3.1655 \(\int \frac{(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=197 \[ -\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{693 e^5 \sqrt{d+e x}}{128 b^6} \]

[Out]

(693*e^5*Sqrt[d + e*x])/(128*b^6) - (231*e^4*(d + e*x)^(3/2))/(128*b^5*(a + b*x)
) - (231*e^3*(d + e*x)^(5/2))/(320*b^4*(a + b*x)^2) - (33*e^2*(d + e*x)^(7/2))/(
80*b^3*(a + b*x)^3) - (11*e*(d + e*x)^(9/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(1
1/2)/(5*b*(a + b*x)^5) - (693*e^5*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x]
)/Sqrt[b*d - a*e]])/(128*b^(13/2))

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Rubi [A]  time = 0.30616, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}-\frac{231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac{231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac{33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac{11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac{693 e^5 \sqrt{d+e x}}{128 b^6} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(693*e^5*Sqrt[d + e*x])/(128*b^6) - (231*e^4*(d + e*x)^(3/2))/(128*b^5*(a + b*x)
) - (231*e^3*(d + e*x)^(5/2))/(320*b^4*(a + b*x)^2) - (33*e^2*(d + e*x)^(7/2))/(
80*b^3*(a + b*x)^3) - (11*e*(d + e*x)^(9/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(1
1/2)/(5*b*(a + b*x)^5) - (693*e^5*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x]
)/Sqrt[b*d - a*e]])/(128*b^(13/2))

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Rubi in Sympy [A]  time = 76.5844, size = 182, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{\frac{11}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{11 e \left (d + e x\right )^{\frac{9}{2}}}{40 b^{2} \left (a + b x\right )^{4}} - \frac{33 e^{2} \left (d + e x\right )^{\frac{7}{2}}}{80 b^{3} \left (a + b x\right )^{3}} - \frac{231 e^{3} \left (d + e x\right )^{\frac{5}{2}}}{320 b^{4} \left (a + b x\right )^{2}} - \frac{231 e^{4} \left (d + e x\right )^{\frac{3}{2}}}{128 b^{5} \left (a + b x\right )} + \frac{693 e^{5} \sqrt{d + e x}}{128 b^{6}} - \frac{693 e^{5} \sqrt{a e - b d} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-(d + e*x)**(11/2)/(5*b*(a + b*x)**5) - 11*e*(d + e*x)**(9/2)/(40*b**2*(a + b*x)
**4) - 33*e**2*(d + e*x)**(7/2)/(80*b**3*(a + b*x)**3) - 231*e**3*(d + e*x)**(5/
2)/(320*b**4*(a + b*x)**2) - 231*e**4*(d + e*x)**(3/2)/(128*b**5*(a + b*x)) + 69
3*e**5*sqrt(d + e*x)/(128*b**6) - 693*e**5*sqrt(a*e - b*d)*atan(sqrt(b)*sqrt(d +
e*x)/sqrt(a*e - b*d))/(128*b**(13/2))

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Mathematica [A]  time = 0.450526, size = 183, normalized size = 0.93 \[ -\frac{693 e^5 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{13/2}}-\frac{\sqrt{d+e x} \left (4215 e^4 (a+b x)^4 (b d-a e)+3590 e^3 (a+b x)^3 (b d-a e)^2+2248 e^2 (a+b x)^2 (b d-a e)^3+816 e (a+b x) (b d-a e)^4+128 (b d-a e)^5-1280 e^5 (a+b x)^5\right )}{640 b^6 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(128*(b*d - a*e)^5 + 816*e*(b*d - a*e)^4*(a + b*x) + 2248*e^2*(b
*d - a*e)^3*(a + b*x)^2 + 3590*e^3*(b*d - a*e)^2*(a + b*x)^3 + 4215*e^4*(b*d - a
*e)*(a + b*x)^4 - 1280*e^5*(a + b*x)^5))/(640*b^6*(a + b*x)^5) - (693*e^5*Sqrt[b
*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(13/2))

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Maple [B]  time = 0.038, size = 673, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*e^5*(e*x+d)^(1/2)/b^6+843/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a-843/128*e^
5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*d+1327/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^
2-1327/32*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a*d+1327/64*e^5/b/(b*e*x+a*e)^5*(e
*x+d)^(7/2)*d^2+131/5*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3-393/5*e^7/b^3/(b*e
*x+a*e)^5*(e*x+d)^(5/2)*a^2*d+393/5*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d^2-13
1/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^3+977/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(
3/2)*a^4-977/16*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3*d+2931/32*e^7/b^3/(b*e*x
+a*e)^5*(e*x+d)^(3/2)*a^2*d^2-977/16*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^3+9
77/64*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^4+437/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+
d)^(1/2)*a^5-2185/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4*d+2185/64*e^8/b^4/
(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d^2-2185/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*
a^2*d^3+2185/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^4-437/128*e^5/b/(b*e*x+
a*e)^5*(e*x+d)^(1/2)*d^5-693/128*e^6/b^6/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2
)*b/(b*(a*e-b*d))^(1/2))*a+693/128*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))*d

