3.1658 \(\int \frac{(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{5/2}}+\frac{3 e^4 \sqrt{d+e x}}{128 b^3 (a+b x) (b d-a e)^2}-\frac{e^3 \sqrt{d+e x}}{64 b^3 (a+b x)^2 (b d-a e)}-\frac{e^2 \sqrt{d+e x}}{16 b^3 (a+b x)^3}-\frac{e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{5/2}}{5 b (a+b x)^5} \]

[Out]

-(e^2*Sqrt[d + e*x])/(16*b^3*(a + b*x)^3) - (e^3*Sqrt[d + e*x])/(64*b^3*(b*d - a
*e)*(a + b*x)^2) + (3*e^4*Sqrt[d + e*x])/(128*b^3*(b*d - a*e)^2*(a + b*x)) - (e*
(d + e*x)^(3/2))/(8*b^2*(a + b*x)^4) - (d + e*x)^(5/2)/(5*b*(a + b*x)^5) - (3*e^
5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(7/2)*(b*d - a*e)^(5/
2))

_______________________________________________________________________________________

Rubi [A]  time = 0.331324, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{5/2}}+\frac{3 e^4 \sqrt{d+e x}}{128 b^3 (a+b x) (b d-a e)^2}-\frac{e^3 \sqrt{d+e x}}{64 b^3 (a+b x)^2 (b d-a e)}-\frac{e^2 \sqrt{d+e x}}{16 b^3 (a+b x)^3}-\frac{e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac{(d+e x)^{5/2}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(e^2*Sqrt[d + e*x])/(16*b^3*(a + b*x)^3) - (e^3*Sqrt[d + e*x])/(64*b^3*(b*d - a
*e)*(a + b*x)^2) + (3*e^4*Sqrt[d + e*x])/(128*b^3*(b*d - a*e)^2*(a + b*x)) - (e*
(d + e*x)^(3/2))/(8*b^2*(a + b*x)^4) - (d + e*x)^(5/2)/(5*b*(a + b*x)^5) - (3*e^
5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(7/2)*(b*d - a*e)^(5/
2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 86.0477, size = 173, normalized size = 0.87 \[ - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{e \left (d + e x\right )^{\frac{3}{2}}}{8 b^{2} \left (a + b x\right )^{4}} + \frac{3 e^{4} \sqrt{d + e x}}{128 b^{3} \left (a + b x\right ) \left (a e - b d\right )^{2}} + \frac{e^{3} \sqrt{d + e x}}{64 b^{3} \left (a + b x\right )^{2} \left (a e - b d\right )} - \frac{e^{2} \sqrt{d + e x}}{16 b^{3} \left (a + b x\right )^{3}} + \frac{3 e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{7}{2}} \left (a e - b d\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-(d + e*x)**(5/2)/(5*b*(a + b*x)**5) - e*(d + e*x)**(3/2)/(8*b**2*(a + b*x)**4)
+ 3*e**4*sqrt(d + e*x)/(128*b**3*(a + b*x)*(a*e - b*d)**2) + e**3*sqrt(d + e*x)/
(64*b**3*(a + b*x)**2*(a*e - b*d)) - e**2*sqrt(d + e*x)/(16*b**3*(a + b*x)**3) +
3*e**5*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*b**(7/2)*(a*e - b*d)**(
5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.365728, size = 171, normalized size = 0.86 \[ -\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} \left (10 e^3 (a+b x)^3 (b d-a e)+248 e^2 (a+b x)^2 (b d-a e)^2+336 e (a+b x) (b d-a e)^3+128 (b d-a e)^4-15 e^4 (a+b x)^4\right )}{640 b^3 (a+b x)^5 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(128*(b*d - a*e)^4 + 336*e*(b*d - a*e)^3*(a + b*x) + 248*e^2*(b*
d - a*e)^2*(a + b*x)^2 + 10*e^3*(b*d - a*e)*(a + b*x)^3 - 15*e^4*(a + b*x)^4))/(
640*b^3*(b*d - a*e)^2*(a + b*x)^5) - (3*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt
[b*d - a*e]])/(128*b^(7/2)*(b*d - a*e)^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 305, normalized size = 1.5 \[{\frac{3\,{e}^{5}b}{128\, \left ( bex+ae \right ) ^{5} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}}+{\frac{7\,{e}^{5}}{64\, \left ( bex+ae \right ) ^{5} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{5}}{5\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{e}^{6}a}{64\, \left ( bex+ae \right ) ^{5}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{e}^{5}d}{64\, \left ( bex+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{a}^{2}{e}^{7}}{128\, \left ( bex+ae \right ) ^{5}{b}^{3}}\sqrt{ex+d}}+{\frac{3\,{e}^{6}ad}{64\, \left ( bex+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}-{\frac{3\,{e}^{5}{d}^{2}}{128\, \left ( bex+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{3\,{e}^{5}}{128\,{b}^{3} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

3/128*e^5/(b*e*x+a*e)^5*b/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)+7/64*e^5/(b*
e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(7/2)-1/5*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)-7/64*e^
6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*a+7/64*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*d-3/1
28*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*a^2+3/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1
/2)*a*d-3/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*d^2+3/128*e^5/b^3/(a^2*e^2-2*a*b
*d*e+b^2*d^2)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))

