3.1709 \(\int \frac{(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=400 \[ -\frac{(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3003 e^4 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3003 e^4 (a+b x) \sqrt{d+e x} (b d-a e)^2}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1001 e^4 (a+b x) (d+e x)^{3/2} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(3003*e^4*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x])/(64*b^7*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) + (1001*e^4*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))/(64*b^6*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) + (3003*e^4*(a + b*x)*(d + e*x)^(5/2))/(320*b^5*Sqrt[a^2 + 2*a
*b*x + b^2*x^2]) - (429*e^3*(d + e*x)^(7/2))/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^
2]) - (143*e^2*(d + e*x)^(9/2))/(96*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
- (13*e*(d + e*x)^(11/2))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(d + e*x)^(13/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3003*e^4*(b*
d - a*e)^(5/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b
^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.690938, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(d+e x)^{13/2}}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{13 e (d+e x)^{11/2}}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3003 e^4 (a+b x) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3003 e^4 (a+b x) \sqrt{d+e x} (b d-a e)^2}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1001 e^4 (a+b x) (d+e x)^{3/2} (b d-a e)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3003 e^4 (a+b x) (d+e x)^{5/2}}{320 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{429 e^3 (d+e x)^{7/2}}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{143 e^2 (d+e x)^{9/2}}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(3003*e^4*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x])/(64*b^7*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) + (1001*e^4*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))/(64*b^6*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) + (3003*e^4*(a + b*x)*(d + e*x)^(5/2))/(320*b^5*Sqrt[a^2 + 2*a
*b*x + b^2*x^2]) - (429*e^3*(d + e*x)^(7/2))/(64*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^
2]) - (143*e^2*(d + e*x)^(9/2))/(96*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
- (13*e*(d + e*x)^(11/2))/(24*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(d + e*x)^(13/2)/(4*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3003*e^4*(b*
d - a*e)^(5/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b
^(15/2)*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)**(13/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.18577, size = 228, normalized size = 0.57 \[ \frac{(a+b x)^5 \left (\frac{\sqrt{d+e x} \left (128 e^4 \left (225 a^2 e^2-475 a b d e+253 b^2 d^2\right )+128 b e^5 x (31 b d-25 a e)+\frac{22155 e^3 (a e-b d)^3}{a+b x}-\frac{7630 e^2 (b d-a e)^4}{(a+b x)^2}+\frac{1960 e (a e-b d)^5}{(a+b x)^3}-\frac{240 (b d-a e)^6}{(a+b x)^4}+384 b^2 e^6 x^2\right )}{15 b^7}-\frac{3003 e^4 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{15/2}}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((a + b*x)^5*((Sqrt[d + e*x]*(128*e^4*(253*b^2*d^2 - 475*a*b*d*e + 225*a^2*e^2)
+ 128*b*e^5*(31*b*d - 25*a*e)*x + 384*b^2*e^6*x^2 - (240*(b*d - a*e)^6)/(a + b*x
)^4 + (1960*e*(-(b*d) + a*e)^5)/(a + b*x)^3 - (7630*e^2*(b*d - a*e)^4)/(a + b*x)
^2 + (22155*e^3*(-(b*d) + a*e)^3)/(a + b*x)))/(15*b^7) - (3003*e^4*(b*d - a*e)^(
5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(15/2)))/(64*((a + b*x)
^2)^(5/2))

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Maple [B]  time = 0.035, size = 2192, normalized size = 5.5 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

