3.1652 \(\int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=229 \[ \frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(-231*e^3)/(40*(b*d - a*e)^4*(d + e*x)^(5/2)) - 1/(3*(b*d - a*e)*(a + b*x)^3*(d
+ e*x)^(5/2)) + (11*e)/(12*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) - (33*e^2)
/(8*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) - (77*b*e^3)/(8*(b*d - a*e)^5*(d +
e*x)^(3/2)) - (231*b^2*e^3)/(8*(b*d - a*e)^6*Sqrt[d + e*x]) + (231*b^(5/2)*e^3*A
rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(13/2))

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Rubi [A]  time = 0.542479, antiderivative size = 229, 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 \[ \frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{231 b^2 e^3}{8 \sqrt{d+e x} (b d-a e)^6}-\frac{77 b e^3}{8 (d+e x)^{3/2} (b d-a e)^5}-\frac{231 e^3}{40 (d+e x)^{5/2} (b d-a e)^4}-\frac{33 e^2}{8 (a+b x) (d+e x)^{5/2} (b d-a e)^3}+\frac{11 e}{12 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(-231*e^3)/(40*(b*d - a*e)^4*(d + e*x)^(5/2)) - 1/(3*(b*d - a*e)*(a + b*x)^3*(d
+ e*x)^(5/2)) + (11*e)/(12*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) - (33*e^2)
/(8*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) - (77*b*e^3)/(8*(b*d - a*e)^5*(d +
e*x)^(3/2)) - (231*b^2*e^3)/(8*(b*d - a*e)^6*Sqrt[d + e*x]) + (231*b^(5/2)*e^3*A
rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(13/2))

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Rubi in Sympy [A]  time = 124.054, size = 226, normalized size = 0.99 \[ - \frac{231 b^{\frac{5}{2}} e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \left (a e - b d\right )^{\frac{13}{2}}} - \frac{231 b^{3} e^{2} \sqrt{d + e x}}{8 \left (a + b x\right ) \left (a e - b d\right )^{6}} - \frac{77 b^{3} e \sqrt{d + e x}}{4 \left (a + b x\right )^{2} \left (a e - b d\right )^{5}} - \frac{77 b^{3} \sqrt{d + e x}}{5 \left (a + b x\right )^{3} \left (a e - b d\right )^{4}} - \frac{66 b^{2}}{5 \left (a + b x\right )^{3} \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{22 b}{15 \left (a + b x\right )^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2}{5 \left (a + b x\right )^{3} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-231*b**(5/2)*e**3*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*(a*e - b*d)**(
13/2)) - 231*b**3*e**2*sqrt(d + e*x)/(8*(a + b*x)*(a*e - b*d)**6) - 77*b**3*e*sq
rt(d + e*x)/(4*(a + b*x)**2*(a*e - b*d)**5) - 77*b**3*sqrt(d + e*x)/(5*(a + b*x)
**3*(a*e - b*d)**4) - 66*b**2/(5*(a + b*x)**3*sqrt(d + e*x)*(a*e - b*d)**3) + 22
*b/(15*(a + b*x)**3*(d + e*x)**(3/2)*(a*e - b*d)**2) - 2/(5*(a + b*x)**3*(d + e*
x)**(5/2)*(a*e - b*d))

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Mathematica [A]  time = 0.97706, size = 193, normalized size = 0.84 \[ \frac{231 b^{5/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{\sqrt{d+e x} \left (-\frac{230 b^3 e (b d-a e)}{(a+b x)^2}+\frac{40 b^3 (b d-a e)^2}{(a+b x)^3}+\frac{1065 b^3 e^2}{a+b x}+\frac{320 b e^3 (b d-a e)}{(d+e x)^2}+\frac{48 e^3 (b d-a e)^2}{(d+e x)^3}+\frac{2400 b^2 e^3}{d+e x}\right )}{120 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-(Sqrt[d + e*x]*((40*b^3*(b*d - a*e)^2)/(a + b*x)^3 - (230*b^3*e*(b*d - a*e))/(a
+ b*x)^2 + (1065*b^3*e^2)/(a + b*x) + (48*e^3*(b*d - a*e)^2)/(d + e*x)^3 + (320
*b*e^3*(b*d - a*e))/(d + e*x)^2 + (2400*b^2*e^3)/(d + e*x)))/(120*(b*d - a*e)^6)
+ (231*b^(5/2)*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d -
a*e)^(13/2))

