3.1660 \(\int \frac{\sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=218 \[ -\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac{7 e^4 \sqrt{d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac{7 e^3 \sqrt{d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac{7 e^2 \sqrt{d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac{e \sqrt{d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac{\sqrt{d+e x}}{5 b (a+b x)^5} \]

[Out]

-Sqrt[d + e*x]/(5*b*(a + b*x)^5) - (e*Sqrt[d + e*x])/(40*b*(b*d - a*e)*(a + b*x)
^4) + (7*e^2*Sqrt[d + e*x])/(240*b*(b*d - a*e)^2*(a + b*x)^3) - (7*e^3*Sqrt[d +
e*x])/(192*b*(b*d - a*e)^3*(a + b*x)^2) + (7*e^4*Sqrt[d + e*x])/(128*b*(b*d - a*
e)^4*(a + b*x)) - (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*
b^(3/2)*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.329045, antiderivative size = 218, 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 \[ -\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}+\frac{7 e^4 \sqrt{d+e x}}{128 b (a+b x) (b d-a e)^4}-\frac{7 e^3 \sqrt{d+e x}}{192 b (a+b x)^2 (b d-a e)^3}+\frac{7 e^2 \sqrt{d+e x}}{240 b (a+b x)^3 (b d-a e)^2}-\frac{e \sqrt{d+e x}}{40 b (a+b x)^4 (b d-a e)}-\frac{\sqrt{d+e x}}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[d + e*x]/(5*b*(a + b*x)^5) - (e*Sqrt[d + e*x])/(40*b*(b*d - a*e)*(a + b*x)
^4) + (7*e^2*Sqrt[d + e*x])/(240*b*(b*d - a*e)^2*(a + b*x)^3) - (7*e^3*Sqrt[d +
e*x])/(192*b*(b*d - a*e)^3*(a + b*x)^2) + (7*e^4*Sqrt[d + e*x])/(128*b*(b*d - a*
e)^4*(a + b*x)) - (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*
b^(3/2)*(b*d - a*e)^(9/2))

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Rubi in Sympy [A]  time = 104.035, size = 187, normalized size = 0.86 \[ \frac{7 e^{4} \sqrt{d + e x}}{128 b \left (a + b x\right ) \left (a e - b d\right )^{4}} + \frac{7 e^{3} \sqrt{d + e x}}{192 b \left (a + b x\right )^{2} \left (a e - b d\right )^{3}} + \frac{7 e^{2} \sqrt{d + e x}}{240 b \left (a + b x\right )^{3} \left (a e - b d\right )^{2}} + \frac{e \sqrt{d + e x}}{40 b \left (a + b x\right )^{4} \left (a e - b d\right )} - \frac{\sqrt{d + e x}}{5 b \left (a + b x\right )^{5}} + \frac{7 e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{3}{2}} \left (a e - b d\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*e**4*sqrt(d + e*x)/(128*b*(a + b*x)*(a*e - b*d)**4) + 7*e**3*sqrt(d + e*x)/(19
2*b*(a + b*x)**2*(a*e - b*d)**3) + 7*e**2*sqrt(d + e*x)/(240*b*(a + b*x)**3*(a*e
- b*d)**2) + e*sqrt(d + e*x)/(40*b*(a + b*x)**4*(a*e - b*d)) - sqrt(d + e*x)/(5
*b*(a + b*x)**5) + 7*e**5*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*b**(3
/2)*(a*e - b*d)**(9/2))

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Mathematica [A]  time = 0.363565, size = 171, normalized size = 0.78 \[ -\frac{7 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{9/2}}-\frac{\sqrt{d+e x} \left (70 e^3 (a+b x)^3 (b d-a e)-56 e^2 (a+b x)^2 (b d-a e)^2+48 e (a+b x) (b d-a e)^3+384 (b d-a e)^4-105 e^4 (a+b x)^4\right )}{1920 b (a+b x)^5 (b d-a e)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(384*(b*d - a*e)^4 + 48*e*(b*d - a*e)^3*(a + b*x) - 56*e^2*(b*d
- a*e)^2*(a + b*x)^2 + 70*e^3*(b*d - a*e)*(a + b*x)^3 - 105*e^4*(a + b*x)^4))/(1
920*b*(b*d - a*e)^4*(a + b*x)^5) - (7*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e]])/(128*b^(3/2)*(b*d - a*e)^(9/2))