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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)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228048, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(3465*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e
^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*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*(1280*b^5*e^5*x^5 - 128*b
^5*d^5 - 176*a*b^4*d^4*e - 264*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*
b*d*e^4 + 3465*a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*x^4 - 10*(359*b^5*d^
2*e^3 + 968*a*b^4*d*e^4 - 2607*a^2*b^3*e^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b
^4*d^2*e^3 + 5247*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*d^4*e + 61
6*a*b^4*d^3*e^2 + 1089*a^2*b^3*d^2*e^3 + 2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)
*sqrt(e*x + d))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a
^4*b^7*x + a^5*b^6), -1/640*(3465*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^
5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt(-(b*d - a*e)/b)*arcta
n(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (1280*b^5*e^5*x^5 - 128*b^5*d^5 - 176*a*
b^4*d^4*e - 264*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*b*d*e^4 + 3465*
a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*x^4 - 10*(359*b^5*d^2*e^3 + 968*a*b
^4*d*e^4 - 2607*a^2*b^3*e^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b^4*d^2*e^3 + 52
47*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*d^4*e + 616*a*b^4*d^3*e^2
+ 1089*a^2*b^3*d^2*e^3 + 2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)*sqrt(e*x + d))
/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*
b^6)]

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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)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.24193, size = 620, normalized size = 3.15 \[ \frac{693 \,{\left (b d e^{5} - a e^{6}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{128 \, \sqrt{-b^{2} d + a b e} b^{6}} + \frac{2 \, \sqrt{x e + d} e^{5}}{b^{6}} - \frac{4215 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{5} d e^{5} - 13270 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d^{2} e^{5} + 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{3} e^{5} - 9770 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{4} e^{5} + 2185 \, \sqrt{x e + d} b^{5} d^{5} e^{5} - 4215 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{4} e^{6} + 26540 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} d e^{6} - 50304 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d^{2} e^{6} + 39080 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{3} e^{6} - 10925 \, \sqrt{x e + d} a b^{4} d^{4} e^{6} - 13270 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{3} e^{7} + 50304 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} d e^{7} - 58620 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d^{2} e^{7} + 21850 \, \sqrt{x e + d} a^{2} b^{3} d^{3} e^{7} - 16768 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{2} e^{8} + 39080 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} d e^{8} - 21850 \, \sqrt{x e + d} a^{3} b^{2} d^{2} e^{8} - 9770 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b e^{9} + 10925 \, \sqrt{x e + d} a^{4} b d e^{9} - 2185 \, \sqrt{x e + d} a^{5} e^{10}}{640 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

693/128*(b*d*e^5 - a*e^6)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^
2*d + a*b*e)*b^6) + 2*sqrt(x*e + d)*e^5/b^6 - 1/640*(4215*(x*e + d)^(9/2)*b^5*d*
e^5 - 13270*(x*e + d)^(7/2)*b^5*d^2*e^5 + 16768*(x*e + d)^(5/2)*b^5*d^3*e^5 - 97
70*(x*e + d)^(3/2)*b^5*d^4*e^5 + 2185*sqrt(x*e + d)*b^5*d^5*e^5 - 4215*(x*e + d)
^(9/2)*a*b^4*e^6 + 26540*(x*e + d)^(7/2)*a*b^4*d*e^6 - 50304*(x*e + d)^(5/2)*a*b
^4*d^2*e^6 + 39080*(x*e + d)^(3/2)*a*b^4*d^3*e^6 - 10925*sqrt(x*e + d)*a*b^4*d^4
*e^6 - 13270*(x*e + d)^(7/2)*a^2*b^3*e^7 + 50304*(x*e + d)^(5/2)*a^2*b^3*d*e^7 -
58620*(x*e + d)^(3/2)*a^2*b^3*d^2*e^7 + 21850*sqrt(x*e + d)*a^2*b^3*d^3*e^7 - 1
6768*(x*e + d)^(5/2)*a^3*b^2*e^8 + 39080*(x*e + d)^(3/2)*a^3*b^2*d*e^8 - 21850*s
qrt(x*e + d)*a^3*b^2*d^2*e^8 - 9770*(x*e + d)^(3/2)*a^4*b*e^9 + 10925*sqrt(x*e +
d)*a^4*b*d*e^9 - 2185*sqrt(x*e + d)*a^5*e^10)/(((x*e + d)*b - b*d + a*e)^5*b^6)