3.1634 \(\int \frac{(d+e x)^{9/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=162 \[ -\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{9 e \sqrt{d+e x} (b d-a e)^3}{b^5}+\frac{3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac{9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{9 e (d+e x)^{7/2}}{7 b^2} \]

[Out]

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

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Rubi [A]  time = 0.431501, antiderivative size = 162, 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 \[ -\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{9 e \sqrt{d+e x} (b d-a e)^3}{b^5}+\frac{3 e (d+e x)^{3/2} (b d-a e)^2}{b^4}+\frac{9 e (d+e x)^{5/2} (b d-a e)}{5 b^3}-\frac{(d+e x)^{9/2}}{b (a+b x)}+\frac{9 e (d+e x)^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.1845, size = 146, normalized size = 0.9 \[ - \frac{\left (d + e x\right )^{\frac{9}{2}}}{b \left (a + b x\right )} + \frac{9 e \left (d + e x\right )^{\frac{7}{2}}}{7 b^{2}} - \frac{9 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )}{5 b^{3}} + \frac{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}}{b^{4}} - \frac{9 e \sqrt{d + e x} \left (a e - b d\right )^{3}}{b^{5}} + \frac{9 e \left (a e - b d\right )^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{11}{2}}} \]

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),x)

[Out]

-(d + e*x)**(9/2)/(b*(a + b*x)) + 9*e*(d + e*x)**(7/2)/(7*b**2) - 9*e*(d + e*x)*
*(5/2)*(a*e - b*d)/(5*b**3) + 3*e*(d + e*x)**(3/2)*(a*e - b*d)**2/b**4 - 9*e*sqr
t(d + e*x)*(a*e - b*d)**3/b**5 + 9*e*(a*e - b*d)**(7/2)*atan(sqrt(b)*sqrt(d + e*
x)/sqrt(a*e - b*d))/b**(11/2)

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Mathematica [A]  time = 0.745431, size = 186, normalized size = 1.15 \[ \frac{\sqrt{d+e x} \left (2 b e^2 x \left (35 a^2 e^2-98 a b d e+78 b^2 d^2\right )+2 e \left (-140 a^3 e^3+455 a^2 b d e^2-504 a b^2 d^2 e+194 b^3 d^3\right )+2 b^2 e^3 x^2 (29 b d-14 a e)-\frac{35 (b d-a e)^4}{a+b x}+10 b^3 e^4 x^3\right )}{35 b^5}-\frac{9 e (b d-a e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(2*e*(194*b^3*d^3 - 504*a*b^2*d^2*e + 455*a^2*b*d*e^2 - 140*a^3*e
^3) + 2*b*e^2*(78*b^2*d^2 - 98*a*b*d*e + 35*a^2*e^2)*x + 2*b^2*e^3*(29*b*d - 14*
a*e)*x^2 + 10*b^3*e^4*x^3 - (35*(b*d - a*e)^4)/(a + b*x)))/(35*b^5) - (9*e*(b*d
- a*e)^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(11/2)

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Maple [B]  time = 0.034, size = 539, normalized size = 3.3 \[{\frac{2\,e}{7\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a{e}^{2}}{5\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{4\,de}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+2\,{\frac{ \left ( ex+d \right ) ^{3/2}{a}^{2}{e}^{3}}{{b}^{4}}}-4\,{\frac{ \left ( ex+d \right ) ^{3/2}ad{e}^{2}}{{b}^{3}}}+2\,{\frac{e \left ( ex+d \right ) ^{3/2}{d}^{2}}{{b}^{2}}}-8\,{\frac{{e}^{4}{a}^{3}\sqrt{ex+d}}{{b}^{5}}}+24\,{\frac{{a}^{2}d{e}^{3}\sqrt{ex+d}}{{b}^{4}}}-24\,{\frac{a{d}^{2}{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+8\,{\frac{e{d}^{3}\sqrt{ex+d}}{{b}^{2}}}-{\frac{{a}^{4}{e}^{5}}{{b}^{5} \left ( bex+ae \right ) }\sqrt{ex+d}}+4\,{\frac{\sqrt{ex+d}{a}^{3}d{e}^{4}}{{b}^{4} \left ( bex+ae \right ) }}-6\,{\frac{\sqrt{ex+d}{a}^{2}{d}^{2}{e}^{3}}{{b}^{3} \left ( bex+ae \right ) }}+4\,{\frac{\sqrt{ex+d}a{d}^{3}{e}^{2}}{{b}^{2} \left ( bex+ae \right ) }}-{\frac{e{d}^{4}}{b \left ( bex+ae \right ) }\sqrt{ex+d}}+9\,{\frac{{a}^{4}{e}^{5}}{{b}^{5}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-36\,{\frac{{a}^{3}d{e}^{4}}{{b}^{4}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+54\,{\frac{{a}^{2}{d}^{2}{e}^{3}}{{b}^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-36\,{\frac{a{d}^{3}{e}^{2}}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+9\,{\frac{e{d}^{4}}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

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),x)

[Out]