1/960*(-45045*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^7*e^7-345600*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a^3*b^3*d*e^5+172800*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(1/2)*x^2*a^2*b^4*d^2*e^4-230400*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^4*b^2*d*e
^5+12800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^3*a*b^5*d*e^4-57600*(b*(a*e-b*d))^(
1/2)*(e*x+d)^(1/2)*x^4*a*b^5*d*e^5+19200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^2*a
^2*b^4*d*e^4-230400*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^3*a^2*b^4*d*e^5+115200*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^3*a*b^5*d^2*e^4-531650*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(3/2)*a^2*b^4*d^3*e^2+265825*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^5*d^4*e
+115200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^5*b*e^6-155070*(b*(a*e-b*d))^(1/2)
*(e*x+d)^(1/2)*a^5*b*d*e^5+272475*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*b^2*d^2*
e^4-324900*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b^3*d^3*e^3+243675*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(1/2)*a^2*b^4*d^4*e^2-97470*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*
b^5*d^5*e+1536*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*x^3*a*b^5*e^4-3200*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(3/2)*x^4*a*b^5*e^5+3200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^4*b
^6*d*e^4+135135*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a^2*b^5*d*e^6-13
5135*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a*b^6*d^2*e^5+2304*(b*(a*e-
b*d))^(1/2)*(e*x+d)^(5/2)*x^2*a^2*b^4*e^4+45045*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2
)*a^6*e^6+16245*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^6*d^6-22155*(b*(a*e-b*d))^(1
/2)*(e*x+d)^(7/2)*b^6*d^3+58835*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^6*d^4-53165*
(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^6*d^5-180180*arctan((e*x+d)^(1/2)*b/(b*(a*e-
b*d))^(1/2))*x*a^6*b*e^7+49965*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^5*b*e^5+13513
5*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^6*b*d*e^6-135135*arctan((e*x+d)^
(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*b^2*d^2*e^5+45045*arctan((e*x+d)^(1/2)*b/(b*(a*
e-b*d))^(1/2))*a^4*b^3*d^3*e^4-45045*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*x^4*a^3*b^4*e^7+45045*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^7*d^3*e
^4-180180*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a^4*b^3*e^7+22155*(b*(
a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a^3*b^3*e^3-270270*arctan((e*x+d)^(1/2)*b/(b*(a*e-
b*d))^(1/2))*x^2*a^5*b^2*e^7+59219*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^4*b^2*e^4
+115200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^3*b^3*d^2*e^4+12800*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(3/2)*x*a^3*b^3*d*e^4+384*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*x^4*b^
6*e^4-12800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^3*a^2*b^4*e^5+28800*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(1/2)*x^4*a^2*b^4*e^6+28800*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^
4*b^6*d^2*e^4+540540*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a^3*b^4*d*e
^6-540540*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a^2*b^5*d^2*e^5+180180
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^6*d^3*e^4-66465*(b*(a*e-b*d
))^(1/2)*(e*x+d)^(7/2)*a^2*b^4*d*e^2+66465*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b
^5*d^2*e+1536*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*x*a^3*b^3*e^4-19200*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(3/2)*x^2*a^3*b^3*e^5+115200*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x
^3*a^3*b^3*e^6+810810*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^4*b^3*d*
e^6-810810*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^4*d^2*e^5+27027
0*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^5*d^3*e^4-235340*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(5/2)*a^3*b^3*d*e^3+353010*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2
)*a^2*b^4*d^2*e^2-235340*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^5*d^3*e-12800*(b*
(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a^4*b^2*e^5+172800*(b*(a*e-b*d))^(1/2)*(e*x+d)^
(1/2)*x^2*a^4*b^2*e^6+540540*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^5*b
^2*d*e^6-540540*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b^3*d^2*e^5+18
0180*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^4*d^3*e^4-262625*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(3/2)*a^4*b^2*d*e^4+531650*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/
2)*a^3*b^3*d^2*e^3)*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^7/((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)^(13/2)/(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 = 0.227394, 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)^(5/2),x, algorithm="fricas")

[Out]

[1/1920*(45045*(a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 + a^6*e^6 + (b^6*d^2*e^4 - 2*a*b
^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 - 2*a^2*b^4*d*e^5 + a^3*b^3*e^6)*
x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 4*(a^3*b^3*d^2*e
^4 - 2*a^4*b^2*d*e^5 + a^5*b*e^6)*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*(384*b^6*e^6*x^6 - 240
*b^6*d^6 - 520*a*b^5*d^5*e - 1430*a^2*b^4*d^4*e^2 - 6435*a^3*b^3*d^3*e^3 + 69069
*a^4*b^2*d^2*e^4 - 105105*a^5*b*d*e^5 + 45045*a^6*e^6 + 128*(31*b^6*d*e^5 - 13*a
*b^5*e^6)*x^5 + 128*(253*b^6*d^2*e^4 - 351*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 -
(22155*b^6*d^3*e^3 - 196001*a*b^5*d^2*e^4 + 285857*a^2*b^4*d*e^5 - 119691*a^3*b^
3*e^6)*x^3 - (7630*b^6*d^4*e^2 + 35945*a*b^5*d^3*e^3 - 347919*a^2*b^4*d^2*e^4 +
517803*a^3*b^3*d*e^5 - 219219*a^4*b^2*e^6)*x^2 - (1960*b^6*d^5*e + 5460*a*b^5*d^
4*e^2 + 25025*a^2*b^4*d^3*e^3 - 256971*a^3*b^3*d^2*e^4 + 387387*a^4*b^2*d*e^5 -
165165*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4
*a^3*b^8*x + a^4*b^7), -1/960*(45045*(a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 + a^6*e^6
+ (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 - 2*a^2*b^4
*d*e^5 + a^3*b^3*e^6)*x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*
x^2 + 4*(a^3*b^3*d^2*e^4 - 2*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)) - (384*b^6*e^6*x^6 - 240*b^6*d^6 - 52
0*a*b^5*d^5*e - 1430*a^2*b^4*d^4*e^2 - 6435*a^3*b^3*d^3*e^3 + 69069*a^4*b^2*d^2*
e^4 - 105105*a^5*b*d*e^5 + 45045*a^6*e^6 + 128*(31*b^6*d*e^5 - 13*a*b^5*e^6)*x^5
+ 128*(253*b^6*d^2*e^4 - 351*a*b^5*d*e^5 + 143*a^2*b^4*e^6)*x^4 - (22155*b^6*d^
3*e^3 - 196001*a*b^5*d^2*e^4 + 285857*a^2*b^4*d*e^5 - 119691*a^3*b^3*e^6)*x^3 -
(7630*b^6*d^4*e^2 + 35945*a*b^5*d^3*e^3 - 347919*a^2*b^4*d^2*e^4 + 517803*a^3*b^
3*d*e^5 - 219219*a^4*b^2*e^6)*x^2 - (1960*b^6*d^5*e + 5460*a*b^5*d^4*e^2 + 25025
*a^2*b^4*d^3*e^3 - 256971*a^3*b^3*d^2*e^4 + 387387*a^4*b^2*d*e^5 - 165165*a^5*b*
e^6)*x)*sqrt(e*x + d))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x +
a^4*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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271796, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^(5/2),x, algorithm="giac")

[Out]

Done