3.2083 \(\int \frac{(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac{1155 e^4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2}}+\frac{1155 e^4 \sqrt{d+e x} (b d-a e)}{64 b^6}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 (a+b x)}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x)^2}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^3}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^4}+\frac{385 e^4 (d+e x)^{3/2}}{64 b^5} \]

[Out]

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

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Rubi [A]  time = 0.310641, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{1155 e^4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2}}+\frac{1155 e^4 \sqrt{d+e x} (b d-a e)}{64 b^6}-\frac{231 e^3 (d+e x)^{5/2}}{64 b^4 (a+b x)}-\frac{33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x)^2}-\frac{11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^3}-\frac{(d+e x)^{11/2}}{4 b (a+b x)^4}+\frac{385 e^4 (d+e x)^{3/2}}{64 b^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 79.4443, size = 182, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{\frac{11}{2}}}{4 b \left (a + b x\right )^{4}} - \frac{11 e \left (d + e x\right )^{\frac{9}{2}}}{24 b^{2} \left (a + b x\right )^{3}} - \frac{33 e^{2} \left (d + e x\right )^{\frac{7}{2}}}{32 b^{3} \left (a + b x\right )^{2}} - \frac{231 e^{3} \left (d + e x\right )^{\frac{5}{2}}}{64 b^{4} \left (a + b x\right )} + \frac{385 e^{4} \left (d + e x\right )^{\frac{3}{2}}}{64 b^{5}} - \frac{1155 e^{4} \sqrt{d + e x} \left (a e - b d\right )}{64 b^{6}} + \frac{1155 e^{4} \left (a e - b d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{64 b^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**(11/2)/(4*b*(a + b*x)**4) - 11*e*(d + e*x)**(9/2)/(24*b**2*(a + b*x)
**3) - 33*e**2*(d + e*x)**(7/2)/(32*b**3*(a + b*x)**2) - 231*e**3*(d + e*x)**(5/
2)/(64*b**4*(a + b*x)) + 385*e**4*(d + e*x)**(3/2)/(64*b**5) - 1155*e**4*sqrt(d
+ e*x)*(a*e - b*d)/(64*b**6) + 1155*e**4*(a*e - b*d)**(3/2)*atan(sqrt(b)*sqrt(d
+ e*x)/sqrt(a*e - b*d))/(64*b**(13/2))

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Mathematica [A]  time = 0.556202, size = 186, normalized size = 0.94 \[ -\frac{1155 e^4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2}}-\frac{\sqrt{d+e x} \left (128 e^4 (a+b x)^4 (15 a e-16 b d)+2295 e^3 (a+b x)^3 (b d-a e)^2+1030 e^2 (a+b x)^2 (b d-a e)^3+328 e (a+b x) (b d-a e)^4+48 (b d-a e)^5-128 b e^5 x (a+b x)^4\right )}{192 b^6 (a+b x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(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))/(192*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]])/(64*b^(13
/2))

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Maple [B]  time = 0.033, size = 701, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*e^4*(e*x+d)^(3/2)/b^5-10*e^5/b^6*(e*x+d)^(1/2)*a+10*e^4/b^5*(e*x+d)^(1/2)*d-
765/64*e^6/b^3/(b*e*x+a*e)^4*(e*x+d)^(7/2)*a^2+765/32*e^5/b^2/(b*e*x+a*e)^4*(e*x
+d)^(7/2)*a*d-765/64*e^4/b/(b*e*x+a*e)^4*(e*x+d)^(7/2)*d^2-5855/192*e^7/b^4/(b*e
*x+a*e)^4*(e*x+d)^(5/2)*a^3+5855/64*e^6/b^3/(b*e*x+a*e)^4*(e*x+d)^(5/2)*a^2*d-58
55/64*e^5/b^2/(b*e*x+a*e)^4*(e*x+d)^(5/2)*a*d^2+5855/192*e^4/b/(b*e*x+a*e)^4*(e*
x+d)^(5/2)*d^3-5153/192*e^8/b^5/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a^4+5153/48*e^7/b^4/
(b*e*x+a*e)^4*(e*x+d)^(3/2)*a^3*d-5153/32*e^6/b^3/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a^
2*d^2+5153/48*e^5/b^2/(b*e*x+a*e)^4*(e*x+d)^(3/2)*a*d^3-5153/192*e^4/b/(b*e*x+a*
e)^4*(e*x+d)^(3/2)*d^4-515/64*e^9/b^6/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a^5+2575/64*e^
8/b^5/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a^4*d-2575/32*e^7/b^4/(b*e*x+a*e)^4*(e*x+d)^(1
/2)*a^3*d^2+2575/32*e^6/b^3/(b*e*x+a*e)^4*(e*x+d)^(1/2)*a^2*d^3-2575/64*e^5/b^2/
(b*e*x+a*e)^4*(e*x+d)^(1/2)*a*d^4+515/64*e^4/b/(b*e*x+a*e)^4*(e*x+d)^(1/2)*d^5+1
155/64*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
^2-1155/32*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*d+1155/64*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))*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((b*x + a)*(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 = 0.316356, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,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((b*x+a)*(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 = 0.333626, size = 643, normalized size = 3.25 \[ \frac{1155 \,{\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \, \sqrt{-b^{2} d + a b e} b^{6}} - \frac{2295 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{5} d^{2} e^{4} - 5855 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} d^{3} e^{4} + 5153 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d^{4} e^{4} - 1545 \, \sqrt{x e + d} b^{5} d^{5} e^{4} - 4590 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{4} d e^{5} + 17565 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{4} d^{2} e^{5} - 20612 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{4} d^{3} e^{5} + 7725 \, \sqrt{x e + d} a b^{4} d^{4} e^{5} + 2295 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{3} e^{6} - 17565 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{3} d e^{6} + 30918 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{3} d^{2} e^{6} - 15450 \, \sqrt{x e + d} a^{2} b^{3} d^{3} e^{6} + 5855 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{2} e^{7} - 20612 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{2} d e^{7} + 15450 \, \sqrt{x e + d} a^{3} b^{2} d^{2} e^{7} + 5153 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b e^{8} - 7725 \, \sqrt{x e + d} a^{4} b d e^{8} + 1545 \, \sqrt{x e + d} a^{5} e^{9}}{192 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{6}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{10} e^{4} + 15 \, \sqrt{x e + d} b^{10} d e^{4} - 15 \, \sqrt{x e + d} a b^{9} e^{5}\right )}}{3 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,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) - 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*sqr
t(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)
+ 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