3.2036 \(\int \frac{(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{(d+e x)^3}{3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.125242, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ -\frac{(d+e x)^3}{3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.2484, size = 34, normalized size = 0.83 \[ \frac{\left (d + e x\right )^{3}}{3 \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0648759, size = 60, normalized size = 1.46 \[ \frac{-a^2 e^2-a b e (d+3 e x)+b^2 \left (-\left (d^2+3 d e x+3 e^2 x^2\right )\right )}{3 b^3 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 69, normalized size = 1.7 \[ -{\frac{ \left ( bx+a \right ) ^{2} \left ( 3\,{x}^{2}{b}^{2}{e}^{2}+3\,xab{e}^{2}+3\,x{b}^{2}de+{a}^{2}{e}^{2}+abde+{b}^{2}{d}^{2} \right ) }{3\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.71498, size = 440, normalized size = 10.73 \[ -\frac{e^{2} x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b} - \frac{2 \, a^{2} e^{2}}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{3}} - \frac{a^{3} b^{2} e^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, a^{2} b e^{2}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{a e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{b d^{2} + 2 \, a d e}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{{\left (2 \, b d e + a e^{2}\right )} a^{2} b^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} - \frac{a d^{2}}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{a^{3} e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b^{2}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \,{\left (2 \, b d e + a e^{2}\right )} a b}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{2 \, b d e + a e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{{\left (b d^{2} + 2 \, a d e\right )} a}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e^2*x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b) - 2/3*a^2*e^2/((b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*b^3) - 1/4*a^3*b^2*e^2/((b^2)^(9/2)*(x + a/b)^4) + 2/3*a^2*b*e^2/((b
^2)^(7/2)*(x + a/b)^3) - 1/2*a*e^2/((b^2)^(5/2)*(x + a/b)^2) - 1/3*(b*d^2 + 2*a*
d*e)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 1/4*(2*b*d*e + a*e^2)*a^2*b^2/((b^2
)^(9/2)*(x + a/b)^4) - 1/4*a*d^2/((b^2)^(5/2)*(x + a/b)^4) + 1/2*a^3*e^2/((b^2)^
(5/2)*b^2*(x + a/b)^4) + 2/3*(2*b*d*e + a*e^2)*a*b/((b^2)^(7/2)*(x + a/b)^3) - 1
/2*(2*b*d*e + a*e^2)/((b^2)^(5/2)*(x + a/b)^2) + 1/4*(b*d^2 + 2*a*d*e)*a/((b^2)^
(5/2)*b*(x + a/b)^4)

_______________________________________________________________________________________

Fricas [A]  time = 0.280954, size = 113, normalized size = 2.76 \[ -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right ) \left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)