2/7*e*(e*x+d)^(7/2)/b^2-4/5/b^3*(e*x+d)^(5/2)*a*e^2+4/5*e/b^2*(e*x+d)^(5/2)*d+2/
b^4*(e*x+d)^(3/2)*a^2*e^3-4/b^3*(e*x+d)^(3/2)*a*d*e^2+2*e/b^2*(e*x+d)^(3/2)*d^2-
8/b^5*e^4*a^3*(e*x+d)^(1/2)+24/b^4*a^2*d*e^3*(e*x+d)^(1/2)-24/b^3*a*d^2*e^2*(e*x
+d)^(1/2)+8*e/b^2*d^3*(e*x+d)^(1/2)-1/b^5*(e*x+d)^(1/2)/(b*e*x+a*e)*a^4*e^5+4/b^
4*(e*x+d)^(1/2)/(b*e*x+a*e)*a^3*d*e^4-6/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*a^2*d^2*e^
3+4/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*a*d^3*e^2-e/b*(e*x+d)^(1/2)/(b*e*x+a*e)*d^4+9/
b^5/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*e^5-36/b
^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*d*e^4+54/
b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*d^2*e^3-
36/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*d^3*e^2
+9*e/b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*d^4

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22353, size = 1, normalized size = 0.01 \[ \left [-\frac{315 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \,{\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{70 \,{\left (b^{6} x + a b^{5}\right )}}, -\frac{315 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (10 \, b^{4} e^{4} x^{4} - 35 \, b^{4} d^{4} + 528 \, a b^{3} d^{3} e - 1218 \, a^{2} b^{2} d^{2} e^{2} + 1050 \, a^{3} b d e^{3} - 315 \, a^{4} e^{4} + 2 \,{\left (29 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (26 \, b^{4} d^{2} e^{2} - 23 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (194 \, b^{4} d^{3} e - 426 \, a b^{3} d^{2} e^{2} + 357 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (b^{6} x + a b^{5}\right )}}\right ] \]

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),x, algorithm="fricas")

[Out]

[-1/70*(315*(a*b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^
3*e - 3*a*b^3*d^2*e^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*(10
*b^4*e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218*a^2*b^2*d^2*e^2 + 1050*a^3*b*
d*e^3 - 315*a^4*e^4 + 2*(29*b^4*d*e^3 - 9*a*b^3*e^4)*x^3 + 6*(26*b^4*d^2*e^2 - 2
3*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(194*b^4*d^3*e - 426*a*b^3*d^2*e^2 + 357*
a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^6*x + a*b^5), -1/35*(315*(a*
b^3*d^3*e - 3*a^2*b^2*d^2*e^2 + 3*a^3*b*d*e^3 - a^4*e^4 + (b^4*d^3*e - 3*a*b^3*d
^2*e^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)) - (10*b^4*e^4*x^4 - 35*b^4*d^4 + 528*a*b^3*d^3*e - 1218
*a^2*b^2*d^2*e^2 + 1050*a^3*b*d*e^3 - 315*a^4*e^4 + 2*(29*b^4*d*e^3 - 9*a*b^3*e^
4)*x^3 + 6*(26*b^4*d^2*e^2 - 23*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 + 2*(194*b^4*d^
3*e - 426*a*b^3*d^2*e^2 + 357*a^2*b^2*d*e^3 - 105*a^3*b*e^4)*x)*sqrt(e*x + d))/(
b^6*x + a*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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221736, size = 522, normalized size = 3.22 \[ \frac{9 \,{\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{5}} - \frac{\sqrt{x e + d} b^{4} d^{4} e - 4 \, \sqrt{x e + d} a b^{3} d^{3} e^{2} + 6 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{3} - 4 \, \sqrt{x e + d} a^{3} b d e^{4} + \sqrt{x e + d} a^{4} e^{5}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac{2 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{12} e + 14 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{12} d e + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{12} d^{2} e + 140 \, \sqrt{x e + d} b^{12} d^{3} e - 14 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{11} e^{2} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{11} d e^{2} - 420 \, \sqrt{x e + d} a b^{11} d^{2} e^{2} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{10} e^{3} + 420 \, \sqrt{x e + d} a^{2} b^{10} d e^{3} - 140 \, \sqrt{x e + d} a^{3} b^{9} e^{4}\right )}}{35 \, b^{14}} \]

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),x, algorithm="giac")

[Out]

9*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*ar
ctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - (sqrt(x*
e + d)*b^4*d^4*e - 4*sqrt(x*e + d)*a*b^3*d^3*e^2 + 6*sqrt(x*e + d)*a^2*b^2*d^2*e
^3 - 4*sqrt(x*e + d)*a^3*b*d*e^4 + sqrt(x*e + d)*a^4*e^5)/(((x*e + d)*b - b*d +
a*e)*b^5) + 2/35*(5*(x*e + d)^(7/2)*b^12*e + 14*(x*e + d)^(5/2)*b^12*d*e + 35*(x
*e + d)^(3/2)*b^12*d^2*e + 140*sqrt(x*e + d)*b^12*d^3*e - 14*(x*e + d)^(5/2)*a*b
^11*e^2 - 70*(x*e + d)^(3/2)*a*b^11*d*e^2 - 420*sqrt(x*e + d)*a*b^11*d^2*e^2 + 3
5*(x*e + d)^(3/2)*a^2*b^10*e^3 + 420*sqrt(x*e + d)*a^2*b^10*d*e^3 - 140*sqrt(x*e
+ d)*a^3*b^9*e^4)/b^14