3.1654 \(\int \frac{(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=224 \[ -\frac{3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}+\frac{3003 e^5 \sqrt{d+e x} (b d-a e)}{128 b^7}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6} \]

[Out]

(3003*e^5*(b*d - a*e)*Sqrt[d + e*x])/(128*b^7) + (1001*e^5*(d + e*x)^(3/2))/(128
*b^6) - (3003*e^4*(d + e*x)^(5/2))/(640*b^5*(a + b*x)) - (429*e^3*(d + e*x)^(7/2
))/(320*b^4*(a + b*x)^2) - (143*e^2*(d + e*x)^(9/2))/(240*b^3*(a + b*x)^3) - (13
*e*(d + e*x)^(11/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(13/2)/(5*b*(a + b*x)^5) -
(3003*e^5*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
128*b^(15/2))

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Rubi [A]  time = 0.408365, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}+\frac{3003 e^5 \sqrt{d+e x} (b d-a e)}{128 b^7}-\frac{3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac{429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac{143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac{13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac{1001 e^5 (d+e x)^{3/2}}{128 b^6} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(3003*e^5*(b*d - a*e)*Sqrt[d + e*x])/(128*b^7) + (1001*e^5*(d + e*x)^(3/2))/(128
*b^6) - (3003*e^4*(d + e*x)^(5/2))/(640*b^5*(a + b*x)) - (429*e^3*(d + e*x)^(7/2
))/(320*b^4*(a + b*x)^2) - (143*e^2*(d + e*x)^(9/2))/(240*b^3*(a + b*x)^3) - (13
*e*(d + e*x)^(11/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(13/2)/(5*b*(a + b*x)^5) -
(3003*e^5*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
128*b^(15/2))

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Rubi in Sympy [A]  time = 91.2708, size = 207, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{\frac{13}{2}}}{5 b \left (a + b x\right )^{5}} - \frac{13 e \left (d + e x\right )^{\frac{11}{2}}}{40 b^{2} \left (a + b x\right )^{4}} - \frac{143 e^{2} \left (d + e x\right )^{\frac{9}{2}}}{240 b^{3} \left (a + b x\right )^{3}} - \frac{429 e^{3} \left (d + e x\right )^{\frac{7}{2}}}{320 b^{4} \left (a + b x\right )^{2}} - \frac{3003 e^{4} \left (d + e x\right )^{\frac{5}{2}}}{640 b^{5} \left (a + b x\right )} + \frac{1001 e^{5} \left (d + e x\right )^{\frac{3}{2}}}{128 b^{6}} - \frac{3003 e^{5} \sqrt{d + e x} \left (a e - b d\right )}{128 b^{7}} + \frac{3003 e^{5} \left (a e - b d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(13/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-(d + e*x)**(13/2)/(5*b*(a + b*x)**5) - 13*e*(d + e*x)**(11/2)/(40*b**2*(a + b*x
)**4) - 143*e**2*(d + e*x)**(9/2)/(240*b**3*(a + b*x)**3) - 429*e**3*(d + e*x)**
(7/2)/(320*b**4*(a + b*x)**2) - 3003*e**4*(d + e*x)**(5/2)/(640*b**5*(a + b*x))
+ 1001*e**5*(d + e*x)**(3/2)/(128*b**6) - 3003*e**5*sqrt(d + e*x)*(a*e - b*d)/(1
28*b**7) + 3003*e**5*(a*e - b*d)**(3/2)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*
d))/(128*b**(15/2))

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Mathematica [A]  time = 0.746738, size = 208, normalized size = 0.93 \[ -\frac{3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}-\frac{\sqrt{d+e x} \left (1280 e^5 (a+b x)^5 (18 a e-19 b d)+35595 e^4 (a+b x)^4 (b d-a e)^2+21070 e^3 (a+b x)^3 (b d-a e)^3+10024 e^2 (a+b x)^2 (b d-a e)^4+2928 e (a+b x) (b d-a e)^5+384 (b d-a e)^6-1280 b e^6 x (a+b x)^5\right )}{1920 b^7 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(384*(b*d - a*e)^6 + 2928*e*(b*d - a*e)^5*(a + b*x) + 10024*e^2*
(b*d - a*e)^4*(a + b*x)^2 + 21070*e^3*(b*d - a*e)^3*(a + b*x)^3 + 35595*e^4*(b*d
- a*e)^2*(a + b*x)^4 + 1280*e^5*(-19*b*d + 18*a*e)*(a + b*x)^5 - 1280*b*e^6*x*(
a + b*x)^5))/(1920*b^7*(a + b*x)^5) - (3003*e^5*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))

