3.1662 \(\int \frac{1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=239 \[ -\frac{693 e^5}{128 \sqrt{d+e x} (b d-a e)^6}+\frac{693 \sqrt{b} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{13/2}}-\frac{231 e^4}{128 (a+b x) \sqrt{d+e x} (b d-a e)^5}+\frac{231 e^3}{320 (a+b x)^2 \sqrt{d+e x} (b d-a e)^4}-\frac{33 e^2}{80 (a+b x)^3 \sqrt{d+e x} (b d-a e)^3}+\frac{11 e}{40 (a+b x)^4 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 \sqrt{d+e x} (b d-a e)} \]

[Out]

(-693*e^5)/(128*(b*d - a*e)^6*Sqrt[d + e*x]) - 1/(5*(b*d - a*e)*(a + b*x)^5*Sqrt
[d + e*x]) + (11*e)/(40*(b*d - a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) - (33*e^2)/(80*
(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) + (231*e^3)/(320*(b*d - a*e)^4*(a + b*x
)^2*Sqrt[d + e*x]) - (231*e^4)/(128*(b*d - a*e)^5*(a + b*x)*Sqrt[d + e*x]) + (69
3*Sqrt[b]*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)
^(13/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.497189, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{693 e^5}{128 \sqrt{d+e x} (b d-a e)^6}+\frac{693 \sqrt{b} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{13/2}}-\frac{231 e^4}{128 (a+b x) \sqrt{d+e x} (b d-a e)^5}+\frac{231 e^3}{320 (a+b x)^2 \sqrt{d+e x} (b d-a e)^4}-\frac{33 e^2}{80 (a+b x)^3 \sqrt{d+e x} (b d-a e)^3}+\frac{11 e}{40 (a+b x)^4 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(-693*e^5)/(128*(b*d - a*e)^6*Sqrt[d + e*x]) - 1/(5*(b*d - a*e)*(a + b*x)^5*Sqrt
[d + e*x]) + (11*e)/(40*(b*d - a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) - (33*e^2)/(80*
(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) + (231*e^3)/(320*(b*d - a*e)^4*(a + b*x
)^2*Sqrt[d + e*x]) - (231*e^4)/(128*(b*d - a*e)^5*(a + b*x)*Sqrt[d + e*x]) + (69
3*Sqrt[b]*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)
^(13/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 133.08, size = 226, normalized size = 0.95 \[ - \frac{693 \sqrt{b} e^{5} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 \left (a e - b d\right )^{\frac{13}{2}}} - \frac{693 b e^{4} \sqrt{d + e x}}{128 \left (a + b x\right ) \left (a e - b d\right )^{6}} - \frac{231 b e^{3} \sqrt{d + e x}}{64 \left (a + b x\right )^{2} \left (a e - b d\right )^{5}} - \frac{231 b e^{2} \sqrt{d + e x}}{80 \left (a + b x\right )^{3} \left (a e - b d\right )^{4}} - \frac{99 b e \sqrt{d + e x}}{40 \left (a + b x\right )^{4} \left (a e - b d\right )^{3}} - \frac{11 b \sqrt{d + e x}}{5 \left (a + b x\right )^{5} \left (a e - b d\right )^{2}} - \frac{2}{\left (a + b x\right )^{5} \sqrt{d + e x} \left (a e - b d\right )} \]

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,x)

[Out]

-693*sqrt(b)*e**5*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*(a*e - b*d)**
(13/2)) - 693*b*e**4*sqrt(d + e*x)/(128*(a + b*x)*(a*e - b*d)**6) - 231*b*e**3*s
qrt(d + e*x)/(64*(a + b*x)**2*(a*e - b*d)**5) - 231*b*e**2*sqrt(d + e*x)/(80*(a
+ b*x)**3*(a*e - b*d)**4) - 99*b*e*sqrt(d + e*x)/(40*(a + b*x)**4*(a*e - b*d)**3
) - 11*b*sqrt(d + e*x)/(5*(a + b*x)**5*(a*e - b*d)**2) - 2/((a + b*x)**5*sqrt(d
+ e*x)*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.975891, size = 187, normalized size = 0.78 \[ \frac{1}{640} \left (\frac{3465 \sqrt{b} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{13/2}}-\frac{\sqrt{d+e x} \left (-\frac{1030 b e^3 (b d-a e)}{(a+b x)^2}+\frac{568 b e^2 (b d-a e)^2}{(a+b x)^3}-\frac{304 b e (b d-a e)^3}{(a+b x)^4}+\frac{128 b (b d-a e)^4}{(a+b x)^5}+\frac{2185 b e^4}{a+b x}+\frac{1280 e^5}{d+e x}\right )}{(b d-a e)^6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[d + e*x]*((128*b*(b*d - a*e)^4)/(a + b*x)^5 - (304*b*e*(b*d - a*e)^3)/(
a + b*x)^4 + (568*b*e^2*(b*d - a*e)^2)/(a + b*x)^3 - (1030*b*e^3*(b*d - a*e))/(a
+ b*x)^2 + (2185*b*e^4)/(a + b*x) + (1280*e^5)/(d + e*x)))/(b*d - a*e)^6) + (34
65*Sqrt[b]*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(13
/2))/640

