3.1700 \(\int \frac{(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=308 \[ -\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{63 e^2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(63*e^2*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x])/(4*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (21*e^2*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))/(4*b^4*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (63*e^2*(a + b*x)*(d + e*x)^(5/2))/(20*b^3*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - (9*e*(d + e*x)^(7/2))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x
)^(9/2)/(2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (63*e^2*(b*d - a*e)^(5/2
)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(11/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.506933, antiderivative size = 308, 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 \[ -\frac{(d+e x)^{9/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{9 e (d+e x)^{7/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{63 e^2 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) \sqrt{d+e x} (b d-a e)^2}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(63*e^2*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x])/(4*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (21*e^2*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))/(4*b^4*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (63*e^2*(a + b*x)*(d + e*x)^(5/2))/(20*b^3*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - (9*e*(d + e*x)^(7/2))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x
)^(9/2)/(2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (63*e^2*(b*d - a*e)^(5/2
)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(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((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.691905, size = 184, normalized size = 0.6 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (8 e^2 \left (30 a^2 e^2-65 a b d e+36 b^2 d^2\right )+8 b e^3 x (7 b d-5 a e)+\frac{85 e (a e-b d)^3}{a+b x}-\frac{10 (b d-a e)^4}{(a+b x)^2}+8 b^2 e^4 x^2\right )}{5 b^5}-\frac{63 e^2 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(9/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((a + b*x)^3*((Sqrt[d + e*x]*(8*e^2*(36*b^2*d^2 - 65*a*b*d*e + 30*a^2*e^2) + 8*b
*e^3*(7*b*d - 5*a*e)*x + 8*b^2*e^4*x^2 - (10*(b*d - a*e)^4)/(a + b*x)^2 + (85*e*
(-(b*d) + a*e)^3)/(a + b*x)))/(5*b^5) - (63*e^2*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)))/(4*((a + b*x)^2)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 1115, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/20*(80*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a*b^3*d*e^2-480*(b*(a*e-b*d))^(1/2)
*(e*x+d)^(1/2)*x^2*a*b^3*d*e^3-960*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^2*b^2*d
*e^3+480*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a*b^3*d^2*e^2-630*arctan((e*x+d)^(1
/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b*e^5+45*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*
b*e^3+945*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*b*d*e^4-945*arctan((e*
x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b^2*d^2*e^3+315*arctan((e*x+d)^(1/2)*b/(b*
(a*e-b*d))^(1/2))*a^2*b^3*d^3*e^2+8*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*x^2*b^4*e^
2-315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^2*e^5+315*arctan((e*
x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^5*d^3*e^2+8*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(5/2)*a^2*b^2*e^2+16*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*x*a*b^3*e^2-40*(b*(a*e-b*
d))^(1/2)*(e*x+d)^(3/2)*x^2*a*b^3*e^3+40*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^2*b
^4*d*e^2+945*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^3*d*e^4-945*a
rctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^4*d^2*e^3-80*(b*(a*e-b*d))^(1
/2)*(e*x+d)^(3/2)*x*a^2*b^2*e^3+240*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^
2*e^4+240*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*b^4*d^2*e^2+1890*arctan((e*x+d)^
(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*d*e^4-1890*arctan((e*x+d)^(1/2)*b/(b*(a*e
-b*d))^(1/2))*x*a^2*b^3*d^2*e^3+630*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*
x*a*b^4*d^3*e^2-215*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*d*e^2+255*(b*(a*e-
b*d))^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e+315*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*
e^4+75*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^4*d^4-85*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(3/2)*b^4*d^3-315*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*e^5+480*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^3*b*e^4-780*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^
3*b*d*e^3+690*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2-300*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e)*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^5/((b*x+a)^2)^(
3/2)

_______________________________________________________________________________________

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((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.223792, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \,{\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \,{\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{40 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac{315 \,{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \,{\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \,{\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} -{\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{20 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

