3.1710 \(\int \frac{(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=346 \[ -\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1155 e^4 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1155 e^4 (a+b x) \sqrt{d+e x} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(1155*e^4*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) + (385*e^4*(a + b*x)*(d + e*x)^(3/2))/(64*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (231*e^3*(d + e*x)^(5/2))/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (33*e^2*
(d + e*x)^(7/2))/(32*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (11*e*(d + e
*x)^(9/2))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(11/2)
/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (1155*e^4*(b*d - a*e)^(3/2)*(
a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(13/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.564344, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1155 e^4 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1155 e^4 (a+b x) \sqrt{d+e x} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(1155*e^4*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) + (385*e^4*(a + b*x)*(d + e*x)^(3/2))/(64*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (231*e^3*(d + e*x)^(5/2))/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (33*e^2*
(d + e*x)^(7/2))/(32*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (11*e*(d + e
*x)^(9/2))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(11/2)
/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (1155*e^4*(b*d - a*e)^(3/2)*(
a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(13/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])

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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)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.831937, size = 206, normalized size = 0.6 \[ \frac{(a+b x)^5 \left (-\frac{1155 e^4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{13/2}}-\frac{\sqrt{d+e x} \left (128 e^4 (a+b x)^4 (15 a e-16 b d)+2295 e^3 (a+b x)^3 (b d-a e)^2+1030 e^2 (a+b x)^2 (b d-a e)^3+328 e (a+b x) (b d-a e)^4+48 (b d-a e)^5-128 b e^5 x (a+b x)^4\right )}{3 b^6 (a+b x)^4}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)^5*(-(Sqrt[d + e*x]*(48*(b*d - a*e)^5 + 328*e*(b*d - a*e)^4*(a + b*x)
+ 1030*e^2*(b*d - a*e)^3*(a + b*x)^2 + 2295*e^3*(b*d - a*e)^2*(a + b*x)^3 + 128*
e^4*(-16*b*d + 15*a*e)*(a + b*x)^4 - 128*b*e^5*x*(a + b*x)^4))/(3*b^6*(a + b*x)^
4) - (1155*e^4*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/b^(13/2)))/(64*((a + b*x)^2)^(5/2))

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Maple [B]  time = 0.034, size = 1471, normalized size = 4.3 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

-1/192*(-128*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^4*b^5*e^4+41580*arctan((e*x+d)^
(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^3*d*e^5-20790*arctan((e*x+d)^(1/2)*b/(b*(
a*e-b*d))^(1/2))*x^2*a^2*b^4*d^2*e^4-17565*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^2
*b^3*d*e^2+17565*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^4*d^2*e-512*(b*(a*e-b*d))
^(1/2)*(e*x+d)^(3/2)*x*a^3*b^2*e^4+11520*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a
^3*b^2*e^5+27720*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b^2*d*e^5-138
60*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^3*d^2*e^4-20612*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(3/2)*a^3*b^2*d*e^3+30918*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a
^2*b^3*d^2*e^2-20612*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^4*d^3*e+7680*(b*(a*e-
b*d))^(1/2)*(e*x+d)^(1/2)*x*a^4*b*e^5-9645*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4
*b*d*e^4+15450*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b^2*d^2*e^3-15450*(b*(a*e-b
*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^3*d^3*e^2+7725*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*
a*b^4*d^4*e+6930*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a*b^5*d*e^5-512
*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^3*a*b^4*e^4+1920*(b*(a*e-b*d))^(1/2)*(e*x+d
)^(1/2)*x^4*a*b^4*e^5-1920*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^4*b^5*d*e^4+27720
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a^2*b^4*d*e^5-13860*arctan((e*x
+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^5*d^2*e^4-4590*(b*(a*e-b*d))^(1/2)*(e*x
+d)^(7/2)*a*b^4*d*e-768*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^2*a^2*b^3*e^4+7680*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^3*a^2*b^3*e^5-7680*(b*(a*e-b*d))^(1/2)*(e*x+d
)^(1/2)*x^3*a*b^4*d*e^4-3465*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^6*e^6
-3465*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a^2*b^4*e^6-3465*arctan((e
*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^6*d^2*e^4-13860*arctan((e*x+d)^(1/2)*b/
(b*(a*e-b*d))^(1/2))*x^3*a^3*b^3*e^6+2295*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a^2*
b^3*e^2-20790*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^4*b^2*e^6+5855*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^3*b^2*e^3-13860*arctan((e*x+d)^(1/2)*b/(b*(a*
e-b*d))^(1/2))*x*a^5*b*e^6+5025*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^4*b*e^4+6930
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*b*d*e^5-3465*arctan((e*x+d)^(1/
2)*b/(b*(a*e-b*d))^(1/2))*a^4*b^2*d^2*e^4+2295*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)
*b^5*d^2-5855*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^5*d^3+5153*(b*(a*e-b*d))^(1/2)
*(e*x+d)^(3/2)*b^5*d^4+3465*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^5*e^5-1545*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^5*d^5-11520*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2
*a^2*b^3*d*e^4-7680*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^3*b^2*d*e^4)*(b*x+a)/(
b*(a*e-b*d))^(1/2)/b^6/((b*x+a)^2)^(5/2)

