3.1571 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=98 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{42 (d+e x)^6 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(42*(b*d - a*e)^2*(d + e*x)^6)

_______________________________________________________________________________________

Rubi [A]  time = 0.115191, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{42 (d+e x)^6 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(42*(b*d - a*e)^2*(d + e*x)^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.0829, size = 82, normalized size = 0.84 \[ \frac{e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{42 \left (d + e x\right )^{7} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 \left (d + e x\right )^{7} \left (a e - b d\right )} \]

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)**8,x)

[Out]

e*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(42*(d + e*x)**7*(a*e - b*d)**2) - (2*a +
2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*(d + e*x)**7*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [B]  time = 0.154636, size = 223, normalized size = 2.28 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^5 e^5+5 a^4 b e^4 (d+7 e x)+4 a^3 b^2 e^3 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a^2 b^3 e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 a b^4 e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{42 e^6 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(6*a^5*e^5 + 5*a^4*b*e^4*(d + 7*e*x) + 4*a^3*b^2*e^3*(d^2 +
7*d*e*x + 21*e^2*x^2) + 3*a^2*b^3*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x
^3) + 2*a*b^4*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) +
b^5*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*
x^5)))/(42*e^6*(a + b*x)*(d + e*x)^7)

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 288, normalized size = 2.9 \[ -{\frac{21\,{x}^{5}{b}^{5}{e}^{5}+70\,{x}^{4}a{b}^{4}{e}^{5}+35\,{x}^{4}{b}^{5}d{e}^{4}+105\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+70\,{x}^{3}a{b}^{4}d{e}^{4}+35\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+84\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+63\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+42\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+21\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+35\,x{a}^{4}b{e}^{5}+28\,x{a}^{3}{b}^{2}d{e}^{4}+21\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+14\,xa{b}^{4}{d}^{3}{e}^{2}+7\,x{b}^{5}{d}^{4}e+6\,{a}^{5}{e}^{5}+5\,{a}^{4}bd{e}^{4}+4\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+3\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{42\,{e}^{6} \left ( ex+d \right ) ^{7} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^8,x)

[Out]

-1/42/e^6*(21*b^5*e^5*x^5+70*a*b^4*e^5*x^4+35*b^5*d*e^4*x^4+105*a^2*b^3*e^5*x^3+
70*a*b^4*d*e^4*x^3+35*b^5*d^2*e^3*x^3+84*a^3*b^2*e^5*x^2+63*a^2*b^3*d*e^4*x^2+42
*a*b^4*d^2*e^3*x^2+21*b^5*d^3*e^2*x^2+35*a^4*b*e^5*x+28*a^3*b^2*d*e^4*x+21*a^2*b
^3*d^2*e^3*x+14*a*b^4*d^3*e^2*x+7*b^5*d^4*e*x+6*a^5*e^5+5*a^4*b*d*e^4+4*a^3*b^2*
d^2*e^3+3*a^2*b^3*d^3*e^2+2*a*b^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)/(e*x+d)^7/(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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.211163, size = 440, normalized size = 4.49 \[ -\frac{21 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 2 \, a b^{4} d^{4} e + 3 \, a^{2} b^{3} d^{3} e^{2} + 4 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} + 6 \, a^{5} e^{5} + 35 \,{\left (b^{5} d e^{4} + 2 \, a b^{4} e^{5}\right )} x^{4} + 35 \,{\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 21 \,{\left (b^{5} d^{3} e^{2} + 2 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} + 4 \, a^{3} b^{2} e^{5}\right )} x^{2} + 7 \,{\left (b^{5} d^{4} e + 2 \, a b^{4} d^{3} e^{2} + 3 \, a^{2} b^{3} d^{2} e^{3} + 4 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x}{42 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

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)^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*e^5*x^5 + b^5*d^5 + 2*a*b^4*d^4*e + 3*a^2*b^3*d^3*e^2 + 4*a^3*b^2*
d^2*e^3 + 5*a^4*b*d*e^4 + 6*a^5*e^5 + 35*(b^5*d*e^4 + 2*a*b^4*e^5)*x^4 + 35*(b^5
*d^2*e^3 + 2*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 21*(b^5*d^3*e^2 + 2*a*b^4*d^2*e^
3 + 3*a^2*b^3*d*e^4 + 4*a^3*b^2*e^5)*x^2 + 7*(b^5*d^4*e + 2*a*b^4*d^3*e^2 + 3*a^
2*b^3*d^2*e^3 + 4*a^3*b^2*d*e^4 + 5*a^4*b*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*
d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x +
d^7*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)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222948, size = 514, normalized size = 5.24 \[ -\frac{{\left (21 \, b^{5} x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 35 \, b^{5} d x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 35 \, b^{5} d^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 21 \, b^{5} d^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 7 \, b^{5} d^{4} x e{\rm sign}\left (b x + a\right ) + b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 70 \, a b^{4} x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 70 \, a b^{4} d x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 42 \, a b^{4} d^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 14 \, a b^{4} d^{3} x e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 105 \, a^{2} b^{3} x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 63 \, a^{2} b^{3} d x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 21 \, a^{2} b^{3} d^{2} x e^{3}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 84 \, a^{3} b^{2} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 28 \, a^{3} b^{2} d x e^{4}{\rm sign}\left (b x + a\right ) + 4 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 35 \, a^{4} b x e^{5}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 6 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{42 \,{\left (x e + d\right )}^{7}} \]

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)^8,x, algorithm="giac")

[Out]

-1/42*(21*b^5*x^5*e^5*sign(b*x + a) + 35*b^5*d*x^4*e^4*sign(b*x + a) + 35*b^5*d^
2*x^3*e^3*sign(b*x + a) + 21*b^5*d^3*x^2*e^2*sign(b*x + a) + 7*b^5*d^4*x*e*sign(
b*x + a) + b^5*d^5*sign(b*x + a) + 70*a*b^4*x^4*e^5*sign(b*x + a) + 70*a*b^4*d*x
^3*e^4*sign(b*x + a) + 42*a*b^4*d^2*x^2*e^3*sign(b*x + a) + 14*a*b^4*d^3*x*e^2*s
ign(b*x + a) + 2*a*b^4*d^4*e*sign(b*x + a) + 105*a^2*b^3*x^3*e^5*sign(b*x + a) +
63*a^2*b^3*d*x^2*e^4*sign(b*x + a) + 21*a^2*b^3*d^2*x*e^3*sign(b*x + a) + 3*a^2
*b^3*d^3*e^2*sign(b*x + a) + 84*a^3*b^2*x^2*e^5*sign(b*x + a) + 28*a^3*b^2*d*x*e
^4*sign(b*x + a) + 4*a^3*b^2*d^2*e^3*sign(b*x + a) + 35*a^4*b*x*e^5*sign(b*x + a
) + 5*a^4*b*d*e^4*sign(b*x + a) + 6*a^5*e^5*sign(b*x + a))*e^(-6)/(x*e + d)^7