3.1557 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=200 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^4 (a+b x) (d+e x)^6}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{8 e^4 (a+b x) (d+e x)^8}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5} \]

[Out]

((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^4*(a + b*x)*(d + e*x)^8) - (3
*b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)^7) +
(b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^4*(a + b*x)*(d + e*x)^6) -
(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)*(d + e*x)^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.243897, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^4 (a+b x) (d+e x)^6}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{8 e^4 (a+b x) (d+e x)^8}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^9,x]

[Out]

((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^4*(a + b*x)*(d + e*x)^8) - (3
*b*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^4*(a + b*x)*(d + e*x)^7) +
(b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^4*(a + b*x)*(d + e*x)^6) -
(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^4*(a + b*x)*(d + e*x)^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.7366, size = 156, normalized size = 0.78 \[ - \frac{3 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{140 e^{3} \left (d + e x\right )^{6}} + \frac{b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{280 e^{4} \left (a + b x\right ) \left (d + e x\right )^{6}} - \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{56 e^{2} \left (d + e x\right )^{7}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 e \left (d + e x\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**9,x)

[Out]

-3*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(140*e**3*(d + e*x)**6) + b**2*(a*e - b
*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(280*e**4*(a + b*x)*(d + e*x)**6) - b*(3*a
+ 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(56*e**2*(d + e*x)**7) - (a**2 + 2*a*b
*x + b**2*x**2)**(3/2)/(8*e*(d + e*x)**8)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0850281, size = 112, normalized size = 0.56 \[ -\frac{\sqrt{(a+b x)^2} \left (35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left (d^2+8 d e x+28 e^2 x^2\right )+b^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )}{280 e^4 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(35*a^3*e^3 + 15*a^2*b*e^2*(d + 8*e*x) + 5*a*b^2*e*(d^2 + 8*
d*e*x + 28*e^2*x^2) + b^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)))/(280*e
^4*(a + b*x)*(d + e*x)^8)

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 131, normalized size = 0.7 \[ -{\frac{56\,{x}^{3}{b}^{3}{e}^{3}+140\,{x}^{2}a{b}^{2}{e}^{3}+28\,{x}^{2}{b}^{3}d{e}^{2}+120\,x{a}^{2}b{e}^{3}+40\,xa{b}^{2}d{e}^{2}+8\,x{b}^{3}{d}^{2}e+35\,{a}^{3}{e}^{3}+15\,{a}^{2}bd{e}^{2}+5\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{280\,{e}^{4} \left ( ex+d \right ) ^{8} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^9,x)

[Out]

-1/280/e^4*(56*b^3*e^3*x^3+140*a*b^2*e^3*x^2+28*b^3*d*e^2*x^2+120*a^2*b*e^3*x+40
*a*b^2*d*e^2*x+8*b^3*d^2*e*x+35*a^3*e^3+15*a^2*b*d*e^2+5*a*b^2*d^2*e+b^3*d^3)*((
b*x+a)^2)^(3/2)/(e*x+d)^8/(b*x+a)^3

_______________________________________________________________________________________

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((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.208656, size = 261, normalized size = 1.3 \[ -\frac{56 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 5 \, a b^{2} d^{2} e + 15 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 28 \,{\left (b^{3} d e^{2} + 5 \, a b^{2} e^{3}\right )} x^{2} + 8 \,{\left (b^{3} d^{2} e + 5 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x}{280 \,{\left (e^{12} x^{8} + 8 \, d e^{11} x^{7} + 28 \, d^{2} e^{10} x^{6} + 56 \, d^{3} e^{9} x^{5} + 70 \, d^{4} e^{8} x^{4} + 56 \, d^{5} e^{7} x^{3} + 28 \, d^{6} e^{6} x^{2} + 8 \, d^{7} e^{5} x + d^{8} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/280*(56*b^3*e^3*x^3 + b^3*d^3 + 5*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 35*a^3*e^3 +
28*(b^3*d*e^2 + 5*a*b^2*e^3)*x^2 + 8*(b^3*d^2*e + 5*a*b^2*d*e^2 + 15*a^2*b*e^3)
*x)/(e^12*x^8 + 8*d*e^11*x^7 + 28*d^2*e^10*x^6 + 56*d^3*e^9*x^5 + 70*d^4*e^8*x^4
+ 56*d^5*e^7*x^3 + 28*d^6*e^6*x^2 + 8*d^7*e^5*x + d^8*e^4)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**9,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215621, size = 228, normalized size = 1.14 \[ -\frac{{\left (56 \, b^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 28 \, b^{3} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 8 \, b^{3} d^{2} x e{\rm sign}\left (b x + a\right ) + b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 140 \, a b^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 40 \, a b^{2} d x e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 120 \, a^{2} b x e^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + 35 \, a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{280 \,{\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^9,x, algorithm="giac")

[Out]

-1/280*(56*b^3*x^3*e^3*sign(b*x + a) + 28*b^3*d*x^2*e^2*sign(b*x + a) + 8*b^3*d^
2*x*e*sign(b*x + a) + b^3*d^3*sign(b*x + a) + 140*a*b^2*x^2*e^3*sign(b*x + a) +
40*a*b^2*d*x*e^2*sign(b*x + a) + 5*a*b^2*d^2*e*sign(b*x + a) + 120*a^2*b*x*e^3*s
ign(b*x + a) + 15*a^2*b*d*e^2*sign(b*x + a) + 35*a^3*e^3*sign(b*x + a))*e^(-4)/(
x*e + d)^8