3.1574 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=308 \[ \frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{4 e^6 (a+b x) (d+e x)^8}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{10 e^6 (a+b x) (d+e x)^{10}}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{6 e^6 (a+b x) (d+e x)^6}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^7} \]

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^6*(a + b*x)*(d + e*x)^10) -
(5*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^9)
+ (5*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)
^8) - (10*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d +
e*x)^7) + (5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^6*(a + b*x)*(d
+ e*x)^6) - (b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^6*(a + b*x)*(d + e*x)^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.419718, antiderivative size = 308, 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 \[ \frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{4 e^6 (a+b x) (d+e x)^8}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{10 e^6 (a+b x) (d+e x)^{10}}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x) (d+e x)^5}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{6 e^6 (a+b x) (d+e x)^6}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^6 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^11,x]

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^6*(a + b*x)*(d + e*x)^10) -
(5*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^9)
+ (5*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)
^8) - (10*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d +
e*x)^7) + (5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^6*(a + b*x)*(d
+ e*x)^6) - (b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^6*(a + b*x)*(d + e*x)^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.9173, size = 235, normalized size = 0.76 \[ - \frac{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{210 e^{5} \left (d + e x\right )^{6}} + \frac{b^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{1260 e^{6} \left (a + b x\right ) \left (d + e x\right )^{6}} - \frac{b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{252 e^{4} \left (d + e x\right )^{7}} - \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{36 e^{3} \left (d + e x\right )^{8}} - \frac{b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{90 e^{2} \left (d + e x\right )^{9}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{10 e \left (d + e x\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**11,x)

[Out]

-b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(210*e**5*(d + e*x)**6) + b**4*(a*e - b*d
)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(1260*e**6*(a + b*x)*(d + e*x)**6) - b**3*(3*
a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(252*e**4*(d + e*x)**7) - b**2*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)/(36*e**3*(d + e*x)**8) - b*(5*a + 5*b*x)*(a**2 +
2*a*b*x + b**2*x**2)**(3/2)/(90*e**2*(d + e*x)**9) - (a**2 + 2*a*b*x + b**2*x**2
)**(5/2)/(10*e*(d + e*x)**10)

_______________________________________________________________________________________

Mathematica [A]  time = 0.157595, size = 223, normalized size = 0.72 \[ -\frac{\sqrt{(a+b x)^2} \left (126 a^5 e^5+70 a^4 b e^4 (d+10 e x)+35 a^3 b^2 e^3 \left (d^2+10 d e x+45 e^2 x^2\right )+15 a^2 b^3 e^2 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+5 a b^4 e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+b^5 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )}{1260 e^6 (a+b x) (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^11,x]

[Out]

-(Sqrt[(a + b*x)^2]*(126*a^5*e^5 + 70*a^4*b*e^4*(d + 10*e*x) + 35*a^3*b^2*e^3*(d
^2 + 10*d*e*x + 45*e^2*x^2) + 15*a^2*b^3*e^2*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 +
120*e^3*x^3) + 5*a*b^4*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 21
0*e^4*x^4) + b^5*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^
4*x^4 + 252*e^5*x^5)))/(1260*e^6*(a + b*x)*(d + e*x)^10)

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 288, normalized size = 0.9 \[ -{\frac{252\,{x}^{5}{b}^{5}{e}^{5}+1050\,{x}^{4}a{b}^{4}{e}^{5}+210\,{x}^{4}{b}^{5}d{e}^{4}+1800\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+600\,{x}^{3}a{b}^{4}d{e}^{4}+120\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+1575\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+675\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+225\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+45\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+700\,x{a}^{4}b{e}^{5}+350\,x{a}^{3}{b}^{2}d{e}^{4}+150\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+50\,xa{b}^{4}{d}^{3}{e}^{2}+10\,x{b}^{5}{d}^{4}e+126\,{a}^{5}{e}^{5}+70\,{a}^{4}bd{e}^{4}+35\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+15\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{1260\,{e}^{6} \left ( ex+d \right ) ^{10} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^11,x)

[Out]

-1/1260/e^6*(252*b^5*e^5*x^5+1050*a*b^4*e^5*x^4+210*b^5*d*e^4*x^4+1800*a^2*b^3*e
^5*x^3+600*a*b^4*d*e^4*x^3+120*b^5*d^2*e^3*x^3+1575*a^3*b^2*e^5*x^2+675*a^2*b^3*
d*e^4*x^2+225*a*b^4*d^2*e^3*x^2+45*b^5*d^3*e^2*x^2+700*a^4*b*e^5*x+350*a^3*b^2*d
*e^4*x+150*a^2*b^3*d^2*e^3*x+50*a*b^4*d^3*e^2*x+10*b^5*d^4*e*x+126*a^5*e^5+70*a^
4*b*d*e^4+35*a^3*b^2*d^2*e^3+15*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e+b^5*d^5)*((b*x+a)^
2)^(5/2)/(e*x+d)^10/(b*x+a)^5