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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)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224831, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(3465*(a^4*b*d*e^4 - a^5*e^5 + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(a*b^4*d*
e^4 - a^2*b^3*e^5)*x^3 + 6*(a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 4*(a^3*b^2*d*e^4
- a^4*b*e^5)*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*(128*b^5*e^5*x^5 - 48*b^5*d^5 - 88*a*b^4*d^
4*e - 198*a^2*b^3*d^3*e^2 - 693*a^3*b^2*d^2*e^3 + 4620*a^4*b*d*e^4 - 3465*a^5*e^
5 + 128*(16*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 - (2295*b^5*d^2*e^3 - 12782*a*b^4*d*e^
4 + 9207*a^2*b^3*e^5)*x^3 - (1030*b^5*d^3*e^2 + 3795*a*b^4*d^2*e^3 - 22968*a^2*b
^3*d*e^4 + 16863*a^3*b^2*e^5)*x^2 - (328*b^5*d^4*e + 748*a*b^4*d^3*e^2 + 2673*a^
2*b^3*d^2*e^3 - 17094*a^3*b^2*d*e^4 + 12705*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^10*x
^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6), -1/192*(3465*(a^4*b*d
*e^4 - a^5*e^5 + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(a*b^4*d*e^4 - a^2*b^3*e^5)*x^3
+ 6*(a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 4*(a^3*b^2*d*e^4 - a^4*b*e^5)*x)*sqrt(-
(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (128*b^5*e^5*x^5 - 4
8*b^5*d^5 - 88*a*b^4*d^4*e - 198*a^2*b^3*d^3*e^2 - 693*a^3*b^2*d^2*e^3 + 4620*a^
4*b*d*e^4 - 3465*a^5*e^5 + 128*(16*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 - (2295*b^5*d^2
*e^3 - 12782*a*b^4*d*e^4 + 9207*a^2*b^3*e^5)*x^3 - (1030*b^5*d^3*e^2 + 3795*a*b^
4*d^2*e^3 - 22968*a^2*b^3*d*e^4 + 16863*a^3*b^2*e^5)*x^2 - (328*b^5*d^4*e + 748*
a*b^4*d^3*e^2 + 2673*a^2*b^3*d^2*e^3 - 17094*a^3*b^2*d*e^4 + 12705*a^4*b*e^5)*x)
*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265513, size = 744, normalized size = 2.15 \[ -\frac{1155 \,{\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \, \sqrt{-b^{2} d + a b e} b^{6}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{2295 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d^{2} e^{4} - 5855 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{3} e^{4} + 5153 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{4} e^{4} - 1545 \, \sqrt{x e + d} b^{5} d^{5} e^{4} - 4590 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} d e^{5} + 17565 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d^{2} e^{5} - 20612 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{3} e^{5} + 7725 \, \sqrt{x e + d} a b^{4} d^{4} e^{5} + 2295 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{3} e^{6} - 17565 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} d e^{6} + 30918 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d^{2} e^{6} - 15450 \, \sqrt{x e + d} a^{2} b^{3} d^{3} e^{6} + 5855 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{2} e^{7} - 20612 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} d e^{7} + 15450 \, \sqrt{x e + d} a^{3} b^{2} d^{2} e^{7} + 5153 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b e^{8} - 7725 \, \sqrt{x e + d} a^{4} b d e^{8} + 1545 \, \sqrt{x e + d} a^{5} e^{9}}{192 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{6}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{10} e^{4} + 15 \, \sqrt{x e + d} b^{10} d e^{4} - 15 \, \sqrt{x e + d} a b^{9} e^{5}\right )}}{3 \, b^{15}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

-1155/64*(b^2*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*
d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/
192*(2295*(x*e + d)^(7/2)*b^5*d^2*e^4 - 5855*(x*e + d)^(5/2)*b^5*d^3*e^4 + 5153*
(x*e + d)^(3/2)*b^5*d^4*e^4 - 1545*sqrt(x*e + d)*b^5*d^5*e^4 - 4590*(x*e + d)^(7
/2)*a*b^4*d*e^5 + 17565*(x*e + d)^(5/2)*a*b^4*d^2*e^5 - 20612*(x*e + d)^(3/2)*a*
b^4*d^3*e^5 + 7725*sqrt(x*e + d)*a*b^4*d^4*e^5 + 2295*(x*e + d)^(7/2)*a^2*b^3*e^
6 - 17565*(x*e + d)^(5/2)*a^2*b^3*d*e^6 + 30918*(x*e + d)^(3/2)*a^2*b^3*d^2*e^6
- 15450*sqrt(x*e + d)*a^2*b^3*d^3*e^6 + 5855*(x*e + d)^(5/2)*a^3*b^2*e^7 - 20612
*(x*e + d)^(3/2)*a^3*b^2*d*e^7 + 15450*sqrt(x*e + d)*a^3*b^2*d^2*e^7 + 5153*(x*e
+ d)^(3/2)*a^4*b*e^8 - 7725*sqrt(x*e + d)*a^4*b*d*e^8 + 1545*sqrt(x*e + d)*a^5*
e^9)/(((x*e + d)*b - b*d + a*e)^4*b^6*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2/
3*((x*e + d)^(3/2)*b^10*e^4 + 15*sqrt(x*e + d)*b^10*d*e^4 - 15*sqrt(x*e + d)*a*b
^9*e^5)/(b^15*sign(-(x*e + d)*b*e + b*d*e - a*e^2))