Optimal. Leaf size=146 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \]
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Rubi [A] time = 0.333072, antiderivative size = 146, 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 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7}{7 e^3 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)}{3 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^2}{5 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 29.1695, size = 119, normalized size = 0.82 \[ \frac{\left (a + b x\right ) \left (d + e x\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7 e} + \frac{\left (d + e x\right )^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 e^{2}} + \frac{\left (d + e x\right )^{5} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105 e^{3} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**4*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.101414, size = 163, normalized size = 1.12 \[ \frac{x \sqrt{(a+b x)^2} \left (21 a^2 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+7 a b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )\right )}{105 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.01, size = 189, normalized size = 1.3 \[{\frac{x \left ( 15\,{b}^{2}{e}^{4}{x}^{6}+35\,{x}^{5}ab{e}^{4}+70\,{x}^{5}{b}^{2}d{e}^{3}+21\,{x}^{4}{a}^{2}{e}^{4}+168\,{x}^{4}abd{e}^{3}+126\,{x}^{4}{b}^{2}{d}^{2}{e}^{2}+105\,{a}^{2}d{e}^{3}{x}^{3}+315\,ab{d}^{2}{e}^{2}{x}^{3}+105\,{b}^{2}{d}^{3}e{x}^{3}+210\,{x}^{2}{a}^{2}{d}^{2}{e}^{2}+280\,{x}^{2}ab{d}^{3}e+35\,{x}^{2}{b}^{2}{d}^{4}+210\,{a}^{2}{d}^{3}ex+105\,ab{d}^{4}x+105\,{a}^{2}{d}^{4} \right ) }{105\,bx+105\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^4*((b*x+a)^2)^(1/2),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(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278088, size = 211, normalized size = 1.45 \[ \frac{1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac{1}{3} \,{\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} +{\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} +{\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.340976, size = 168, normalized size = 1.15 \[ a^{2} d^{4} x + \frac{b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac{a b e^{4}}{3} + \frac{2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac{a^{2} e^{4}}{5} + \frac{8 a b d e^{3}}{5} + \frac{6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac{8 a b d^{3} e}{3} + \frac{b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**4*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.29186, size = 343, normalized size = 2.35 \[ \frac{1}{7} \, b^{2} x^{7} e^{4}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, b^{2} d x^{6} e^{3}{\rm sign}\left (b x + a\right ) + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2}{\rm sign}\left (b x + a\right ) + b^{2} d^{3} x^{4} e{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, b^{2} d^{4} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a b x^{6} e^{4}{\rm sign}\left (b x + a\right ) + \frac{8}{5} \, a b d x^{5} e^{3}{\rm sign}\left (b x + a\right ) + 3 \, a b d^{2} x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{8}{3} \, a b d^{3} x^{3} e{\rm sign}\left (b x + a\right ) + a b d^{4} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, a^{2} x^{5} e^{4}{\rm sign}\left (b x + a\right ) + a^{2} d x^{4} e^{3}{\rm sign}\left (b x + a\right ) + 2 \, a^{2} d^{2} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a^{2} d^{3} x^{2} e{\rm sign}\left (b x + a\right ) + a^{2} d^{4} x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^4,x, algorithm="giac")
[Out]