3.1636 \(\int \frac{(d+e x)^{5/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=110 \[ -\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{5 e \sqrt{d+e x} (b d-a e)}{b^3}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{5 e (d+e x)^{3/2}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.17287, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{5 e \sqrt{d+e x} (b d-a e)}{b^3}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{5 e (d+e x)^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.5242, size = 97, normalized size = 0.88 \[ - \frac{\left (d + e x\right )^{\frac{5}{2}}}{b \left (a + b x\right )} + \frac{5 e \left (d + e x\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{5 e \sqrt{d + e x} \left (a e - b d\right )}{b^{3}} + \frac{5 e \left (a e - b d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{7}{2}}} \]

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),x)

[Out]

-(d + e*x)**(5/2)/(b*(a + b*x)) + 5*e*(d + e*x)**(3/2)/(3*b**2) - 5*e*sqrt(d + e
*x)*(a*e - b*d)/b**3 + 5*e*(a*e - b*d)**(3/2)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*
e - b*d))/b**(7/2)

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Mathematica [A]  time = 0.207581, size = 104, normalized size = 0.95 \[ \frac{\sqrt{d+e x} \left (-\frac{3 (b d-a e)^2}{a+b x}+2 e (7 b d-6 a e)+2 b e^2 x\right )}{3 b^3}-\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(2*e*(7*b*d - 6*a*e) + 2*b*e^2*x - (3*(b*d - a*e)^2)/(a + b*x)))/
(3*b^3) - (5*e*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/b^(7/2)

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Maple [B]  time = 0.023, size = 258, normalized size = 2.4 \[{\frac{2\,e}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{\sqrt{ex+d}a{e}^{2}}{{b}^{3}}}+4\,{\frac{e\sqrt{ex+d}d}{{b}^{2}}}-{\frac{{a}^{2}{e}^{3}}{{b}^{3} \left ( bex+ae \right ) }\sqrt{ex+d}}+2\,{\frac{\sqrt{ex+d}ad{e}^{2}}{{b}^{2} \left ( bex+ae \right ) }}-{\frac{e{d}^{2}}{b \left ( bex+ae \right ) }\sqrt{ex+d}}+5\,{\frac{{a}^{2}{e}^{3}}{{b}^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-10\,{\frac{ad{e}^{2}}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+5\,{\frac{e{d}^{2}}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

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),x)

[Out]

2/3*e*(e*x+d)^(3/2)/b^2-4/b^3*(e*x+d)^(1/2)*a*e^2+4*e/b^2*(e*x+d)^(1/2)*d-1/b^3*
(e*x+d)^(1/2)/(b*e*x+a*e)*a^2*e^3+2/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*a*d*e^2-e/b*(e
*x+d)^(1/2)/(b*e*x+a*e)*d^2+5/b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*
(a*e-b*d))^(1/2))*a^2*e^3-10/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(
a*e-b*d))^(1/2))*a*d*e^2+5*e/b/(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((e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220957, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, -\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]

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),x, algorithm="fricas")

[Out]

[-1/6*(15*(a*b*d*e - a^2*e^2 + (b^2*d*e - a*b*e^2)*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*(2*b^
2*e^2*x^2 - 3*b^2*d^2 + 20*a*b*d*e - 15*a^2*e^2 + 2*(7*b^2*d*e - 5*a*b*e^2)*x)*s
qrt(e*x + d))/(b^4*x + a*b^3), -1/3*(15*(a*b*d*e - a^2*e^2 + (b^2*d*e - a*b*e^2)
*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (2*b^2*e^2
*x^2 - 3*b^2*d^2 + 20*a*b*d*e - 15*a^2*e^2 + 2*(7*b^2*d*e - 5*a*b*e^2)*x)*sqrt(e
*x + d))/(b^4*x + a*b^3)]

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Sympy [A]  time = 105.094, size = 1622, normalized size = 14.75 \[ \text{result too large to display} \]

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),x)

[Out]

-2*a**3*e**4*sqrt(d + e*x)/(2*a**2*b**3*e**2 - 2*a*b**4*d*e + 2*a*b**4*e**2*x -
2*b**5*d*e*x) + a**3*e**4*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*
(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*
(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**3) - a**3*e**4*sqrt(-1/(b*(a*e - b*d)**3
))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)*
*3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**3) + 6*a**2*
d*e**3*sqrt(d + e*x)/(2*a**2*b**2*e**2 - 2*a*b**3*d*e + 2*a*b**3*e**2*x - 2*b**4
*d*e*x) - 3*a**2*d*e**3*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a
*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a
*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) + 3*a**2*d*e**3*sqrt(-1/(b*(a*e - b*d)*
*3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d
)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) + 6*a**
2*e**3*Piecewise((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*e/b
- d > 0), (-acoth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b
- d < 0) & (d + e*x > -a*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*s
qrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b**3 - 6*a*d**2*e**
2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3*d*e*x)
+ 3*a*d**2*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d
)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d
)**3)) + sqrt(d + e*x))/(2*b) - 3*a*d**2*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a*
*2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b*
*2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - 12*a*d*e**2*Piecewi
se((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*e/b - d > 0), (-a
coth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d
+ e*x > -a*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d
)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b**2 - 4*a*e**2*sqrt(d + e*x)/b**
3 - d**3*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)
) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)
) + sqrt(d + e*x))/2 + d**3*e*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/
(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/
(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + 2*d**3*e*sqrt(d + e*x)/(2*a**2*e**2 - 2
*a*b*d*e + 2*a*b*e**2*x - 2*b**2*d*e*x) + 6*d**2*e*Piecewise((atan(sqrt(d + e*x)
/sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*e/b - d > 0), (-acoth(sqrt(d + e*x)/sqr
t(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x > -a*e/b + d)),
(-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) &
(d + e*x < -a*e/b + d)))/b + 4*d*e*sqrt(d + e*x)/b**2 + 2*e*(d + e*x)**(3/2)/(3*
b**2)

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GIAC/XCAS [A]  time = 0.216501, size = 258, normalized size = 2.35 \[ \frac{5 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} - \frac{\sqrt{x e + d} b^{2} d^{2} e - 2 \, \sqrt{x e + d} a b d e^{2} + \sqrt{x e + d} a^{2} e^{3}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{3}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{4} e + 6 \, \sqrt{x e + d} b^{4} d e - 6 \, \sqrt{x e + d} a b^{3} e^{2}\right )}}{3 \, b^{6}} \]

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),x, algorithm="giac")

[Out]

5*(b^2*d^2*e - 2*a*b*d*e^2 + a^2*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e
))/(sqrt(-b^2*d + a*b*e)*b^3) - (sqrt(x*e + d)*b^2*d^2*e - 2*sqrt(x*e + d)*a*b*d
*e^2 + sqrt(x*e + d)*a^2*e^3)/(((x*e + d)*b - b*d + a*e)*b^3) + 2/3*((x*e + d)^(
3/2)*b^4*e + 6*sqrt(x*e + d)*b^4*d*e - 6*sqrt(x*e + d)*a*b^3*e^2)/b^6