Optimal. Leaf size=146 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.220937, 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 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^3 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{6 e^3 (a+b x) (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^7,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 28.1736, size = 114, normalized size = 0.78 \[ - \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 e^{2} \left (d + e x\right )^{5}} + \frac{b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{60 e^{3} \left (a + b x\right ) \left (d + e x\right )^{5}} - \frac{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e \left (d + e x\right )^{6}} \]
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)**7,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0571813, size = 73, normalized size = 0.5 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^7,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 78, normalized size = 0.5 \[ -{\frac{15\,{x}^{2}{b}^{2}{e}^{2}+24\,xab{e}^{2}+6\,x{b}^{2}de+10\,{a}^{2}{e}^{2}+4\,abde+{b}^{2}{d}^{2}}{60\,{e}^{3} \left ( ex+d \right ) ^{6} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^7,x)
[Out]
_______________________________________________________________________________________
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)^7,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.276305, size = 162, normalized size = 1.11 \[ -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d)^7,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 5.23898, size = 128, normalized size = 0.88 \[ - \frac{10 a^{2} e^{2} + 4 a b d e + b^{2} d^{2} + 15 b^{2} e^{2} x^{2} + x \left (24 a b e^{2} + 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**7,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.285179, size = 130, normalized size = 0.89 \[ -\frac{{\left (15 \, b^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, b^{2} d x e{\rm sign}\left (b x + a\right ) + b^{2} d^{2}{\rm sign}\left (b x + a\right ) + 24 \, a b x e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a b d e{\rm sign}\left (b x + a\right ) + 10 \, a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d)^7,x, algorithm="giac")
[Out]