3.2028 \(\int \frac{(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{e (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{d+e x}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.108439, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{e (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{d+e x}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.0373, size = 63, normalized size = 0.94 \[ - \frac{d + e x}{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{e \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0340996, size = 36, normalized size = 0.54 \[ \frac{e (a+b x) \log (a+b x)+a e-b d}{b^2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 48, normalized size = 0.7 \[{\frac{ \left ( \ln \left ( bx+a \right ) xbe+\ln \left ( bx+a \right ) ae+ae-bd \right ) \left ( bx+a \right ) ^{2}}{{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.698073, size = 185, normalized size = 2.76 \[ \frac{b e \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, a^{2} b^{3} e}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a b^{2} e x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{b d + a e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{a d}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{{\left (b d + a e\right )} a}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*e*log(x + a/b)/(b^2)^(3/2) + 3/2*a^2*b^3*e/((b^2)^(7/2)*(x + a/b)^2) + 2*a*b^2
*e*x/((b^2)^(5/2)*(x + a/b)^2) - (b*d + a*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2)
- 1/2*a*d/((b^2)^(3/2)*(x + a/b)^2) + 1/2*a*(b*d + a*e)/((b^2)^(3/2)*b*(x + a/b
)^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.280407, size = 53, normalized size = 0.79 \[ -\frac{b d - a e -{\left (b e x + a e\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right ) \left (d + e x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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