3.1717 \(\int \frac{1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=329 \[ \frac{315 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{315 \sqrt{b} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac{105 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{21 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

[Out]

(105*e^3)/(64*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(4*
(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e)/(8*
(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (21*e^2
)/(32*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (31
5*e^4*(a + b*x))/(64*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
- (315*Sqrt[b]*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
64*(b*d - a*e)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.631191, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{315 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{315 \sqrt{b} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac{105 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{21 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(105*e^3)/(64*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(4*
(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e)/(8*
(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (21*e^2
)/(32*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (31
5*e^4*(a + b*x))/(64*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
- (315*Sqrt[b]*e^4*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
64*(b*d - a*e)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

_______________________________________________________________________________________

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

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 1.10726, size = 181, normalized size = 0.55 \[ \frac{(a+b x)^5 \left (\frac{\sqrt{d+e x} \left (-\frac{82 b e^2 (b d-a e)}{(a+b x)^2}+\frac{40 b e (b d-a e)^2}{(a+b x)^3}-\frac{16 b (b d-a e)^3}{(a+b x)^4}+\frac{187 b e^3}{a+b x}+\frac{128 e^4}{d+e x}\right )}{(b d-a e)^5}-\frac{315 \sqrt{b} e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{11/2}}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((a + b*x)^5*((Sqrt[d + e*x]*((-16*b*(b*d - a*e)^3)/(a + b*x)^4 + (40*b*e*(b*d -
a*e)^2)/(a + b*x)^3 - (82*b*e^2*(b*d - a*e))/(a + b*x)^2 + (187*b*e^3)/(a + b*x
) + (128*e^4)/(d + e*x)))/(b*d - a*e)^5 - (315*Sqrt[b]*e^4*ArcTanh[(Sqrt[b]*Sqrt
[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(11/2)))/(64*((a + b*x)^2)^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 602, normalized size = 1.8 \[ -{\frac{bx+a}{64\, \left ( ae-bd \right ) ^{5}} \left ( 315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{4}{b}^{5}{e}^{4}+1260\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{3}a{b}^{4}{e}^{4}+315\,\sqrt{b \left ( ae-bd \right ) }{x}^{4}{b}^{4}{e}^{4}+1890\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}{a}^{2}{b}^{3}{e}^{4}+1155\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}a{b}^{3}{e}^{4}+105\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}{b}^{4}d{e}^{3}+1260\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}x{a}^{3}{b}^{2}{e}^{4}+1533\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{a}^{2}{b}^{2}{e}^{4}+399\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}a{b}^{3}d{e}^{3}-42\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{4}{d}^{2}{e}^{2}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{4}b{e}^{4}+837\,\sqrt{b \left ( ae-bd \right ) }x{a}^{3}b{e}^{4}+555\,\sqrt{b \left ( ae-bd \right ) }x{a}^{2}{b}^{2}d{e}^{3}-156\,\sqrt{b \left ( ae-bd \right ) }xa{b}^{3}{d}^{2}{e}^{2}+24\,\sqrt{b \left ( ae-bd \right ) }x{b}^{4}{d}^{3}e+128\,\sqrt{b \left ( ae-bd \right ) }{a}^{4}{e}^{4}+325\,\sqrt{b \left ( ae-bd \right ) }{a}^{3}bd{e}^{3}-210\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}{b}^{2}{d}^{2}{e}^{2}+88\,\sqrt{b \left ( ae-bd \right ) }a{b}^{3}{d}^{3}e-16\,\sqrt{b \left ( ae-bd \right ) }{b}^{4}{d}^{4} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

-1/64*(315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^4*b^5*e^4
+1260*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^3*a*b^4*e^4+31
5*(b*(a*e-b*d))^(1/2)*x^4*b^4*e^4+1890*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2
))*(e*x+d)^(1/2)*x^2*a^2*b^3*e^4+1155*(b*(a*e-b*d))^(1/2)*x^3*a*b^3*e^4+105*(b*(
a*e-b*d))^(1/2)*x^3*b^4*d*e^3+1260*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(
e*x+d)^(1/2)*x*a^3*b^2*e^4+1533*(b*(a*e-b*d))^(1/2)*x^2*a^2*b^2*e^4+399*(b*(a*e-
b*d))^(1/2)*x^2*a*b^3*d*e^3-42*(b*(a*e-b*d))^(1/2)*x^2*b^4*d^2*e^2+315*arctan((e
*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^4*b*e^4+837*(b*(a*e-b*d))^(1/
2)*x*a^3*b*e^4+555*(b*(a*e-b*d))^(1/2)*x*a^2*b^2*d*e^3-156*(b*(a*e-b*d))^(1/2)*x
*a*b^3*d^2*e^2+24*(b*(a*e-b*d))^(1/2)*x*b^4*d^3*e+128*(b*(a*e-b*d))^(1/2)*a^4*e^
4+325*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3-210*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2+88
*(b*(a*e-b*d))^(1/2)*a*b^3*d^3*e-16*(b*(a*e-b*d))^(1/2)*b^4*d^4)*(b*x+a)/(b*(a*e
-b*d))^(1/2)/(e*x+d)^(1/2)/(a*e-b*d)^5/((b*x+a)^2)^(5/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.253839, 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)^(3/2)),x, algorithm="fricas")

