3.1719 \(\int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=435 \[ \frac{3003 b^2 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^7}+\frac{1001 b e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac{3003 e^4 (a+b x)}{320 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac{429 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac{143 e^2}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac{13 e}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}} \]

[Out]

(429*e^3)/(64*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(
4*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (13*e
)/(24*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(143*e^2)/(96*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) + (3003*e^4*(a + b*x))/(320*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (1001*b*e^4*(a + b*x))/(64*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) + (3003*b^2*e^4*(a + b*x))/(64*(b*d - a*e)^7*Sqrt[d + e
*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3003*b^(5/2)*e^4*(a + b*x)*ArcTanh[(Sqrt[b
]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d - a*e)^(15/2)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])

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Rubi [A]  time = 0.892805, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3003 b^2 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^7}+\frac{1001 b e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac{3003 e^4 (a+b x)}{320 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac{429 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac{143 e^2}{96 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac{13 e}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(429*e^3)/(64*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(
4*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (13*e
)/(24*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(143*e^2)/(96*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) + (3003*e^4*(a + b*x))/(320*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) + (1001*b*e^4*(a + b*x))/(64*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]) + (3003*b^2*e^4*(a + b*x))/(64*(b*d - a*e)^7*Sqrt[d + e
*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3003*b^(5/2)*e^4*(a + b*x)*ArcTanh[(Sqrt[b
]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d - a*e)^(15/2)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

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)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 2.27047, size = 238, normalized size = 0.55 \[ \frac{(a+b x)^5 \left (\frac{\sqrt{d+e x} \left (-\frac{4430 b^3 e^2 (b d-a e)}{(a+b x)^2}+\frac{1240 b^3 e (b d-a e)^2}{(a+b x)^3}-\frac{240 b^3 (b d-a e)^3}{(a+b x)^4}+\frac{16245 b^3 e^3}{a+b x}+\frac{3200 b e^4 (b d-a e)}{(d+e x)^2}+\frac{384 e^4 (b d-a e)^2}{(d+e x)^3}+\frac{28800 b^2 e^4}{d+e x}\right )}{15 (b d-a e)^7}-\frac{3003 b^{5/2} e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{15/2}}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((a + b*x)^5*((Sqrt[d + e*x]*((-240*b^3*(b*d - a*e)^3)/(a + b*x)^4 + (1240*b^3*e
*(b*d - a*e)^2)/(a + b*x)^3 - (4430*b^3*e^2*(b*d - a*e))/(a + b*x)^2 + (16245*b^
3*e^3)/(a + b*x) + (384*e^4*(b*d - a*e)^2)/(d + e*x)^3 + (3200*b*e^4*(b*d - a*e)
)/(d + e*x)^2 + (28800*b^2*e^4)/(d + e*x)))/(15*(b*d - a*e)^7) - (3003*b^(5/2)*e
^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(15/2)))/(64*((
a + b*x)^2)^(5/2))

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Maple [B]  time = 0.041, size = 951, normalized size = 2.2 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

