Optimal. Leaf size=158 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e) (B d-A e)}{4 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^3 (a+b x)} \]
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Rubi [A] time = 0.406071, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e) (B d-A e)}{4 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 29.2498, size = 158, normalized size = 1. \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 b e} + \frac{\left (d + e x\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - 2 B a e - B b d\right )}{15 b e^{2}} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (3 A b e - 2 B a e - B b d\right )}{60 b e^{3} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**3*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.114635, size = 163, normalized size = 1.03 \[ \frac{x \sqrt{(a+b x)^2} \left (3 a \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )\right )}{60 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 180, normalized size = 1.1 \[{\frac{x \left ( 10\,Bb{e}^{3}{x}^{5}+12\,{x}^{4}Ab{e}^{3}+12\,{x}^{4}Ba{e}^{3}+36\,{x}^{4}Bbd{e}^{2}+15\,{x}^{3}aA{e}^{3}+45\,{x}^{3}Abd{e}^{2}+45\,{x}^{3}aBd{e}^{2}+45\,{x}^{3}Bb{d}^{2}e+60\,{x}^{2}Aad{e}^{2}+60\,{x}^{2}Ab{d}^{2}e+60\,{x}^{2}Ba{d}^{2}e+20\,{x}^{2}Bb{d}^{3}+90\,xaA{d}^{2}e+30\,xAb{d}^{3}+30\,xBa{d}^{3}+60\,aA{d}^{3} \right ) }{60\,bx+60\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^3*((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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282807, size = 181, normalized size = 1.15 \[ \frac{1}{6} \, B b e^{3} x^{6} + A a d^{3} x + \frac{1}{5} \,{\left (3 \, B b d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B b d^{2} e + A a e^{3} + 3 \,{\left (B a + A b\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{3} + 3 \, A a d e^{2} + 3 \,{\left (B a + A b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a d^{2} e +{\left (B a + A b\right )} d^{3}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.332467, size = 168, normalized size = 1.06 \[ A a d^{3} x + \frac{B b e^{3} x^{6}}{6} + x^{5} \left (\frac{A b e^{3}}{5} + \frac{B a e^{3}}{5} + \frac{3 B b d e^{2}}{5}\right ) + x^{4} \left (\frac{A a e^{3}}{4} + \frac{3 A b d e^{2}}{4} + \frac{3 B a d e^{2}}{4} + \frac{3 B b d^{2} e}{4}\right ) + x^{3} \left (A a d e^{2} + A b d^{2} e + B a d^{2} e + \frac{B b d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a d^{2} e}{2} + \frac{A b d^{3}}{2} + \frac{B a d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**3*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.302917, size = 344, normalized size = 2.18 \[ \frac{1}{6} \, B b x^{6} e^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, B b d x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, B b d^{2} x^{4} e{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, B b d^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, B a x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, A b x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, B a d x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, A b d x^{4} e^{2}{\rm sign}\left (b x + a\right ) + B a d^{2} x^{3} e{\rm sign}\left (b x + a\right ) + A b d^{2} x^{3} e{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B a d^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A b d^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, A a x^{4} e^{3}{\rm sign}\left (b x + a\right ) + A a d x^{3} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, A a d^{2} x^{2} e{\rm sign}\left (b x + a\right ) + A a d^{3} x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*(e*x + d)^3,x, algorithm="giac")
[Out]