[1/40*(315*(a^2*b^2*d^2*e^2 - 2*a^3*b*d*e^3 + a^4*e^4 + (b^4*d^2*e^2 - 2*a*b^3*d
*e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 - 2*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*sqr
t((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b
))/(b*x + a)) + 2*(8*b^4*e^4*x^4 - 10*b^4*d^4 - 45*a*b^3*d^3*e + 483*a^2*b^2*d^2
*e^2 - 735*a^3*b*d*e^3 + 315*a^4*e^4 + 8*(7*b^4*d*e^3 - 3*a*b^3*e^4)*x^3 + 24*(1
2*b^4*d^2*e^2 - 17*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 - (85*b^4*d^3*e - 831*a*b^3*
d^2*e^2 + 1239*a^2*b^2*d*e^3 - 525*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b
^6*x + a^2*b^5), -1/20*(315*(a^2*b^2*d^2*e^2 - 2*a^3*b*d*e^3 + a^4*e^4 + (b^4*d^
2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 - 2*a^2*b^2*d*e^3 +
a^3*b*e^4)*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) -
(8*b^4*e^4*x^4 - 10*b^4*d^4 - 45*a*b^3*d^3*e + 483*a^2*b^2*d^2*e^2 - 735*a^3*b*d
*e^3 + 315*a^4*e^4 + 8*(7*b^4*d*e^3 - 3*a*b^3*e^4)*x^3 + 24*(12*b^4*d^2*e^2 - 17
*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 - (85*b^4*d^3*e - 831*a*b^3*d^2*e^2 + 1239*a^2
*b^2*d*e^3 - 525*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)]

_______________________________________________________________________________________

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)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.251657, size = 606, normalized size = 1.97 \[ -\frac{63 \,{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{17 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{2} - 15 \, \sqrt{x e + d} b^{4} d^{4} e^{2} - 51 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{3} + 60 \, \sqrt{x e + d} a b^{3} d^{3} e^{3} + 51 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{4} - 90 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{4} - 17 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{5} + 60 \, \sqrt{x e + d} a^{3} b d e^{5} - 15 \, \sqrt{x e + d} a^{4} e^{6}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{5}{2}} b^{12} e^{2} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{12} d e^{2} + 30 \, \sqrt{x e + d} b^{12} d^{2} e^{2} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{11} e^{3} - 60 \, \sqrt{x e + d} a b^{11} d e^{3} + 30 \, \sqrt{x e + d} a^{2} b^{10} e^{4}\right )}}{5 \, b^{15}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

-63/4*(b^3*d^3*e^2 - 3*a*b^2*d^2*e^3 + 3*a^2*b*d*e^4 - a^3*e^5)*arctan(sqrt(x*e
+ d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5*sign(-(x*e + d)*b*e + b*d
*e - a*e^2)) + 1/4*(17*(x*e + d)^(3/2)*b^4*d^3*e^2 - 15*sqrt(x*e + d)*b^4*d^4*e^
2 - 51*(x*e + d)^(3/2)*a*b^3*d^2*e^3 + 60*sqrt(x*e + d)*a*b^3*d^3*e^3 + 51*(x*e
+ d)^(3/2)*a^2*b^2*d*e^4 - 90*sqrt(x*e + d)*a^2*b^2*d^2*e^4 - 17*(x*e + d)^(3/2)
*a^3*b*e^5 + 60*sqrt(x*e + d)*a^3*b*d*e^5 - 15*sqrt(x*e + d)*a^4*e^6)/(((x*e + d
)*b - b*d + a*e)^2*b^5*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2/5*((x*e + d)^(5
/2)*b^12*e^2 + 5*(x*e + d)^(3/2)*b^12*d*e^2 + 30*sqrt(x*e + d)*b^12*d^2*e^2 - 5*
(x*e + d)^(3/2)*a*b^11*e^3 - 60*sqrt(x*e + d)*a*b^11*d*e^3 + 30*sqrt(x*e + d)*a^
2*b^10*e^4)/(b^15*sign(-(x*e + d)*b*e + b*d*e - a*e^2))