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Maple [B]  time = 0.039, size = 908, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

4401/64*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^5*d-22005/128*e^9/b^5/(b*e*x+a*e)
^5*(e*x+d)^(1/2)*a^4*d^2+7335/32*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d^3-220
05/128*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^4*a^2+4401/64*e^6/b^2/(b*e*x+a*e)^5
*(e*x+d)^(1/2)*d^5*a+12131/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^2*d-12131/64
*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a*d^2+2373/64*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)
^(9/2)*a*d-3003/64*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*d+5012/15*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3*d-2506/5*e^7/b^3/
(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^2*d^2+5012/15*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*
a*d^3+48145/192*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^4*d-48145/96*e^8/b^4/(b*e*
x+a*e)^5*(e*x+d)^(3/2)*a^3*d^2+48145/96*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*
d^3-48145/192*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^4+2/3*e^5*(e*x+d)^(3/2)/b^
6-1253/15*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^4-9629/192*e^10/b^6/(b*e*x+a*e)^
5*(e*x+d)^(3/2)*a^5-12131/192*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^3+3003/128*e
^7/b^7/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2-1467/
128*e^11/b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^6+3003/128*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^2-2373/128*e^5/b/(b*e*x+a*e)^5*(
e*x+d)^(9/2)*d^2-1253/15*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^4+9629/192*e^5/b/(b
*e*x+a*e)^5*(e*x+d)^(3/2)*d^5-1467/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^6+121
31/192*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^3-2373/128*e^7/b^3/(b*e*x+a*e)^5*(e*x
+d)^(9/2)*a^2-12*e^6/b^7*(e*x+d)^(1/2)*a+12*e^5/b^6*(e*x+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)^(13/2)/(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 = 0.22956, 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)^(13/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*b^5*
d*e^5 - a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*
e^5 - a^4*b^2*e^6)*x^2 + 5*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sqrt((b*d - a*e)/b)*lo
g((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^6*e^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e - 1144*a^2*b^4*d^4*e^2 - 2574*a
^3*b^3*d^3*e^3 - 9009*a^4*b^2*d^2*e^4 + 60060*a^5*b*d*e^5 - 45045*a^6*e^6 + 1280
*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^2*e^4 - 38558*a*b^5*d*e^5 + 2
7599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917*a*b^5*d^2*e^4 - 46475*a^2*b^
4*d*e^5 + 33891*a^3*b^3*e^6)*x^3 - 2*(5012*b^6*d^4*e^2 + 11557*a*b^5*d^3*e^3 + 4
2042*a^2*b^4*d^2*e^4 - 260403*a^3*b^3*d*e^5 + 192192*a^4*b^2*e^6)*x^2 - 2*(1464*
b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e^3 + 21879*a^3*b^3*d^2*e^4 -
141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*
x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7), -1/1920*(45045*
(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 - a^2*b^4*
e^6)*x^4 + 10*(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a^4*b^2*e^
6)*x^2 + 5*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x +
d)/sqrt(-(b*d - a*e)/b)) - (1280*b^6*e^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e -
1144*a^2*b^4*d^4*e^2 - 2574*a^3*b^3*d^3*e^3 - 9009*a^4*b^2*d^2*e^4 + 60060*a^5*b
*d*e^5 - 45045*a^6*e^6 + 1280*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^
2*e^4 - 38558*a*b^5*d*e^5 + 27599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917
*a*b^5*d^2*e^4 - 46475*a^2*b^4*d*e^5 + 33891*a^3*b^3*e^6)*x^3 - 2*(5012*b^6*d^4*
e^2 + 11557*a*b^5*d^3*e^3 + 42042*a^2*b^4*d^2*e^4 - 260403*a^3*b^3*d*e^5 + 19219
2*a^4*b^2*e^6)*x^2 - 2*(1464*b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e
^3 + 21879*a^3*b^3*d^2*e^4 - 141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*
x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*
x + a^5*b^7)]

