3.2097 \(\int \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^3 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x) \sqrt{d+e x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.212565, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^3 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.4299, size = 122, normalized size = 0.82 \[ \frac{8 b \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{2}} + \frac{16 b \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{3} \left (a + b x\right )} - \frac{2 \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**(3/2),x)

[Out]

8*b*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**2) + 16*b*sqrt(d + e*x)
*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**3*(a + b*x)) - 2*(a + b*x)*s
qrt(a**2 + 2*a*b*x + b**2*x**2)/(e*sqrt(d + e*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0873054, size = 78, normalized size = 0.53 \[ -\frac{2 \sqrt{(a+b x)^2} \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{3 e^3 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(3*a^2*e^2 - 6*a*b*e*(2*d + e*x) + b^2*(8*d^2 + 4*d*e*x -
e^2*x^2)))/(3*e^3*(a + b*x)*Sqrt[d + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 79, normalized size = 0.5 \[ -{\frac{-2\,{x}^{2}{b}^{2}{e}^{2}-12\,xab{e}^{2}+8\,x{b}^{2}de+6\,{a}^{2}{e}^{2}-24\,abde+16\,{b}^{2}{d}^{2}}{3\, \left ( bx+a \right ){e}^{3}}\sqrt{ \left ( bx+a \right ) ^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x)

[Out]

-2/3/(e*x+d)^(1/2)*(-b^2*e^2*x^2-6*a*b*e^2*x+4*b^2*d*e*x+3*a^2*e^2-12*a*b*d*e+8*
b^2*d^2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

_______________________________________________________________________________________

Maxima [A]  time = 0.723293, size = 101, normalized size = 0.68 \[ \frac{2 \,{\left (b e x + 2 \, b d - a e\right )} a}{\sqrt{e x + d} e^{2}} + \frac{2 \,{\left (b e^{2} x^{2} - 8 \, b d^{2} + 6 \, a d e -{\left (4 \, b d e - 3 \, a e^{2}\right )} x\right )} b}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b*e*x + 2*b*d - a*e)*a/(sqrt(e*x + d)*e^2) + 2/3*(b*e^2*x^2 - 8*b*d^2 + 6*a*d
*e - (4*b*d*e - 3*a*e^2)*x)*b/(sqrt(e*x + d)*e^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.275487, size = 85, normalized size = 0.57 \[ \frac{2 \,{\left (b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 12 \, a b d e - 3 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 3 \, a b e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^2*e^2*x^2 - 8*b^2*d^2 + 12*a*b*d*e - 3*a^2*e^2 - 2*(2*b^2*d*e - 3*a*b*e^2
)*x)/(sqrt(e*x + d)*e^3)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.313282, size = 161, normalized size = 1.09 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{6}{\rm sign}\left (b x + a\right ) - 6 \, \sqrt{x e + d} b^{2} d e^{6}{\rm sign}\left (b x + a\right ) + 6 \, \sqrt{x e + d} a b e^{7}{\rm sign}\left (b x + a\right )\right )} e^{\left (-9\right )} - \frac{2 \,{\left (b^{2} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b d e{\rm sign}\left (b x + a\right ) + a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((x*e + d)^(3/2)*b^2*e^6*sign(b*x + a) - 6*sqrt(x*e + d)*b^2*d*e^6*sign(b*x
+ a) + 6*sqrt(x*e + d)*a*b*e^7*sign(b*x + a))*e^(-9) - 2*(b^2*d^2*sign(b*x + a)
- 2*a*b*d*e*sign(b*x + a) + a^2*e^2*sign(b*x + a))*e^(-3)/sqrt(x*e + d)