3.1967 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^6}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^3 (a+b x) (d+e x)^7} \]

[Out]

-((b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3*(a + b*x)*(d + e*x)^7) + (
b*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a + b*x)*(d + e*x)^6) - (b^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)*(d + e*x)^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.224701, 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 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^6}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^3 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^8,x]

[Out]

-((b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^3*(a + b*x)*(d + e*x)^7) + (
b*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a + b*x)*(d + e*x)^6) - (b^
2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*x)*(d + e*x)^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.1976, size = 116, normalized size = 0.79 \[ - \frac{2 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{2} \left (d + e x\right )^{6}} + \frac{b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{105 e^{3} \left (a + b x\right ) \left (d + e x\right )^{6}} - \frac{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7 e \left (d + e x\right )^{7}} \]

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)**8,x)

[Out]

-2*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**2*(d + e*x)**6) + b*(a*e - b*d)*sqr
t(a**2 + 2*a*b*x + b**2*x**2)/(105*e**3*(a + b*x)*(d + e*x)**6) - (a + b*x)*sqrt
(a**2 + 2*a*b*x + b**2*x**2)/(7*e*(d + e*x)**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0647383, size = 73, normalized size = 0.5 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )}{105 e^3 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(15*a^2*e^2 + 5*a*b*e*(d + 7*e*x) + b^2*(d^2 + 7*d*e*x + 21*
e^2*x^2)))/(105*e^3*(a + b*x)*(d + e*x)^7)

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 78, normalized size = 0.5 \[ -{\frac{21\,{x}^{2}{b}^{2}{e}^{2}+35\,xab{e}^{2}+7\,x{b}^{2}de+15\,{a}^{2}{e}^{2}+5\,abde+{b}^{2}{d}^{2}}{105\,{e}^{3} \left ( ex+d \right ) ^{7} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^8,x)

[Out]

-1/105/e^3*(21*b^2*e^2*x^2+35*a*b*e^2*x+7*b^2*d*e*x+15*a^2*e^2+5*a*b*d*e+b^2*d^2
)*((b*x+a)^2)^(1/2)/(e*x+d)^7/(b*x+a)

_______________________________________________________________________________________

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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.276663, size = 177, normalized size = 1.21 \[ -\frac{21 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 5 \, a b d e + 15 \, a^{2} e^{2} + 7 \,{\left (b^{2} d e + 5 \, a b e^{2}\right )} x}{105 \,{\left (e^{10} x^{7} + 7 \, d e^{9} x^{6} + 21 \, d^{2} e^{8} x^{5} + 35 \, d^{3} e^{7} x^{4} + 35 \, d^{4} e^{6} x^{3} + 21 \, d^{5} e^{5} x^{2} + 7 \, d^{6} e^{4} x + d^{7} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/105*(21*b^2*e^2*x^2 + b^2*d^2 + 5*a*b*d*e + 15*a^2*e^2 + 7*(b^2*d*e + 5*a*b*e
^2)*x)/(e^10*x^7 + 7*d*e^9*x^6 + 21*d^2*e^8*x^5 + 35*d^3*e^7*x^4 + 35*d^4*e^6*x^
3 + 21*d^5*e^5*x^2 + 7*d^6*e^4*x + d^7*e^3)

_______________________________________________________________________________________

Sympy [A]  time = 6.16977, size = 139, normalized size = 0.95 \[ - \frac{15 a^{2} e^{2} + 5 a b d e + b^{2} d^{2} + 21 b^{2} e^{2} x^{2} + x \left (35 a b e^{2} + 7 b^{2} d e\right )}{105 d^{7} e^{3} + 735 d^{6} e^{4} x + 2205 d^{5} e^{5} x^{2} + 3675 d^{4} e^{6} x^{3} + 3675 d^{3} e^{7} x^{4} + 2205 d^{2} e^{8} x^{5} + 735 d e^{9} x^{6} + 105 e^{10} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**8,x)

[Out]

-(15*a**2*e**2 + 5*a*b*d*e + b**2*d**2 + 21*b**2*e**2*x**2 + x*(35*a*b*e**2 + 7*
b**2*d*e))/(105*d**7*e**3 + 735*d**6*e**4*x + 2205*d**5*e**5*x**2 + 3675*d**4*e*
*6*x**3 + 3675*d**3*e**7*x**4 + 2205*d**2*e**8*x**5 + 735*d*e**9*x**6 + 105*e**1
0*x**7)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.287338, size = 130, normalized size = 0.89 \[ -\frac{{\left (21 \, b^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 7 \, b^{2} d x e{\rm sign}\left (b x + a\right ) + b^{2} d^{2}{\rm sign}\left (b x + a\right ) + 35 \, a b x e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a b d e{\rm sign}\left (b x + a\right ) + 15 \, a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{105 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(b*x + a)/(e*x + d)^8,x, algorithm="giac")

[Out]

-1/105*(21*b^2*x^2*e^2*sign(b*x + a) + 7*b^2*d*x*e*sign(b*x + a) + b^2*d^2*sign(
b*x + a) + 35*a*b*x*e^2*sign(b*x + a) + 5*a*b*d*e*sign(b*x + a) + 15*a^2*e^2*sig
n(b*x + a))*e^(-3)/(x*e + d)^7