_______________________________________________________________________________________

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((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229048, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(2*(15*b^4*e^4*x^4 - 128*b^4*d^4 + 176*a*b^3*d^3*e - 8*a^2*b^2*d^2*e^2 -
10*a^3*b*d*e^3 - 15*a^4*e^4 - 10*(b^4*d*e^3 - 7*a*b^3*e^4)*x^3 - 2*(124*b^4*d^2
*e^2 - 233*a*b^3*d*e^3 + 64*a^2*b^2*e^4)*x^2 - 2*(168*b^4*d^3*e - 256*a*b^3*d^2*
e^2 + 23*a^2*b^2*d*e^3 + 35*a^3*b*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 15
*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^
4*b*e^5*x + a^5*e^5)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d -
a*b*e)*sqrt(e*x + d))/(b*x + a)))/((a^5*b^5*d^2 - 2*a^6*b^4*d*e + a^7*b^3*e^2 +
(b^10*d^2 - 2*a*b^9*d*e + a^2*b^8*e^2)*x^5 + 5*(a*b^9*d^2 - 2*a^2*b^8*d*e + a^3
*b^7*e^2)*x^4 + 10*(a^2*b^8*d^2 - 2*a^3*b^7*d*e + a^4*b^6*e^2)*x^3 + 10*(a^3*b^7
*d^2 - 2*a^4*b^6*d*e + a^5*b^5*e^2)*x^2 + 5*(a^4*b^6*d^2 - 2*a^5*b^5*d*e + a^6*b
^4*e^2)*x)*sqrt(b^2*d - a*b*e)), 1/640*((15*b^4*e^4*x^4 - 128*b^4*d^4 + 176*a*b^
3*d^3*e - 8*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 - 15*a^4*e^4 - 10*(b^4*d*e^3 - 7*a*
b^3*e^4)*x^3 - 2*(124*b^4*d^2*e^2 - 233*a*b^3*d*e^3 + 64*a^2*b^2*e^4)*x^2 - 2*(1
68*b^4*d^3*e - 256*a*b^3*d^2*e^2 + 23*a^2*b^2*d*e^3 + 35*a^3*b*e^4)*x)*sqrt(-b^2
*d + a*b*e)*sqrt(e*x + d) - 15*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x
^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*arctan(-(b*d - a*e)/(sqrt(-b^
2*d + a*b*e)*sqrt(e*x + d))))/((a^5*b^5*d^2 - 2*a^6*b^4*d*e + a^7*b^3*e^2 + (b^1
0*d^2 - 2*a*b^9*d*e + a^2*b^8*e^2)*x^5 + 5*(a*b^9*d^2 - 2*a^2*b^8*d*e + a^3*b^7*
e^2)*x^4 + 10*(a^2*b^8*d^2 - 2*a^3*b^7*d*e + a^4*b^6*e^2)*x^3 + 10*(a^3*b^7*d^2
- 2*a^4*b^6*d*e + a^5*b^5*e^2)*x^2 + 5*(a^4*b^6*d^2 - 2*a^5*b^5*d*e + a^6*b^4*e^
2)*x)*sqrt(-b^2*d + a*b*e))]

_______________________________________________________________________________________

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((e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229508, size = 518, normalized size = 2.62 \[ \frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \sqrt{-b^{2} d + a b e}} + \frac{15 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 70 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} - 128 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} - 15 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 70 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} + 256 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} + 60 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} - 128 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} - 90 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} + 60 \, \sqrt{x e + d} a^{3} b d e^{8} - 15 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

3/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^5*d^2 - 2*a*b^4*d*e +
a^2*b^3*e^2)*sqrt(-b^2*d + a*b*e)) + 1/640*(15*(x*e + d)^(9/2)*b^4*e^5 - 70*(x*
e + d)^(7/2)*b^4*d*e^5 - 128*(x*e + d)^(5/2)*b^4*d^2*e^5 + 70*(x*e + d)^(3/2)*b^
4*d^3*e^5 - 15*sqrt(x*e + d)*b^4*d^4*e^5 + 70*(x*e + d)^(7/2)*a*b^3*e^6 + 256*(x
*e + d)^(5/2)*a*b^3*d*e^6 - 210*(x*e + d)^(3/2)*a*b^3*d^2*e^6 + 60*sqrt(x*e + d)
*a*b^3*d^3*e^6 - 128*(x*e + d)^(5/2)*a^2*b^2*e^7 + 210*(x*e + d)^(3/2)*a^2*b^2*d
*e^7 - 90*sqrt(x*e + d)*a^2*b^2*d^2*e^7 - 70*(x*e + d)^(3/2)*a^3*b*e^8 + 60*sqrt
(x*e + d)*a^3*b*d*e^8 - 15*sqrt(x*e + d)*a^4*e^9)/((b^5*d^2 - 2*a*b^4*d*e + a^2*
b^3*e^2)*((x*e + d)*b - b*d + a*e)^5)