3.1644 \(\int \frac{(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=172 \[ -\frac{105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}+\frac{105 e^3 \sqrt{d+e x} (b d-a e)}{8 b^5}-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4} \]

[Out]

(105*e^3*(b*d - a*e)*Sqrt[d + e*x])/(8*b^5) + (35*e^3*(d + e*x)^(3/2))/(8*b^4) -
(21*e^2*(d + e*x)^(5/2))/(8*b^3*(a + b*x)) - (3*e*(d + e*x)^(7/2))/(4*b^2*(a +
b*x)^2) - (d + e*x)^(9/2)/(3*b*(a + b*x)^3) - (105*e^3*(b*d - a*e)^(3/2)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(11/2))

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Rubi [A]  time = 0.28388, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}+\frac{105 e^3 \sqrt{d+e x} (b d-a e)}{8 b^5}-\frac{21 e^2 (d+e x)^{5/2}}{8 b^3 (a+b x)}-\frac{3 e (d+e x)^{7/2}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{9/2}}{3 b (a+b x)^3}+\frac{35 e^3 (d+e x)^{3/2}}{8 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(105*e^3*(b*d - a*e)*Sqrt[d + e*x])/(8*b^5) + (35*e^3*(d + e*x)^(3/2))/(8*b^4) -
(21*e^2*(d + e*x)^(5/2))/(8*b^3*(a + b*x)) - (3*e*(d + e*x)^(7/2))/(4*b^2*(a +
b*x)^2) - (d + e*x)^(9/2)/(3*b*(a + b*x)^3) - (105*e^3*(b*d - a*e)^(3/2)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(11/2))

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Rubi in Sympy [A]  time = 65.5213, size = 156, normalized size = 0.91 \[ - \frac{\left (d + e x\right )^{\frac{9}{2}}}{3 b \left (a + b x\right )^{3}} - \frac{3 e \left (d + e x\right )^{\frac{7}{2}}}{4 b^{2} \left (a + b x\right )^{2}} - \frac{21 e^{2} \left (d + e x\right )^{\frac{5}{2}}}{8 b^{3} \left (a + b x\right )} + \frac{35 e^{3} \left (d + e x\right )^{\frac{3}{2}}}{8 b^{4}} - \frac{105 e^{3} \sqrt{d + e x} \left (a e - b d\right )}{8 b^{5}} + \frac{105 e^{3} \left (a e - b d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{11}{2}}} \]

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)**2,x)

[Out]

-(d + e*x)**(9/2)/(3*b*(a + b*x)**3) - 3*e*(d + e*x)**(7/2)/(4*b**2*(a + b*x)**2
) - 21*e**2*(d + e*x)**(5/2)/(8*b**3*(a + b*x)) + 35*e**3*(d + e*x)**(3/2)/(8*b*
*4) - 105*e**3*sqrt(d + e*x)*(a*e - b*d)/(8*b**5) + 105*e**3*(a*e - b*d)**(3/2)*
atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*b**(11/2))

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Mathematica [A]  time = 0.421303, size = 164, normalized size = 0.95 \[ -\frac{105 e^3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{11/2}}-\frac{\sqrt{d+e x} \left (16 e^3 (a+b x)^3 (12 a e-13 b d)+165 e^2 (a+b x)^2 (b d-a e)^2+50 e (a+b x) (b d-a e)^3+8 (b d-a e)^4-16 b e^4 x (a+b x)^3\right )}{24 b^5 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(8*(b*d - a*e)^4 + 50*e*(b*d - a*e)^3*(a + b*x) + 165*e^2*(b*d -
a*e)^2*(a + b*x)^2 + 16*e^3*(-13*b*d + 12*a*e)*(a + b*x)^3 - 16*b*e^4*x*(a + b*
x)^3))/(24*b^5*(a + b*x)^3) - (105*e^3*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d
+ e*x])/Sqrt[b*d - a*e]])/(8*b^(11/2))

