Optimal. Leaf size=146 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^3 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^3 (a+b x) (d+e x)^5} \]
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Rubi [A] time = 0.224175, 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 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^3 (a+b x) (d+e x)^4}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^3 (a+b x) (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 41.9242, size = 114, normalized size = 0.78 \[ - \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{30 \left (d + e x\right )^{3} \left (a e - b d\right )^{3}} + \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{10 \left (d + e x\right )^{4} \left (a e - b d\right )^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{5 \left (d + e x\right )^{5} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**6,x)
[Out]
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Mathematica [A] time = 0.0699505, size = 73, normalized size = 0.5 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{30 e^3 (a+b x) (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^6,x]
[Out]
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Maple [A] time = 0.011, size = 78, normalized size = 0.5 \[ -{\frac{10\,{x}^{2}{b}^{2}{e}^{2}+15\,xab{e}^{2}+5\,x{b}^{2}de+6\,{a}^{2}{e}^{2}+3\,abde+{b}^{2}{d}^{2}}{30\,{e}^{3} \left ( ex+d \right ) ^{5} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^6,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)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282353, size = 147, normalized size = 1.01 \[ -\frac{10 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 3 \, a b d e + 6 \, a^{2} e^{2} + 5 \,{\left (b^{2} d e + 3 \, a b e^{2}\right )} x}{30 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.4895, size = 116, normalized size = 0.79 \[ - \frac{6 a^{2} e^{2} + 3 a b d e + b^{2} d^{2} + 10 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} + 5 b^{2} d e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.28631, size = 130, normalized size = 0.89 \[ -\frac{{\left (10 \, b^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 5 \, b^{2} d x e{\rm sign}\left (b x + a\right ) + b^{2} d^{2}{\rm sign}\left (b x + a\right ) + 15 \, a b x e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a b d e{\rm sign}\left (b x + a\right ) + 6 \, a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{30 \,{\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d)^6,x, algorithm="giac")
[Out]