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Maple [A]  time = 0.035, size = 344, normalized size = 1.5 \[ -{\frac{2\,{e}^{3}}{5\, \left ( ae-bd \right ) ^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-20\,{\frac{{e}^{3}{b}^{2}}{ \left ( ae-bd \right ) ^{6}\sqrt{ex+d}}}+{\frac{8\,{e}^{3}b}{3\, \left ( ae-bd \right ) ^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{71\,{e}^{3}{b}^{5}}{8\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{59\,{b}^{4}{e}^{4}a}{3\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{59\,{e}^{3}{b}^{5}d}{3\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{89\,{e}^{5}{b}^{3}{a}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{89\,{b}^{4}{e}^{4}ad}{4\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{89\,{e}^{3}{b}^{5}{d}^{2}}{8\, \left ( ae-bd \right ) ^{6} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{231\,{b}^{3}{e}^{3}}{8\, \left ( ae-bd \right ) ^{6}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-2/5*e^3/(a*e-b*d)^4/(e*x+d)^(5/2)-20*e^3/(a*e-b*d)^6*b^2/(e*x+d)^(1/2)+8/3*e^3/
(a*e-b*d)^5*b/(e*x+d)^(3/2)-71/8*e^3/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(5/2)
-59/3*e^4/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*(e*x+d)^(3/2)*a+59/3*e^3/(a*e-b*d)^6*b^5
/(b*e*x+a*e)^3*(e*x+d)^(3/2)*d-89/8*e^5/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*(e*x+d)^(1
/2)*a^2+89/4*e^4/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a*d-89/8*e^3/(a*e-b
*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*d^2-231/8*e^3/(a*e-b*d)^6*b^3/(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(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251486, 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)^2*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[-1/240*(6930*b^5*e^5*x^5 + 80*b^5*d^5 - 620*a*b^4*d^4*e + 2670*a^2*b^3*d^3*e^2
+ 5536*a^3*b^2*d^2*e^3 - 832*a^4*b*d*e^4 + 96*a^5*e^5 + 2310*(7*b^5*d*e^4 + 8*a*
b^4*e^5)*x^4 + 462*(23*b^5*d^2*e^3 + 94*a*b^4*d*e^4 + 33*a^2*b^3*e^5)*x^3 + 198*
(5*b^5*d^3*e^2 + 146*a*b^4*d^2*e^3 + 183*a^2*b^3*d*e^4 + 16*a^3*b^2*e^5)*x^2 - 3
465*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^5*d^2*
e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 +
a^3*b^2*e^5)*x^2 + (3*a^2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*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*(10*b^5*d^4*e - 130*a*b^4*d^3*e^2 - 1119*a^2*b^3*d^2*e
^3 - 352*a^3*b^2*d*e^4 + 16*a^4*b*e^5)*x)/((a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a
^5*b^4*d^6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9
*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*d^5*e^3 + 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*
e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^7 + a^6*b^3*e^8)*x^5 + (2*b^9*d^7*e - 9
*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3*e^5 + 3
3*a^5*b^4*d^2*e^6 - 16*a^6*b^3*d*e^7 + 3*a^7*b^2*e^8)*x^4 + (b^9*d^8 - 18*a^2*b^
7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 60*a^4*b^5*d^4*e^4 + 24*a^5*b^4*d^3*e^5 + 10*a^
6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*b*e^8)*x^3 + (3*a*b^8*d^8 - 12*a^2*b^7*
d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d^5*e^3 - 60*a^5*b^4*d^4*e^4 + 52*a^6*b^
3*d^3*e^5 - 18*a^7*b^2*d^2*e^6 + a^9*e^8)*x^2 + (3*a^2*b^7*d^8 - 16*a^3*b^6*d^7*
e + 33*a^4*b^5*d^6*e^2 - 30*a^5*b^4*d^5*e^3 + 5*a^6*b^3*d^4*e^4 + 12*a^7*b^2*d^3
*e^5 - 9*a^8*b*d^2*e^6 + 2*a^9*d*e^7)*x)*sqrt(e*x + d)), -1/120*(3465*b^5*e^5*x^
5 + 40*b^5*d^5 - 310*a*b^4*d^4*e + 1335*a^2*b^3*d^3*e^2 + 2768*a^3*b^2*d^2*e^3 -
416*a^4*b*d*e^4 + 48*a^5*e^5 + 1155*(7*b^5*d*e^4 + 8*a*b^4*e^5)*x^4 + 231*(23*b
^5*d^2*e^3 + 94*a*b^4*d*e^4 + 33*a^2*b^3*e^5)*x^3 + 99*(5*b^5*d^3*e^2 + 146*a*b^
4*d^2*e^3 + 183*a^2*b^3*d*e^4 + 16*a^3*b^2*e^5)*x^2 - 3465*(b^5*e^5*x^5 + a^3*b^
2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a
^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^2
*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*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*(10*b^5*d^4*e - 130*a*b^4
*d^3*e^2 - 1119*a^2*b^3*d^2*e^3 - 352*a^3*b^2*d*e^4 + 16*a^4*b*e^5)*x)/((a^3*b^6
*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*a^7*b^2*d^
4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*d^5*e^3 + 15*a^2*
b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^7 + a^6*b^
3*e^8)*x^5 + (2*b^9*d^7*e - 9*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a^3*b^6*d^4
*e^4 - 30*a^4*b^5*d^3*e^5 + 33*a^5*b^4*d^2*e^6 - 16*a^6*b^3*d*e^7 + 3*a^7*b^2*e^
8)*x^4 + (b^9*d^8 - 18*a^2*b^7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 60*a^4*b^5*d^4*e^4
+ 24*a^5*b^4*d^3*e^5 + 10*a^6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*b*e^8)*x^3
+ (3*a*b^8*d^8 - 12*a^2*b^7*d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d^5*e^3 - 6
0*a^5*b^4*d^4*e^4 + 52*a^6*b^3*d^3*e^5 - 18*a^7*b^2*d^2*e^6 + a^9*e^8)*x^2 + (3*
a^2*b^7*d^8 - 16*a^3*b^6*d^7*e + 33*a^4*b^5*d^6*e^2 - 30*a^5*b^4*d^5*e^3 + 5*a^6
*b^3*d^4*e^4 + 12*a^7*b^2*d^3*e^5 - 9*a^8*b*d^2*e^6 + 2*a^9*d*e^7)*x)*sqrt(e*x +
d))]