[Out]

[1/128*(630*b^4*e^4*x^4 - 32*b^4*d^4 + 176*a*b^3*d^3*e - 420*a^2*b^2*d^2*e^2 + 6
50*a^3*b*d*e^3 + 256*a^4*e^4 + 210*(b^4*d*e^3 + 11*a*b^3*e^4)*x^3 - 42*(2*b^4*d^
2*e^2 - 19*a*b^3*d*e^3 - 73*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^
3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*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)) + 6*(8*b^4*d^3*e - 52*a*b^3*d^2*e^2 + 185*a^2*b^2*d*e^3 + 279*a^3*b*e^
4)*x)/((a^4*b^5*d^5 - 5*a^5*b^4*d^4*e + 10*a^6*b^3*d^3*e^2 - 10*a^7*b^2*d^2*e^3
+ 5*a^8*b*d*e^4 - a^9*e^5 + (b^9*d^5 - 5*a*b^8*d^4*e + 10*a^2*b^7*d^3*e^2 - 10*a
^3*b^6*d^2*e^3 + 5*a^4*b^5*d*e^4 - a^5*b^4*e^5)*x^4 + 4*(a*b^8*d^5 - 5*a^2*b^7*d
^4*e + 10*a^3*b^6*d^3*e^2 - 10*a^4*b^5*d^2*e^3 + 5*a^5*b^4*d*e^4 - a^6*b^3*e^5)*
x^3 + 6*(a^2*b^7*d^5 - 5*a^3*b^6*d^4*e + 10*a^4*b^5*d^3*e^2 - 10*a^5*b^4*d^2*e^3
+ 5*a^6*b^3*d*e^4 - a^7*b^2*e^5)*x^2 + 4*(a^3*b^6*d^5 - 5*a^4*b^5*d^4*e + 10*a^
5*b^4*d^3*e^2 - 10*a^6*b^3*d^2*e^3 + 5*a^7*b^2*d*e^4 - a^8*b*e^5)*x)*sqrt(e*x +
d)), 1/64*(315*b^4*e^4*x^4 - 16*b^4*d^4 + 88*a*b^3*d^3*e - 210*a^2*b^2*d^2*e^2 +
325*a^3*b*d*e^3 + 128*a^4*e^4 + 105*(b^4*d*e^3 + 11*a*b^3*e^4)*x^3 - 21*(2*b^4*
d^2*e^2 - 19*a*b^3*d*e^3 - 73*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x^4 + 4*a*b^3*e^4*
x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*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)) + 3*(8*b^4*d^3
*e - 52*a*b^3*d^2*e^2 + 185*a^2*b^2*d*e^3 + 279*a^3*b*e^4)*x)/((a^4*b^5*d^5 - 5*
a^5*b^4*d^4*e + 10*a^6*b^3*d^3*e^2 - 10*a^7*b^2*d^2*e^3 + 5*a^8*b*d*e^4 - a^9*e^
5 + (b^9*d^5 - 5*a*b^8*d^4*e + 10*a^2*b^7*d^3*e^2 - 10*a^3*b^6*d^2*e^3 + 5*a^4*b
^5*d*e^4 - a^5*b^4*e^5)*x^4 + 4*(a*b^8*d^5 - 5*a^2*b^7*d^4*e + 10*a^3*b^6*d^3*e^
2 - 10*a^4*b^5*d^2*e^3 + 5*a^5*b^4*d*e^4 - a^6*b^3*e^5)*x^3 + 6*(a^2*b^7*d^5 - 5
*a^3*b^6*d^4*e + 10*a^4*b^5*d^3*e^2 - 10*a^5*b^4*d^2*e^3 + 5*a^6*b^3*d*e^4 - a^7
*b^2*e^5)*x^2 + 4*(a^3*b^6*d^5 - 5*a^4*b^5*d^4*e + 10*a^5*b^4*d^3*e^2 - 10*a^6*b
^3*d^2*e^3 + 5*a^7*b^2*d*e^4 - a^8*b*e^5)*x)*sqrt(e*x + d))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral(1/((d + e*x)**(3/2)*((a + b*x)**2)**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260311, 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)^(3/2)),x, algorithm="giac")

[Out]

Done