3.2015 \(\int \frac{(a+b x) (d+e x)^4}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(a+b x) (d+e x)^5}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.131605, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(a+b x) (d+e x)^5}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(d + e*x)^4)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.6335, size = 34, normalized size = 0.87 \[ \frac{\left (a + b x\right ) \left (d + e x\right )^{5}}{5 e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

(a + b*x)*(d + e*x)**5/(5*e*sqrt(a**2 + 2*a*b*x + b**2*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.040442, size = 30, normalized size = 0.77 \[ \frac{(a+b x) (d+e x)^5}{5 e \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(d + e*x)^4)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.007, size = 58, normalized size = 1.5 \[{\frac{x \left ({e}^{4}{x}^{4}+5\,d{e}^{3}{x}^{3}+10\,{d}^{2}{e}^{2}{x}^{2}+10\,{d}^{3}ex+5\,{d}^{4} \right ) \left ( bx+a \right ) }{5}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^4/((b*x+a)^2)^(1/2),x)

[Out]

1/5*x*(e^4*x^4+5*d*e^3*x^3+10*d^2*e^2*x^2+10*d^3*e*x+5*d^4)*(b*x+a)/((b*x+a)^2)^
(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.704949, size = 1129, normalized size = 28.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^4/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

1/5*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^4*x^4/b - 77/30*a^5*e^4*log(x + a/b)/(b^2)^(
5/2) + 77/30*a^4*e^4*x/((b^2)^(3/2)*b) - 77/60*a^3*e^4*x^2/(sqrt(b^2)*b^2) - 9/2
0*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*e^4*x^3/b^2 + a*sqrt(b^(-2))*d^4*log(x + a/b)
+ 47/30*a^5*sqrt(b^(-2))*e^4*log(x + a/b)/b^4 + 47/60*sqrt(b^2*x^2 + 2*a*b*x + a
^2)*a^2*e^4*x^2/b^3 - 47/30*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4*e^4/b^5 + 13/6*(4*
b*d*e^3 + a*e^4)*a^4*log(x + a/b)/(b^2)^(5/2) - 10/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a
^3*b*log(x + a/b)/(b^2)^(5/2) + 2*(2*b*d^3*e + 3*a*d^2*e^2)*a^2*b^2*log(x + a/b)
/(b^2)^(5/2) + 10/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a^2*x/(b^2)^(3/2) - 13/6*(4*b*d*e^
3 + a*e^4)*a^3*x/((b^2)^(3/2)*b) - 2*(2*b*d^3*e + 3*a*d^2*e^2)*a*b*x/(b^2)^(3/2)
+ (2*b*d^3*e + 3*a*d^2*e^2)*x^2/sqrt(b^2) + 13/12*(4*b*d*e^3 + a*e^4)*a^2*x^2/(
sqrt(b^2)*b^2) - 5/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a*x^2/(sqrt(b^2)*b) + 1/4*(4*b*d*
e^3 + a*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^3/b^2 - 7/6*(4*b*d*e^3 + a*e^4)*a^4
*sqrt(b^(-2))*log(x + a/b)/b^4 + 4/3*(3*b*d^2*e^2 + 2*a*d*e^3)*a^3*sqrt(b^(-2))*
log(x + a/b)/b^3 - (b*d^4 + 4*a*d^3*e)*a*sqrt(b^(-2))*log(x + a/b)/b - 7/12*(4*b
*d*e^3 + a*e^4)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a*x^2/b^3 + 2/3*(3*b*d^2*e^2 + 2*a
*d*e^3)*sqrt(b^2*x^2 + 2*a*b*x + a^2)*x^2/b^2 + 7/6*(4*b*d*e^3 + a*e^4)*sqrt(b^2
*x^2 + 2*a*b*x + a^2)*a^3/b^5 - 4/3*(3*b*d^2*e^2 + 2*a*d*e^3)*sqrt(b^2*x^2 + 2*a
*b*x + a^2)*a^2/b^4 + (b*d^4 + 4*a*d^3*e)*sqrt(b^2*x^2 + 2*a*b*x + a^2)/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.281325, size = 57, normalized size = 1.46 \[ \frac{1}{5} \, e^{4} x^{5} + d e^{3} x^{4} + 2 \, d^{2} e^{2} x^{3} + 2 \, d^{3} e x^{2} + d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^4/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

1/5*e^4*x^5 + d*e^3*x^4 + 2*d^2*e^2*x^3 + 2*d^3*e*x^2 + d^4*x

_______________________________________________________________________________________

Sympy [A]  time = 0.23577, size = 42, normalized size = 1.08 \[ d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac{e^{4} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**4/((b*x+a)**2)**(1/2),x)

[Out]

d**4*x + 2*d**3*e*x**2 + 2*d**2*e**2*x**3 + d*e**3*x**4 + e**4*x**5/5

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.272906, size = 24, normalized size = 0.62 \[ \frac{1}{5} \,{\left (x e + d\right )}^{5} e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^4/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

1/5*(x*e + d)^5*e^(-1)*sign(b*x + a)