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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(1/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221782, size = 635, normalized size = 2.77 \[ -\frac{231 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} b^{2} e^{3} + 20 \,{\left (x e + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3} - 20 \,{\left (x e + d\right )} a b e^{4} - 6 \, a b d e^{4} + 3 \, a^{2} e^{5}\right )}}{15 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} - \frac{213 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{3} - 472 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{3} + 267 \, \sqrt{x e + d} b^{5} d^{2} e^{3} + 472 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} e^{4} - 534 \, \sqrt{x e + d} a b^{4} d e^{4} + 267 \, \sqrt{x e + d} a^{2} b^{3} e^{5}}{24 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

-231/8*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((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)) - 2/15*(150*(x*e + d)^2*b^2*e^3 + 20*(x*e
+ d)*b^2*d*e^3 + 3*b^2*d^2*e^3 - 20*(x*e + d)*a*b*e^4 - 6*a*b*d*e^4 + 3*a^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)*(x*e + d)^(5/2)) - 1/24*(213*(x*e + d)^(5/2
)*b^5*e^3 - 472*(x*e + d)^(3/2)*b^5*d*e^3 + 267*sqrt(x*e + d)*b^5*d^2*e^3 + 472*
(x*e + d)^(3/2)*a*b^4*e^4 - 534*sqrt(x*e + d)*a*b^4*d*e^4 + 267*sqrt(x*e + d)*a^
2*b^3*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)*((x*e + d)*b - b*d + a*e)^3)