3.2017 \(\int \frac{(a+b x) (d+e x)^2}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(a+b x) (d+e x)^3}{3 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.129752, 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 \[ \frac{(a+b x) (d+e x)^3}{3 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.294, size = 34, normalized size = 0.87 \[ \frac{\left (a + b x\right ) \left (d + e x\right )^{3}}{3 e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)*(d + e*x)**3/(3*e*sqrt(a**2 + 2*a*b*x + b**2*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0216712, size = 30, normalized size = 0.77 \[ \frac{(a+b x) (d+e x)^3}{3 e \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 36, normalized size = 0.9 \[{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) \left ( bx+a \right ) }{3}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.694098, size = 392, normalized size = 10.05 \[ -\frac{5 \, a^{3} b^{2} e^{2} \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, a^{2} b e^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{5 \, a e^{2} x^{2}}{6 \, \sqrt{b^{2}}} + a \sqrt{\frac{1}{b^{2}}} d^{2} \log \left (x + \frac{a}{b}\right ) + \frac{2 \, a^{3} \sqrt{\frac{1}{b^{2}}} e^{2} \log \left (x + \frac{a}{b}\right )}{3 \, b^{2}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{2} x^{2}}{3 \, b} + \frac{{\left (2 \, b d e + a e^{2}\right )} a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{{\left (2 \, b d e + a e^{2}\right )} a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{{\left (2 \, b d e + a e^{2}\right )} x^{2}}{2 \, \sqrt{b^{2}}} - \frac{{\left (b d^{2} + 2 \, a d e\right )} a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} - \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e^{2}}{3 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}{\left (b d^{2} + 2 \, a d e\right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/3*a^3*b^2*e^2*log(x + a/b)/(b^2)^(5/2) + 5/3*a^2*b*e^2*x/(b^2)^(3/2) - 5/6*a*
e^2*x^2/sqrt(b^2) + a*sqrt(b^(-2))*d^2*log(x + a/b) + 2/3*a^3*sqrt(b^(-2))*e^2*l
og(x + a/b)/b^2 + 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*e^2*x^2/b + (2*b*d*e + a*e^2
)*a^2*b^2*log(x + a/b)/(b^2)^(5/2) - (2*b*d*e + a*e^2)*a*b*x/(b^2)^(3/2) + 1/2*(
2*b*d*e + a*e^2)*x^2/sqrt(b^2) - (b*d^2 + 2*a*d*e)*a*sqrt(b^(-2))*log(x + a/b)/b
- 2/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^2*e^2/b^3 + sqrt(b^2*x^2 + 2*a*b*x + a^2)
*(b*d^2 + 2*a*d*e)/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.272436, size = 27, normalized size = 0.69 \[ \frac{1}{3} \, e^{2} x^{3} + d e x^{2} + d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*e^2*x^3 + d*e*x^2 + d^2*x

_______________________________________________________________________________________

Sympy [A]  time = 0.187792, size = 19, normalized size = 0.49 \[ d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*x + d*e*x**2 + e**2*x**3/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278157, size = 24, normalized size = 0.62 \[ \frac{1}{3} \,{\left (x e + d\right )}^{3} e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x*e + d)^3*e^(-1)*sign(b*x + a)