4} x^{8} e^{3}{\rm sign}\left (b x + a\right ) + \frac{45}{7} \, a^{2} b^{4} d x^{7} e^{2}{\rm sign}\left (b x + a\right ) + \frac{15}{2} \, a^{2} b^{4} d^{2} x^{6} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{20}{7} \, a^{3} b^{3} x^{7} e^{3}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{3} d x^{6} e^{2}{\rm sign}\left (b x + a\right ) + 12 \, a^{3} b^{3} d^{2} x^{5} e{\rm sign}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{3} x^{4}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b^{2} x^{6} e^{3}{\rm sign}\left (b x + a\right ) + 9 \, a^{4} b^{2} d x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{45}{4} \, a^{4} b^{2} d^{2} x^{4} e{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{6}{5} \, a^{5} b x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{9}{2} \, a^{5} b d x^{4} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{5} b d^{2} x^{3} e{\rm sign}\left (b x + a\right ) + 3 \, a^{5} b d^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a^{6} x^{4} e^{3}{\rm sign}\left (b x + a\right ) + a^{6} d x^{3} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{6} d^{2} x^{2} e{\rm sign}\left (b x + a\right ) + a^{6} d^{3} x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^3,x, algorithm="giac")
[Out]