-1/960*(22155*(b*(a*e-b*d))^(1/2)*a^3*b^3*d^3*e^3-7630*(b*(a*e-b*d))^(1/2)*a^2*b
^4*d^4*e^2+1960*(b*(a*e-b*d))^(1/2)*a*b^5*d^5*e+384*(b*(a*e-b*d))^(1/2)*a^6*e^6-
240*(b*(a*e-b*d))^(1/2)*b^6*d^6+45045*(b*(a*e-b*d))^(1/2)*x^6*b^6*e^6-3968*(b*(a
*e-b*d))^(1/2)*a^5*b*d*e^5+32384*(b*(a*e-b*d))^(1/2)*a^4*b^2*d^2*e^4-1664*(b*(a*
e-b*d))^(1/2)*x*a^5*b*e^6+387387*(b*(a*e-b*d))^(1/2)*x^4*a*b^5*d*e^5+517803*(b*(
a*e-b*d))^(1/2)*x^3*a^2*b^4*d*e^5+256971*(b*(a*e-b*d))^(1/2)*x^3*a*b^5*d^2*e^4+2
85857*(b*(a*e-b*d))^(1/2)*x^2*a^3*b^3*d*e^5+347919*(b*(a*e-b*d))^(1/2)*x^2*a^2*b
^4*d^2*e^4+44928*(b*(a*e-b*d))^(1/2)*x*a^4*b^2*d*e^5+196001*(b*(a*e-b*d))^(1/2)*
x*a^3*b^3*d^2*e^4+180180*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/
2)*x^3*a*b^6*e^4+25025*(b*(a*e-b*d))^(1/2)*x^2*a*b^5*d^3*e^3+35945*(b*(a*e-b*d))
^(1/2)*x*a^2*b^4*d^3*e^3-5460*(b*(a*e-b*d))^(1/2)*x*a*b^5*d^4*e^2+270270*arctan(
(e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*x^2*a^2*b^5*e^4+180180*arctan
((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*x*a^3*b^4*e^4+45045*arctan((
e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*x^4*b^7*e^4+165165*(b*(a*e-b*d
))^(1/2)*x^5*a*b^5*e^6+105105*(b*(a*e-b*d))^(1/2)*x^5*b^6*d*e^5+6435*(b*(a*e-b*d
))^(1/2)*x^3*b^6*d^3*e^3-1430*(b*(a*e-b*d))^(1/2)*x^2*b^6*d^4*e^2+520*(b*(a*e-b*
d))^(1/2)*x*b^6*d^5*e+219219*(b*(a*e-b*d))^(1/2)*x^4*a^2*b^4*e^6+69069*(b*(a*e-b
*d))^(1/2)*x^4*b^6*d^2*e^4+45045*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*
x+d)^(5/2)*a^4*b^3*e^4+119691*(b*(a*e-b*d))^(1/2)*x^3*a^3*b^3*e^6+18304*(b*(a*e-
b*d))^(1/2)*x^2*a^4*b^2*e^6)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(5/2)/(a*e-b*d)
^7/((b*x+a)^2)^(5/2)