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Maple [A]  time = 0.026, size = 337, normalized size = 1.6 \[{\frac{7\,{e}^{5}{b}^{3}}{128\, \left ( bex+ae \right ) ^{5} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}}+{\frac{49\,{e}^{5}{b}^{2}}{192\, \left ( bex+ae \right ) ^{5} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{e}^{5}b}{15\, \left ( bex+ae \right ) ^{5} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{79\,{e}^{5}}{192\, \left ( bex+ae \right ) ^{5} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{5}}{128\, \left ( bex+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{7\,{e}^{5}}{128\,b \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/128*e^5/(b*e*x+a*e)^5*b^3/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3
*e+b^4*d^4)*(e*x+d)^(9/2)+49/192*e^5/(b*e*x+a*e)^5*b^2/(a^3*e^3-3*a^2*b*d*e^2+3*
a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(7/2)+7/15*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)^(5/2)+79/192*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(3/2)-7/128*e^5
/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)+7/128*e^5/b/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*
e^2-4*a*b^3*d^3*e+b^4*d^4)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*
d))^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232411, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(2*(105*b^4*e^4*x^4 - 384*b^4*d^4 + 1488*a*b^3*d^3*e - 2104*a^2*b^2*d^2*
e^2 + 1210*a^3*b*d*e^3 - 105*a^4*e^4 - 70*(b^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 14*(4*
b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 64*a^2*b^2*e^4)*x^2 - 2*(24*b^4*d^3*e - 128*a*b^3
*d^2*e^2 + 289*a^2*b^2*d*e^3 - 395*a^3*b*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x +
d) + 105*(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^4 - 4*a^6*b^4*d^3*e + 6*a
^7*b^3*d^2*e^2 - 4*a^8*b^2*d*e^3 + a^9*b*e^4 + (b^10*d^4 - 4*a*b^9*d^3*e + 6*a^2
*b^8*d^2*e^2 - 4*a^3*b^7*d*e^3 + a^4*b^6*e^4)*x^5 + 5*(a*b^9*d^4 - 4*a^2*b^8*d^3
*e + 6*a^3*b^7*d^2*e^2 - 4*a^4*b^6*d*e^3 + a^5*b^5*e^4)*x^4 + 10*(a^2*b^8*d^4 -
4*a^3*b^7*d^3*e + 6*a^4*b^6*d^2*e^2 - 4*a^5*b^5*d*e^3 + a^6*b^4*e^4)*x^3 + 10*(a
^3*b^7*d^4 - 4*a^4*b^6*d^3*e + 6*a^5*b^5*d^2*e^2 - 4*a^6*b^4*d*e^3 + a^7*b^3*e^4
)*x^2 + 5*(a^4*b^6*d^4 - 4*a^5*b^5*d^3*e + 6*a^6*b^4*d^2*e^2 - 4*a^7*b^3*d*e^3 +
a^8*b^2*e^4)*x)*sqrt(b^2*d - a*b*e)), 1/1920*((105*b^4*e^4*x^4 - 384*b^4*d^4 +
1488*a*b^3*d^3*e - 2104*a^2*b^2*d^2*e^2 + 1210*a^3*b*d*e^3 - 105*a^4*e^4 - 70*(b
^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 14*(4*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 64*a^2*b^2*e^
4)*x^2 - 2*(24*b^4*d^3*e - 128*a*b^3*d^2*e^2 + 289*a^2*b^2*d*e^3 - 395*a^3*b*e^4
)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 105*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 1
0*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^4 - 4*a^6*b^4*d^3*e + 6
*a^7*b^3*d^2*e^2 - 4*a^8*b^2*d*e^3 + a^9*b*e^4 + (b^10*d^4 - 4*a*b^9*d^3*e + 6*a
^2*b^8*d^2*e^2 - 4*a^3*b^7*d*e^3 + a^4*b^6*e^4)*x^5 + 5*(a*b^9*d^4 - 4*a^2*b^8*d
^3*e + 6*a^3*b^7*d^2*e^2 - 4*a^4*b^6*d*e^3 + a^5*b^5*e^4)*x^4 + 10*(a^2*b^8*d^4
- 4*a^3*b^7*d^3*e + 6*a^4*b^6*d^2*e^2 - 4*a^5*b^5*d*e^3 + a^6*b^4*e^4)*x^3 + 10*
(a^3*b^7*d^4 - 4*a^4*b^6*d^3*e + 6*a^5*b^5*d^2*e^2 - 4*a^6*b^4*d*e^3 + a^7*b^3*e
^4)*x^2 + 5*(a^4*b^6*d^4 - 4*a^5*b^5*d^3*e + 6*a^6*b^4*d^2*e^2 - 4*a^7*b^3*d*e^3
+ a^8*b^2*e^4)*x)*sqrt(-b^2*d + a*b*e))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224185, size = 583, normalized size = 2.67 \[ \frac{7 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{105 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 490 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 896 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} - 790 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 490 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 1792 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} + 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 896 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} - 2370 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} + 790 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} + 420 \, \sqrt{x e + d} a^{3} b d e^{8} - 105 \, \sqrt{x e + d} a^{4} e^{9}}{1920 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/128*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^5*d^4 - 4*a*b^4*d^3*e
+ 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*sqrt(-b^2*d + a*b*e)) + 1/19
20*(105*(x*e + d)^(9/2)*b^4*e^5 - 490*(x*e + d)^(7/2)*b^4*d*e^5 + 896*(x*e + d)^
(5/2)*b^4*d^2*e^5 - 790*(x*e + d)^(3/2)*b^4*d^3*e^5 - 105*sqrt(x*e + d)*b^4*d^4*
e^5 + 490*(x*e + d)^(7/2)*a*b^3*e^6 - 1792*(x*e + d)^(5/2)*a*b^3*d*e^6 + 2370*(x
*e + d)^(3/2)*a*b^3*d^2*e^6 + 420*sqrt(x*e + d)*a*b^3*d^3*e^6 + 896*(x*e + d)^(5
/2)*a^2*b^2*e^7 - 2370*(x*e + d)^(3/2)*a^2*b^2*d*e^7 - 630*sqrt(x*e + d)*a^2*b^2
*d^2*e^7 + 790*(x*e + d)^(3/2)*a^3*b*e^8 + 420*sqrt(x*e + d)*a^3*b*d*e^8 - 105*s
qrt(x*e + d)*a^4*e^9)/((b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*
d*e^3 + a^4*b*e^4)*((x*e + d)*b - b*d + a*e)^5)