_______________________________________________________________________________________

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)^(5/2)/(e*x + d)^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.210743, size = 485, normalized size = 1.57 \[ -\frac{252 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 15 \, a^{2} b^{3} d^{3} e^{2} + 35 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 126 \, a^{5} e^{5} + 210 \,{\left (b^{5} d e^{4} + 5 \, a b^{4} e^{5}\right )} x^{4} + 120 \,{\left (b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 45 \,{\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 15 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 10 \,{\left (b^{5} d^{4} e + 5 \, a b^{4} d^{3} e^{2} + 15 \, a^{2} b^{3} d^{2} e^{3} + 35 \, a^{3} b^{2} d e^{4} + 70 \, a^{4} b e^{5}\right )} x}{1260 \,{\left (e^{16} x^{10} + 10 \, d e^{15} x^{9} + 45 \, d^{2} e^{14} x^{8} + 120 \, d^{3} e^{13} x^{7} + 210 \, d^{4} e^{12} x^{6} + 252 \, d^{5} e^{11} x^{5} + 210 \, d^{6} e^{10} x^{4} + 120 \, d^{7} e^{9} x^{3} + 45 \, d^{8} e^{8} x^{2} + 10 \, d^{9} e^{7} x + d^{10} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^11,x, algorithm="fricas")

[Out]

-1/1260*(252*b^5*e^5*x^5 + b^5*d^5 + 5*a*b^4*d^4*e + 15*a^2*b^3*d^3*e^2 + 35*a^3
*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 126*a^5*e^5 + 210*(b^5*d*e^4 + 5*a*b^4*e^5)*x^4
+ 120*(b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 + 45*(b^5*d^3*e^2 + 5*a
*b^4*d^2*e^3 + 15*a^2*b^3*d*e^4 + 35*a^3*b^2*e^5)*x^2 + 10*(b^5*d^4*e + 5*a*b^4*
d^3*e^2 + 15*a^2*b^3*d^2*e^3 + 35*a^3*b^2*d*e^4 + 70*a^4*b*e^5)*x)/(e^16*x^10 +
10*d*e^15*x^9 + 45*d^2*e^14*x^8 + 120*d^3*e^13*x^7 + 210*d^4*e^12*x^6 + 252*d^5*
e^11*x^5 + 210*d^6*e^10*x^4 + 120*d^7*e^9*x^3 + 45*d^8*e^8*x^2 + 10*d^9*e^7*x +
d^10*e^6)

_______________________________________________________________________________________

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)**(5/2)/(e*x+d)**11,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222381, size = 514, normalized size = 1.67 \[ -\frac{{\left (252 \, b^{5} x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 210 \, b^{5} d x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 120 \, b^{5} d^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 45 \, b^{5} d^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 10 \, b^{5} d^{4} x e{\rm sign}\left (b x + a\right ) + b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 1050 \, a b^{4} x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 600 \, a b^{4} d x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 225 \, a b^{4} d^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 50 \, a b^{4} d^{3} x e^{2}{\rm sign}\left (b x + a\right ) + 5 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 1800 \, a^{2} b^{3} x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 675 \, a^{2} b^{3} d x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 150 \, a^{2} b^{3} d^{2} x e^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 1575 \, a^{3} b^{2} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 350 \, a^{3} b^{2} d x e^{4}{\rm sign}\left (b x + a\right ) + 35 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 700 \, a^{4} b x e^{5}{\rm sign}\left (b x + a\right ) + 70 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 126 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{1260 \,{\left (x e + d\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^11,x, algorithm="giac")

[Out]

-1/1260*(252*b^5*x^5*e^5*sign(b*x + a) + 210*b^5*d*x^4*e^4*sign(b*x + a) + 120*b
^5*d^2*x^3*e^3*sign(b*x + a) + 45*b^5*d^3*x^2*e^2*sign(b*x + a) + 10*b^5*d^4*x*e
*sign(b*x + a) + b^5*d^5*sign(b*x + a) + 1050*a*b^4*x^4*e^5*sign(b*x + a) + 600*
a*b^4*d*x^3*e^4*sign(b*x + a) + 225*a*b^4*d^2*x^2*e^3*sign(b*x + a) + 50*a*b^4*d
^3*x*e^2*sign(b*x + a) + 5*a*b^4*d^4*e*sign(b*x + a) + 1800*a^2*b^3*x^3*e^5*sign
(b*x + a) + 675*a^2*b^3*d*x^2*e^4*sign(b*x + a) + 150*a^2*b^3*d^2*x*e^3*sign(b*x
+ a) + 15*a^2*b^3*d^3*e^2*sign(b*x + a) + 1575*a^3*b^2*x^2*e^5*sign(b*x + a) +
350*a^3*b^2*d*x*e^4*sign(b*x + a) + 35*a^3*b^2*d^2*e^3*sign(b*x + a) + 700*a^4*b
*x*e^5*sign(b*x + a) + 70*a^4*b*d*e^4*sign(b*x + a) + 126*a^5*e^5*sign(b*x + a))
*e^(-6)/(x*e + d)^10