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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)^(5/2)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278079, 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)^(5/2)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/1920*(90090*b^6*e^6*x^6 - 480*b^6*d^6 + 3920*a*b^5*d^5*e - 15260*a^2*b^4*d^4*
e^2 + 44310*a^3*b^3*d^3*e^3 + 64768*a^4*b^2*d^2*e^4 - 7936*a^5*b*d*e^5 + 768*a^6
*e^6 + 30030*(7*b^6*d*e^5 + 11*a*b^5*e^6)*x^5 + 6006*(23*b^6*d^2*e^4 + 129*a*b^5
*d*e^5 + 73*a^2*b^4*e^6)*x^4 + 858*(15*b^6*d^3*e^3 + 599*a*b^5*d^2*e^4 + 1207*a^
2*b^4*d*e^5 + 279*a^3*b^3*e^6)*x^3 - 286*(10*b^6*d^4*e^2 - 175*a*b^5*d^3*e^3 - 2
433*a^2*b^4*d^2*e^4 - 1999*a^3*b^3*d*e^5 - 128*a^4*b^2*e^6)*x^2 - 45045*(b^6*e^6
*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*a*b^
5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)
*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^
2*e^4 + a^4*b^2*d*e^5)*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)) + 26*(40*b^6*
d^5*e - 420*a*b^5*d^4*e^2 + 2765*a^2*b^4*d^3*e^3 + 15077*a^3*b^3*d^2*e^4 + 3456*
a^4*b^2*d*e^5 - 128*a^5*b*e^6)*x)/((a^4*b^7*d^9 - 7*a^5*b^6*d^8*e + 21*a^6*b^5*d
^7*e^2 - 35*a^7*b^4*d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 + 7*a^10*b
*d^3*e^6 - a^11*d^2*e^7 + (b^11*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9*d^5*e^4
- 35*a^3*b^8*d^4*e^5 + 35*a^4*b^7*d^3*e^6 - 21*a^5*b^6*d^2*e^7 + 7*a^6*b^5*d*e^8
- a^7*b^4*e^9)*x^6 + 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^2*b^9*d^6*e^3 + 7*a
^3*b^8*d^5*e^4 - 35*a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d^2*e^7 +
13*a^7*b^4*d*e^8 - 2*a^8*b^3*e^9)*x^5 + (b^11*d^9 + a*b^10*d^8*e - 29*a^2*b^9*d^
7*e^2 + 91*a^3*b^8*d^6*e^3 - 119*a^4*b^7*d^5*e^4 + 49*a^5*b^6*d^4*e^5 + 49*a^6*b
^5*d^3*e^6 - 71*a^7*b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4 + 4*(a*b
^10*d^9 - 4*a^2*b^9*d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a^5*b^6*d^
5*e^4 + 49*a^6*b^5*d^4*e^5 - 21*a^7*b^4*d^3*e^6 - a^8*b^3*d^2*e^7 + 4*a^9*b^2*d*
e^8 - a^10*b*e^9)*x^3 + (6*a^2*b^9*d^9 - 34*a^3*b^8*d^8*e + 71*a^4*b^7*d^7*e^2 -
49*a^5*b^6*d^6*e^3 - 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8*b^3*d^3*
e^6 + 29*a^9*b^2*d^2*e^7 - a^10*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*d^9 - 13*
a^4*b^7*d^8*e + 35*a^5*b^6*d^7*e^2 - 49*a^6*b^5*d^6*e^3 + 35*a^7*b^4*d^5*e^4 - 7
*a^8*b^3*d^4*e^5 - 7*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^11*d*e^8)*x)*sqrt(e*
x + d)), 1/960*(45045*b^6*e^6*x^6 - 240*b^6*d^6 + 1960*a*b^5*d^5*e - 7630*a^2*b^
4*d^4*e^2 + 22155*a^3*b^3*d^3*e^3 + 32384*a^4*b^2*d^2*e^4 - 3968*a^5*b*d*e^5 + 3
84*a^6*e^6 + 15015*(7*b^6*d*e^5 + 11*a*b^5*e^6)*x^5 + 3003*(23*b^6*d^2*e^4 + 129
*a*b^5*d*e^5 + 73*a^2*b^4*e^6)*x^4 + 429*(15*b^6*d^3*e^3 + 599*a*b^5*d^2*e^4 + 1
207*a^2*b^4*d*e^5 + 279*a^3*b^3*e^6)*x^3 - 143*(10*b^6*d^4*e^2 - 175*a*b^5*d^3*e
^3 - 2433*a^2*b^4*d^2*e^4 - 1999*a^3*b^3*d*e^5 - 128*a^4*b^2*e^6)*x^2 - 45045*(b
^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 +
8*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^
3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*
b^3*d^2*e^4 + a^4*b^2*d*e^5)*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)) + 13*(40*b^6*d^5*e - 420*a*b^5*d^
4*e^2 + 2765*a^2*b^4*d^3*e^3 + 15077*a^3*b^3*d^2*e^4 + 3456*a^4*b^2*d*e^5 - 128*
a^5*b*e^6)*x)/((a^4*b^7*d^9 - 7*a^5*b^6*d^8*e + 21*a^6*b^5*d^7*e^2 - 35*a^7*b^4*
d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 + 7*a^10*b*d^3*e^6 - a^11*d^2*
e^7 + (b^11*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9*d^5*e^4 - 35*a^3*b^8*d^4*e^5
+ 35*a^4*b^7*d^3*e^6 - 21*a^5*b^6*d^2*e^7 + 7*a^6*b^5*d*e^8 - a^7*b^4*e^9)*x^6
+ 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^2*b^9*d^6*e^3 + 7*a^3*b^8*d^5*e^4 - 35*
a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d^2*e^7 + 13*a^7*b^4*d*e^8 - 2
*a^8*b^3*e^9)*x^5 + (b^11*d^9 + a*b^10*d^8*e - 29*a^2*b^9*d^7*e^2 + 91*a^3*b^8*d
^6*e^3 - 119*a^4*b^7*d^5*e^4 + 49*a^5*b^6*d^4*e^5 + 49*a^6*b^5*d^3*e^6 - 71*a^7*
b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4 + 4*(a*b^10*d^9 - 4*a^2*b^9*
d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a^5*b^6*d^5*e^4 + 49*a^6*b^5*d
^4*e^5 - 21*a^7*b^4*d^3*e^6 - a^8*b^3*d^2*e^7 + 4*a^9*b^2*d*e^8 - a^10*b*e^9)*x^
3 + (6*a^2*b^9*d^9 - 34*a^3*b^8*d^8*e + 71*a^4*b^7*d^7*e^2 - 49*a^5*b^6*d^6*e^3
- 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8*b^3*d^3*e^6 + 29*a^9*b^2*d^2
*e^7 - a^10*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*d^9 - 13*a^4*b^7*d^8*e + 35*a
^5*b^6*d^7*e^2 - 49*a^6*b^5*d^6*e^3 + 35*a^7*b^4*d^5*e^4 - 7*a^8*b^3*d^4*e^5 - 7
*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^11*d*e^8)*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28403, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

Done