3.1702 \(\int \frac{(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=202 \[ -\frac{(d+e x)^{5/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (d+e x)^{3/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 e^2 (a+b x) \sqrt{d+e x}}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(15*e^2*(a + b*x)*Sqrt[d + e*x])/(4*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*e*(d
+ e*x)^(3/2))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(5/2)/(2*b*(a +
b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*e^2*Sqrt[b*d - a*e]*(a + b*x)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])

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Rubi [A]  time = 0.279261, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(d+e x)^{5/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (d+e x)^{3/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 e^2 (a+b x) \sqrt{d+e x}}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(15*e^2*(a + b*x)*Sqrt[d + e*x])/(4*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*e*(d
+ e*x)^(3/2))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(5/2)/(2*b*(a +
b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*e^2*Sqrt[b*d - a*e]*(a + b*x)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(7/2)*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.245714, size = 138, normalized size = 0.68 \[ \frac{\sqrt{b} \sqrt{d+e x} \left (15 a^2 e^2-5 a b e (d-5 e x)+b^2 \left (-2 d^2-9 d e x+8 e^2 x^2\right )\right )-15 e^2 (a+b x)^2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(Sqrt[b]*Sqrt[d + e*x]*(15*a^2*e^2 - 5*a*b*e*(d - 5*e*x) + b^2*(-2*d^2 - 9*d*e*x
+ 8*e^2*x^2)) - 15*e^2*Sqrt[b*d - a*e]*(a + b*x)^2*ArcTanh[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[b*d - a*e]])/(4*b^(7/2)*(a + b*x)*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.024, size = 413, normalized size = 2. \[{\frac{bx+a}{4\,{b}^{3}} \left ( -15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}a{b}^{2}{e}^{3}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{b}^{3}d{e}^{2}+8\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}{b}^{2}{e}^{2}-30\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}b{e}^{3}+30\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xa{b}^{2}d{e}^{2}+9\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}abe-9\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{2}d+16\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xab{e}^{2}-15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}bd{e}^{2}+15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}-14\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde+7\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/4*(-15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^2*e^3+15*arctan((e*
x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^3*d*e^2+8*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1
/2)*x^2*b^2*e^2-30*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^2*b*e^3+30*ar
ctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b^2*d*e^2+9*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(3/2)*a*b*e-9*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^2*d+16*(b*(a*e-b*d))^(1/
2)*(e*x+d)^(1/2)*x*a*b*e^2-15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*e^
3+15*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b*d*e^2+15*(b*(a*e-b*d))^(1
/2)*(e*x+d)^(1/2)*a^2*e^2-14*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b*d*e+7*(b*(a*e
-b*d))^(1/2)*(e*x+d)^(1/2)*b^2*d^2)*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^3/((b*x+a)^2)^
(3/2)

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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)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221997, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (8 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} - 5 \, a b d e + 15 \, a^{2} e^{2} -{\left (9 \, b^{2} d e - 25 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (8 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} - 5 \, a b d e + 15 \, a^{2} e^{2} -{\left (9 \, b^{2} d e - 25 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(15*(b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*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*(8*b^2*e^2*x
^2 - 2*b^2*d^2 - 5*a*b*d*e + 15*a^2*e^2 - (9*b^2*d*e - 25*a*b*e^2)*x)*sqrt(e*x +
d))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3), -1/4*(15*(b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*
e^2)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (8*b^2*e^
2*x^2 - 2*b^2*d^2 - 5*a*b*d*e + 15*a^2*e^2 - (9*b^2*d*e - 25*a*b*e^2)*x)*sqrt(e*
x + d))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)]

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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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233052, size = 336, normalized size = 1.66 \[ -\frac{15 \,{\left (b d e^{2} - a e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \, \sqrt{x e + d} e^{2}}{b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{9 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{2} - 7 \, \sqrt{x e + d} b^{2} d^{2} e^{2} - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{3} + 14 \, \sqrt{x e + d} a b d e^{3} - 7 \, \sqrt{x e + d} a^{2} e^{4}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

-15/4*(b*d*e^2 - a*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*
d + a*b*e)*b^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2*sqrt(x*e + d)*e^2/(b^3*
sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/4*(9*(x*e + d)^(3/2)*b^2*d*e^2 - 7*sqr
t(x*e + d)*b^2*d^2*e^2 - 9*(x*e + d)^(3/2)*a*b*e^3 + 14*sqrt(x*e + d)*a*b*d*e^3
- 7*sqrt(x*e + d)*a^2*e^4)/(((x*e + d)*b - b*d + a*e)^2*b^3*sign(-(x*e + d)*b*e
+ b*d*e - a*e^2))