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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)**(13/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247106, size = 828, normalized size = 3.7 \[ \frac{3003 \,{\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\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^{7}} - \frac{35595 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} d^{2} e^{5} - 121310 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d^{3} e^{5} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{4} e^{5} - 96290 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{5} e^{5} + 22005 \, \sqrt{x e + d} b^{6} d^{6} e^{5} - 71190 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{5} d e^{6} + 363930 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} d^{2} e^{6} - 641536 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d^{3} e^{6} + 481450 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{4} e^{6} - 132030 \, \sqrt{x e + d} a b^{5} d^{5} e^{6} + 35595 \,{\left (x e + d\right )}^{\frac{9}{2}} a^{2} b^{4} e^{7} - 363930 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{4} d e^{7} + 962304 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} d^{2} e^{7} - 962900 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d^{3} e^{7} + 330075 \, \sqrt{x e + d} a^{2} b^{4} d^{4} e^{7} + 121310 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{3} b^{3} e^{8} - 641536 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{3} d e^{8} + 962900 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} d^{2} e^{8} - 440100 \, \sqrt{x e + d} a^{3} b^{3} d^{3} e^{8} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{4} b^{2} e^{9} - 481450 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b^{2} d e^{9} + 330075 \, \sqrt{x e + d} a^{4} b^{2} d^{2} e^{9} + 96290 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{5} b e^{10} - 132030 \, \sqrt{x e + d} a^{5} b d e^{10} + 22005 \, \sqrt{x e + d} a^{6} e^{11}}{1920 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{7}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{12} e^{5} + 18 \, \sqrt{x e + d} b^{12} d e^{5} - 18 \, \sqrt{x e + d} a b^{11} e^{6}\right )}}{3 \, b^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(13/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

3003/128*(b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*
d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^7) - 1/1920*(35595*(x*e + d)^(9/2)*b^6*d^2*e
^5 - 121310*(x*e + d)^(7/2)*b^6*d^3*e^5 + 160384*(x*e + d)^(5/2)*b^6*d^4*e^5 - 9
6290*(x*e + d)^(3/2)*b^6*d^5*e^5 + 22005*sqrt(x*e + d)*b^6*d^6*e^5 - 71190*(x*e
+ d)^(9/2)*a*b^5*d*e^6 + 363930*(x*e + d)^(7/2)*a*b^5*d^2*e^6 - 641536*(x*e + d)
^(5/2)*a*b^5*d^3*e^6 + 481450*(x*e + d)^(3/2)*a*b^5*d^4*e^6 - 132030*sqrt(x*e +
d)*a*b^5*d^5*e^6 + 35595*(x*e + d)^(9/2)*a^2*b^4*e^7 - 363930*(x*e + d)^(7/2)*a^
2*b^4*d*e^7 + 962304*(x*e + d)^(5/2)*a^2*b^4*d^2*e^7 - 962900*(x*e + d)^(3/2)*a^
2*b^4*d^3*e^7 + 330075*sqrt(x*e + d)*a^2*b^4*d^4*e^7 + 121310*(x*e + d)^(7/2)*a^
3*b^3*e^8 - 641536*(x*e + d)^(5/2)*a^3*b^3*d*e^8 + 962900*(x*e + d)^(3/2)*a^3*b^
3*d^2*e^8 - 440100*sqrt(x*e + d)*a^3*b^3*d^3*e^8 + 160384*(x*e + d)^(5/2)*a^4*b^
2*e^9 - 481450*(x*e + d)^(3/2)*a^4*b^2*d*e^9 + 330075*sqrt(x*e + d)*a^4*b^2*d^2*
e^9 + 96290*(x*e + d)^(3/2)*a^5*b*e^10 - 132030*sqrt(x*e + d)*a^5*b*d*e^10 + 220
05*sqrt(x*e + d)*a^6*e^11)/(((x*e + d)*b - b*d + a*e)^5*b^7) + 2/3*((x*e + d)^(3
/2)*b^12*e^5 + 18*sqrt(x*e + d)*b^12*d*e^5 - 18*sqrt(x*e + d)*a*b^11*e^6)/b^18