3.2076 \(\int \frac{(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=119 \[ -\frac{15 e^2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2}}-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{15 e^2 \sqrt{d+e x}}{4 b^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.142669, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{15 e^2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2}}-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{15 e^2 \sqrt{d+e x}}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.5674, size = 105, normalized size = 0.88 \[ - \frac{\left (d + e x\right )^{\frac{5}{2}}}{2 b \left (a + b x\right )^{2}} - \frac{5 e \left (d + e x\right )^{\frac{3}{2}}}{4 b^{2} \left (a + b x\right )} + \frac{15 e^{2} \sqrt{d + e x}}{4 b^{3}} - \frac{15 e^{2} \sqrt{a e - b d} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**(5/2)/(2*b*(a + b*x)**2) - 5*e*(d + e*x)**(3/2)/(4*b**2*(a + b*x)) +
15*e**2*sqrt(d + e*x)/(4*b**3) - 15*e**2*sqrt(a*e - b*d)*atan(sqrt(b)*sqrt(d +
e*x)/sqrt(a*e - b*d))/(4*b**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.257058, size = 119, normalized size = 1. \[ \sqrt{d+e x} \left (-\frac{9 e (b d-a e)}{4 b^3 (a+b x)}-\frac{(b d-a e)^2}{2 b^3 (a+b x)^2}+\frac{2 e^2}{b^3}\right )-\frac{15 e^2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.025, size = 238, normalized size = 2. \[ 2\,{\frac{{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+{\frac{9\,a{e}^{3}}{4\,{b}^{2} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,d{e}^{2}}{4\,b \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{a}^{2}{e}^{4}}{4\,{b}^{3} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{7\,ad{e}^{3}}{2\,{b}^{2} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{7\,{d}^{2}{e}^{2}}{4\,b \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,a{e}^{3}}{4\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{15\,d{e}^{2}}{4\,{b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.305415, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (8 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} - 5 \, a b d e + 15 \, a^{2} e^{2} -{\left (9 \, b^{2} d e - 25 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (8 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} - 5 \, a b d e + 15 \, a^{2} e^{2} -{\left (9 \, b^{2} d e - 25 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.343009, size = 235, normalized size = 1.97 \[ \frac{15 \,{\left (b d e^{2} - a e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{3}} + \frac{2 \, \sqrt{x e + d} e^{2}}{b^{3}} - \frac{9 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{2} - 7 \, \sqrt{x e + d} b^{2} d^{2} e^{2} - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{3} + 14 \, \sqrt{x e + d} a b d e^{3} - 7 \, \sqrt{x e + d} a^{2} e^{4}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/4*(b*d*e^2 - a*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d
+ a*b*e)*b^3) + 2*sqrt(x*e + d)*e^2/b^3 - 1/4*(9*(x*e + d)^(3/2)*b^2*d*e^2 - 7*
sqrt(x*e + d)*b^2*d^2*e^2 - 9*(x*e + d)^(3/2)*a*b*e^3 + 14*sqrt(x*e + d)*a*b*d*e
^3 - 7*sqrt(x*e + d)*a^2*e^4)/(((x*e + d)*b - b*d + a*e)^2*b^3)