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Maple [B]  time = 0.029, size = 525, normalized size = 3.1 \[{\frac{2\,{e}^{3}}{3\,{b}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-8\,{\frac{{e}^{4}\sqrt{ex+d}a}{{b}^{5}}}+8\,{\frac{{e}^{3}\sqrt{ex+d}d}{{b}^{4}}}-{\frac{55\,{a}^{2}{e}^{5}}{8\,{b}^{3} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{55\,{e}^{4}ad}{4\,{b}^{2} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{55\,{e}^{3}{d}^{2}}{8\,b \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{35\,{a}^{3}{e}^{6}}{3\,{b}^{4} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+35\,{\frac{{e}^{5} \left ( ex+d \right ) ^{3/2}{a}^{2}d}{{b}^{3} \left ( bex+ae \right ) ^{3}}}-35\,{\frac{{e}^{4} \left ( ex+d \right ) ^{3/2}a{d}^{2}}{{b}^{2} \left ( bex+ae \right ) ^{3}}}+{\frac{35\,{e}^{3}{d}^{3}}{3\,b \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{41\,{e}^{7}{a}^{4}}{8\,{b}^{5} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{41\,{a}^{3}{e}^{6}d}{2\,{b}^{4} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{123\,{a}^{2}{e}^{5}{d}^{2}}{4\,{b}^{3} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{41\,{e}^{4}a{d}^{3}}{2\,{b}^{2} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{41\,{e}^{3}{d}^{4}}{8\,b \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{105\,{a}^{2}{e}^{5}}{8\,{b}^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}-{\frac{105\,{e}^{4}ad}{4\,{b}^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{105\,{e}^{3}{d}^{2}}{8\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*e^3*(e*x+d)^(3/2)/b^4-8*e^4/b^5*(e*x+d)^(1/2)*a+8*e^3/b^4*(e*x+d)^(1/2)*d-55
/8*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*a^2+55/4*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(5
/2)*a*d-55/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*d^2-35/3*e^6/b^4/(b*e*x+a*e)^3*(e
*x+d)^(3/2)*a^3+35*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(3/2)*a^2*d-35*e^4/b^2/(b*e*x+a
*e)^3*(e*x+d)^(3/2)*a*d^2+35/3*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(3/2)*d^3-41/8*e^7/b^
5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a^4+41/2*e^6/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a^3*d
-123/4*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a^2*d^2+41/2*e^4/b^2/(b*e*x+a*e)^3*(e
*x+d)^(1/2)*a*d^3-41/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*d^4+105/8*e^5/b^5/(b*(a
*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2-105/4*e^4/b^4/(b*
(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*d+105/8*e^3/b^3/(
b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*d^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((e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225513, size = 1, normalized size = 0.01 \[ \left [-\frac{315 \,{\left (a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (16 \, b^{4} e^{4} x^{4} - 8 \, b^{4} d^{4} - 18 \, a b^{3} d^{3} e - 63 \, a^{2} b^{2} d^{2} e^{2} + 420 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 16 \,{\left (13 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (55 \, b^{4} d^{2} e^{2} - 318 \, a b^{3} d e^{3} + 231 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (25 \, b^{4} d^{3} e + 90 \, a b^{3} d^{2} e^{2} - 567 \, a^{2} b^{2} d e^{3} + 420 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}, -\frac{315 \,{\left (a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (16 \, b^{4} e^{4} x^{4} - 8 \, b^{4} d^{4} - 18 \, a b^{3} d^{3} e - 63 \, a^{2} b^{2} d^{2} e^{2} + 420 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 16 \,{\left (13 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} - 3 \,{\left (55 \, b^{4} d^{2} e^{2} - 318 \, a b^{3} d e^{3} + 231 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (25 \, b^{4} d^{3} e + 90 \, a b^{3} d^{2} e^{2} - 567 \, a^{2} b^{2} d e^{3} + 420 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

