3.1706 \(\int \frac{1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=223 \[ \frac{15 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{15 \sqrt{b} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}+\frac{5 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

[Out]

(5*e)/(4*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(2*(b*d
- a*e)*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*e^2*(a + b*x
))/(4*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*Sqrt[b]*e
^2*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(7
/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.353665, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{15 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{15 \sqrt{b} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}+\frac{5 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(5*e)/(4*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(2*(b*d
- a*e)*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*e^2*(a + b*x
))/(4*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*Sqrt[b]*e
^2*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(7
/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

_______________________________________________________________________________________

Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.46585, size = 144, normalized size = 0.65 \[ \frac{(a+b x)^3 \left (\frac{8 a^2 e^2+a b e (9 d+25 e x)+b^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )}{(a+b x)^2 \sqrt{d+e x} (b d-a e)^3}-\frac{15 \sqrt{b} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^3*((8*a^2*e^2 + a*b*e*(9*d + 25*e*x) + b^2*(-2*d^2 + 5*d*e*x + 15*e^2
*x^2))/((b*d - a*e)^3*(a + b*x)^2*Sqrt[d + e*x]) - (15*Sqrt[b]*e^2*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)))/(4*((a + b*x)^2)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.03, size = 285, normalized size = 1.3 \[ -{\frac{bx+a}{4\, \left ( ae-bd \right ) ^{3}} \left ( 15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}{b}^{3}{e}^{2}+30\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}xa{b}^{2}{e}^{2}+15\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}{e}^{2}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{2}b{e}^{2}+25\,\sqrt{b \left ( ae-bd \right ) }xab{e}^{2}+5\,\sqrt{b \left ( ae-bd \right ) }x{b}^{2}de+8\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}{e}^{2}+9\,\sqrt{b \left ( ae-bd \right ) }abde-2\,\sqrt{b \left ( ae-bd \right ) }{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^2*b^3*e^2+3
0*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x*a*b^2*e^2+15*(b*(a
*e-b*d))^(1/2)*x^2*b^2*e^2+15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d
)^(1/2)*a^2*b*e^2+25*(b*(a*e-b*d))^(1/2)*x*a*b*e^2+5*(b*(a*e-b*d))^(1/2)*x*b^2*d
*e+8*(b*(a*e-b*d))^(1/2)*a^2*e^2+9*(b*(a*e-b*d))^(1/2)*a*b*d*e-2*(b*(a*e-b*d))^(
1/2)*b^2*d^2)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(1/2)/(a*e-b*d)^3/((b*x+a)^2)^
(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.247927, size = 1, normalized size = 0. \[ \left [\frac{30 \, b^{2} e^{2} x^{2} - 4 \, b^{2} d^{2} + 18 \, a b d e + 16 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 10 \,{\left (b^{2} d e + 5 \, a b e^{2}\right )} x}{8 \,{\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} +{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )} \sqrt{e x + d}}, \frac{15 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} + 9 \, a b d e + 8 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 5 \,{\left (b^{2} d e + 5 \, a b e^{2}\right )} x}{4 \,{\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} +{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/8*(30*b^2*e^2*x^2 - 4*b^2*d^2 + 18*a*b*d*e + 16*a^2*e^2 - 15*(b^2*e^2*x^2 + 2
*a*b*e^2*x + a^2*e^2)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e
+ 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 10*(b^2*d*e + 5
*a*b*e^2)*x)/((a^2*b^3*d^3 - 3*a^3*b^2*d^2*e + 3*a^4*b*d*e^2 - a^5*e^3 + (b^5*d^
3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*x^2 + 2*(a*b^4*d^3 - 3*a^2*b^
3*d^2*e + 3*a^3*b^2*d*e^2 - a^4*b*e^3)*x)*sqrt(e*x + d)), 1/4*(15*b^2*e^2*x^2 -
2*b^2*d^2 + 9*a*b*d*e + 8*a^2*e^2 - 15*(b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*sqr
t(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e
*x + d)*b)) + 5*(b^2*d*e + 5*a*b*e^2)*x)/((a^2*b^3*d^3 - 3*a^3*b^2*d^2*e + 3*a^4
*b*d*e^2 - a^5*e^3 + (b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*x
^2 + 2*(a*b^4*d^3 - 3*a^2*b^3*d^2*e + 3*a^3*b^2*d*e^2 - a^4*b*e^3)*x)*sqrt(e*x +
d))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.241532, size = 690, normalized size = 3.09 \[ -\frac{15 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{3} d^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \, e^{2}}{{\left (b^{3} d^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{x e + d}} - \frac{7 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{2} - 9 \, \sqrt{x e + d} b^{2} d e^{2} + 9 \, \sqrt{x e + d} a b e^{3}}{4 \,{\left (b^{3} d^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

-15/4*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((b^3*d^3*sign(-(x*e +
d)*b*e + b*d*e - a*e^2) - 3*a*b^2*d^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 3
*a^2*b*d*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a^3*e^3*sign(-(x*e + d)*b*e
+ b*d*e - a*e^2))*sqrt(-b^2*d + a*b*e)) - 2*e^2/((b^3*d^3*sign(-(x*e + d)*b*e +
b*d*e - a*e^2) - 3*a*b^2*d^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 3*a^2*b*d*
e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a^3*e^3*sign(-(x*e + d)*b*e + b*d*e -
a*e^2))*sqrt(x*e + d)) - 1/4*(7*(x*e + d)^(3/2)*b^2*e^2 - 9*sqrt(x*e + d)*b^2*d
*e^2 + 9*sqrt(x*e + d)*a*b*e^3)/((b^3*d^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2) -
3*a*b^2*d^2*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 3*a^2*b*d*e^2*sign(-(x*e +
d)*b*e + b*d*e - a*e^2) - a^3*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e +
d)*b - b*d + a*e)^2)