_______________________________________________________________________________________

Maple [B]  time = 0.04, size = 641, normalized size = 2.7 \[ \text{result too large to display} \]

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,x)

[Out]

-2*e^5/(a*e-b*d)^6/(e*x+d)^(1/2)-437/128*e^5/(a*e-b*d)^6*b^5/(b*e*x+a*e)^5*(e*x+
d)^(9/2)-977/64*e^6/(a*e-b*d)^6*b^4/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a+977/64*e^5/(a*
e-b*d)^6*b^5/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d-131/5*e^7/(a*e-b*d)^6*b^3/(b*e*x+a*e)
^5*(e*x+d)^(5/2)*a^2+262/5*e^6/(a*e-b*d)^6*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d-1
31/5*e^5/(a*e-b*d)^6*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^2-1327/64*e^8/(a*e-b*d)^6
*b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3+3981/64*e^7/(a*e-b*d)^6*b^3/(b*e*x+a*e)^5*(
e*x+d)^(3/2)*a^2*d-3981/64*e^6/(a*e-b*d)^6*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^2
+1327/64*e^5/(a*e-b*d)^6*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^3-843/128*e^9/(a*e-b*
d)^6*b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4+843/32*e^8/(a*e-b*d)^6*b^2/(b*e*x+a*e)^5*
(e*x+d)^(1/2)*a^3*d-2529/64*e^7/(a*e-b*d)^6*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^2*
d^2+843/32*e^6/(a*e-b*d)^6*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3-843/128*e^5/(a*
e-b*d)^6*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^4-693/128*e^5/(a*e-b*d)^6*b/(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(1/((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.293032, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/1280*(6930*b^5*e^5*x^5 + 256*b^5*d^5 - 1632*a*b^4*d^4*e + 4496*a^2*b^3*d^3*e
^2 - 7180*a^3*b^2*d^2*e^3 + 8430*a^4*b*d*e^4 + 2560*a^5*e^5 + 2310*(b^5*d*e^4 +
14*a*b^4*e^5)*x^4 - 924*(b^5*d^2*e^3 - 12*a*b^4*d*e^4 - 64*a^2*b^3*e^5)*x^3 + 13
2*(4*b^5*d^3*e^2 - 33*a*b^4*d^2*e^3 + 159*a^2*b^3*d*e^4 + 395*a^3*b^2*e^5)*x^2 -
3465*(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)*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)) - 22*(16*b^5*
d^4*e - 112*a*b^4*d^3*e^2 + 366*a^2*b^3*d^2*e^3 - 880*a^3*b^2*d*e^4 - 965*a^4*b*
e^5)*x)/((a^5*b^6*d^6 - 6*a^6*b^5*d^5*e + 15*a^7*b^4*d^4*e^2 - 20*a^8*b^3*d^3*e^
3 + 15*a^9*b^2*d^2*e^4 - 6*a^10*b*d*e^5 + a^11*e^6 + (b^11*d^6 - 6*a*b^10*d^5*e
+ 15*a^2*b^9*d^4*e^2 - 20*a^3*b^8*d^3*e^3 + 15*a^4*b^7*d^2*e^4 - 6*a^5*b^6*d*e^5
+ a^6*b^5*e^6)*x^5 + 5*(a*b^10*d^6 - 6*a^2*b^9*d^5*e + 15*a^3*b^8*d^4*e^2 - 20*
a^4*b^7*d^3*e^3 + 15*a^5*b^6*d^2*e^4 - 6*a^6*b^5*d*e^5 + a^7*b^4*e^6)*x^4 + 10*(
a^2*b^9*d^6 - 6*a^3*b^8*d^5*e + 15*a^4*b^7*d^4*e^2 - 20*a^5*b^6*d^3*e^3 + 15*a^6
*b^5*d^2*e^4 - 6*a^7*b^4*d*e^5 + a^8*b^3*e^6)*x^3 + 10*(a^3*b^8*d^6 - 6*a^4*b^7*
d^5*e + 15*a^5*b^6*d^4*e^2 - 20*a^6*b^5*d^3*e^3 + 15*a^7*b^4*d^2*e^4 - 6*a^8*b^3
*d*e^5 + a^9*b^2*e^6)*x^2 + 5*(a^4*b^7*d^6 - 6*a^5*b^6*d^5*e + 15*a^6*b^5*d^4*e^
2 - 20*a^7*b^4*d^3*e^3 + 15*a^8*b^3*d^2*e^4 - 6*a^9*b^2*d*e^5 + a^10*b*e^6)*x)*s
qrt(e*x + d)), -1/640*(3465*b^5*e^5*x^5 + 128*b^5*d^5 - 816*a*b^4*d^4*e + 2248*a
^2*b^3*d^3*e^2 - 3590*a^3*b^2*d^2*e^3 + 4215*a^4*b*d*e^4 + 1280*a^5*e^5 + 1155*(
b^5*d*e^4 + 14*a*b^4*e^5)*x^4 - 462*(b^5*d^2*e^3 - 12*a*b^4*d*e^4 - 64*a^2*b^3*e
^5)*x^3 + 66*(4*b^5*d^3*e^2 - 33*a*b^4*d^2*e^3 + 159*a^2*b^3*d*e^4 + 395*a^3*b^2
*e^5)*x^2 - 3465*(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)*sqrt(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)) - 11*(16*b^5*d^4*e - 112*a*b
^4*d^3*e^2 + 366*a^2*b^3*d^2*e^3 - 880*a^3*b^2*d*e^4 - 965*a^4*b*e^5)*x)/((a^5*b
^6*d^6 - 6*a^6*b^5*d^5*e + 15*a^7*b^4*d^4*e^2 - 20*a^8*b^3*d^3*e^3 + 15*a^9*b^2*
d^2*e^4 - 6*a^10*b*d*e^5 + a^11*e^6 + (b^11*d^6 - 6*a*b^10*d^5*e + 15*a^2*b^9*d^
4*e^2 - 20*a^3*b^8*d^3*e^3 + 15*a^4*b^7*d^2*e^4 - 6*a^5*b^6*d*e^5 + a^6*b^5*e^6)
*x^5 + 5*(a*b^10*d^6 - 6*a^2*b^9*d^5*e + 15*a^3*b^8*d^4*e^2 - 20*a^4*b^7*d^3*e^3
+ 15*a^5*b^6*d^2*e^4 - 6*a^6*b^5*d*e^5 + a^7*b^4*e^6)*x^4 + 10*(a^2*b^9*d^6 - 6
*a^3*b^8*d^5*e + 15*a^4*b^7*d^4*e^2 - 20*a^5*b^6*d^3*e^3 + 15*a^6*b^5*d^2*e^4 -
6*a^7*b^4*d*e^5 + a^8*b^3*e^6)*x^3 + 10*(a^3*b^8*d^6 - 6*a^4*b^7*d^5*e + 15*a^5*
b^6*d^4*e^2 - 20*a^6*b^5*d^3*e^3 + 15*a^7*b^4*d^2*e^4 - 6*a^8*b^3*d*e^5 + a^9*b^
2*e^6)*x^2 + 5*(a^4*b^7*d^6 - 6*a^5*b^6*d^5*e + 15*a^6*b^5*d^4*e^2 - 20*a^7*b^4*
d^3*e^3 + 15*a^8*b^3*d^2*e^4 - 6*a^9*b^2*d*e^5 + a^10*b*e^6)*x)*sqrt(e*x + d))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{6} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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,x)