[-1/48*(315*(a^3*b*d*e^3 - a^4*e^4 + (b^4*d*e^3 - a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^
3 - a^2*b^2*e^4)*x^2 + 3*(a^2*b^2*d*e^3 - a^3*b*e^4)*x)*sqrt((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*(16
*b^4*e^4*x^4 - 8*b^4*d^4 - 18*a*b^3*d^3*e - 63*a^2*b^2*d^2*e^2 + 420*a^3*b*d*e^3
- 315*a^4*e^4 + 16*(13*b^4*d*e^3 - 9*a*b^3*e^4)*x^3 - 3*(55*b^4*d^2*e^2 - 318*a
*b^3*d*e^3 + 231*a^2*b^2*e^4)*x^2 - 2*(25*b^4*d^3*e + 90*a*b^3*d^2*e^2 - 567*a^2
*b^2*d*e^3 + 420*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6
*x + a^3*b^5), -1/24*(315*(a^3*b*d*e^3 - a^4*e^4 + (b^4*d*e^3 - a*b^3*e^4)*x^3 +
3*(a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 3*(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)) - (16*b^4*e^4*x^4 - 8*b^4*
d^4 - 18*a*b^3*d^3*e - 63*a^2*b^2*d^2*e^2 + 420*a^3*b*d*e^3 - 315*a^4*e^4 + 16*(
13*b^4*d*e^3 - 9*a*b^3*e^4)*x^3 - 3*(55*b^4*d^2*e^2 - 318*a*b^3*d*e^3 + 231*a^2*
b^2*e^4)*x^2 - 2*(25*b^4*d^3*e + 90*a*b^3*d^2*e^2 - 567*a^2*b^2*d*e^3 + 420*a^3*
b*e^4)*x)*sqrt(e*x + d))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5)]

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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((e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227597, size = 486, normalized size = 2.83 \[ \frac{105 \,{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{165 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{3} - 280 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{3} + 123 \, \sqrt{x e + d} b^{4} d^{4} e^{3} - 330 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{4} + 840 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{4} - 492 \, \sqrt{x e + d} a b^{3} d^{3} e^{4} + 165 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{5} - 840 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{5} + 738 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{5} + 280 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{6} - 492 \, \sqrt{x e + d} a^{3} b d e^{6} + 123 \, \sqrt{x e + d} a^{4} e^{7}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{5}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{8} e^{3} + 12 \, \sqrt{x e + d} b^{8} d e^{3} - 12 \, \sqrt{x e + d} a b^{7} e^{4}\right )}}{3 \, b^{12}} \]

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)^2,x, algorithm="giac")

[Out]

105/8*(b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d +
a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - 1/24*(165*(x*e + d)^(5/2)*b^4*d^2*e^3 - 28
0*(x*e + d)^(3/2)*b^4*d^3*e^3 + 123*sqrt(x*e + d)*b^4*d^4*e^3 - 330*(x*e + d)^(5
/2)*a*b^3*d*e^4 + 840*(x*e + d)^(3/2)*a*b^3*d^2*e^4 - 492*sqrt(x*e + d)*a*b^3*d^
3*e^4 + 165*(x*e + d)^(5/2)*a^2*b^2*e^5 - 840*(x*e + d)^(3/2)*a^2*b^2*d*e^5 + 73
8*sqrt(x*e + d)*a^2*b^2*d^2*e^5 + 280*(x*e + d)^(3/2)*a^3*b*e^6 - 492*sqrt(x*e +
d)*a^3*b*d*e^6 + 123*sqrt(x*e + d)*a^4*e^7)/(((x*e + d)*b - b*d + a*e)^3*b^5) +
2/3*((x*e + d)^(3/2)*b^8*e^3 + 12*sqrt(x*e + d)*b^8*d*e^3 - 12*sqrt(x*e + d)*a*
b^7*e^4)/b^12