[Out]

Integral(1/((a + b*x)**6*(d + e*x)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.226401, size = 771, normalized size = 3.23 \[ -\frac{693 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \, e^{5}}{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt{x e + d}} - \frac{2185 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} e^{5} - 9770 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d e^{5} + 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{2} e^{5} - 13270 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{3} e^{5} + 4215 \, \sqrt{x e + d} b^{5} d^{4} e^{5} + 9770 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} e^{6} - 33536 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d e^{6} + 39810 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{2} e^{6} - 16860 \, \sqrt{x e + d} a b^{4} d^{3} e^{6} + 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} e^{7} - 39810 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d e^{7} + 25290 \, \sqrt{x e + d} a^{2} b^{3} d^{2} e^{7} + 13270 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} e^{8} - 16860 \, \sqrt{x e + d} a^{3} b^{2} d e^{8} + 4215 \, \sqrt{x e + d} a^{4} b e^{9}}{640 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-693/128*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^6*d^6 - 6*a*b^5*
d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d
*e^5 + a^6*e^6)*sqrt(-b^2*d + a*b*e)) - 2*e^5/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2
*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6
)*sqrt(x*e + d)) - 1/640*(2185*(x*e + d)^(9/2)*b^5*e^5 - 9770*(x*e + d)^(7/2)*b^
5*d*e^5 + 16768*(x*e + d)^(5/2)*b^5*d^2*e^5 - 13270*(x*e + d)^(3/2)*b^5*d^3*e^5
+ 4215*sqrt(x*e + d)*b^5*d^4*e^5 + 9770*(x*e + d)^(7/2)*a*b^4*e^6 - 33536*(x*e +
d)^(5/2)*a*b^4*d*e^6 + 39810*(x*e + d)^(3/2)*a*b^4*d^2*e^6 - 16860*sqrt(x*e + d
)*a*b^4*d^3*e^6 + 16768*(x*e + d)^(5/2)*a^2*b^3*e^7 - 39810*(x*e + d)^(3/2)*a^2*
b^3*d*e^7 + 25290*sqrt(x*e + d)*a^2*b^3*d^2*e^7 + 13270*(x*e + d)^(3/2)*a^3*b^2*
e^8 - 16860*sqrt(x*e + d)*a^3*b^2*d*e^8 + 4215*sqrt(x*e + d)*a^4*b*e^9)/((b^6*d^
6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4
- 6*a^5*b*d*e^5 + a^6*e^6)*((x*e + d)*b - b*d + a*e)^5)