3.1637 \(\int \frac{(d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{3 e \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}-\frac{(d+e x)^{3/2}}{b (a+b x)}+\frac{3 e \sqrt{d+e x}}{b^2} \]

[Out]

(3*e*Sqrt[d + e*x])/b^2 - (d + e*x)^(3/2)/(b*(a + b*x)) - (3*e*Sqrt[b*d - a*e]*A
rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(5/2)

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Rubi [A]  time = 0.12022, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{3 e \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}-\frac{(d+e x)^{3/2}}{b (a+b x)}+\frac{3 e \sqrt{d+e x}}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*Sqrt[d + e*x])/b^2 - (d + e*x)^(3/2)/(b*(a + b*x)) - (3*e*Sqrt[b*d - a*e]*A
rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(5/2)

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Rubi in Sympy [A]  time = 31.7374, size = 73, normalized size = 0.86 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{b \left (a + b x\right )} + \frac{3 e \sqrt{d + e x}}{b^{2}} - \frac{3 e \sqrt{a e - b d} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.17085, size = 85, normalized size = 1. \[ \sqrt{d+e x} \left (\frac{a e-b d}{b^2 (a+b x)}+\frac{2 e}{b^2}\right )-\frac{3 e \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 148, normalized size = 1.7 \[ 2\,{\frac{e\sqrt{ex+d}}{{b}^{2}}}+{\frac{a{e}^{2}}{{b}^{2} \left ( bex+ae \right ) }\sqrt{ex+d}}-{\frac{de}{b \left ( bex+ae \right ) }\sqrt{ex+d}}-3\,{\frac{a{e}^{2}}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+3\,{\frac{de}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217817, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b e x + a e\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (2 \, b e x - b d + 3 \, a e\right )} \sqrt{e x + d}}{2 \,{\left (b^{3} x + a b^{2}\right )}}, -\frac{3 \,{\left (b e x + a e\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (2 \, b e x - b d + 3 \, a e\right )} \sqrt{e x + d}}{b^{3} x + a b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(3*(b*e*x + a*e)*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*e*x - b*d + 3*a*e)*sqrt(e*x + d)
)/(b^3*x + a*b^2), -(3*(b*e*x + a*e)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/s
qrt(-(b*d - a*e)/b)) - (2*b*e*x - b*d + 3*a*e)*sqrt(e*x + d))/(b^3*x + a*b^2)]

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Sympy [A]  time = 57.2858, size = 1129, normalized size = 13.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*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) - a**2*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) + a**2*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) - 4*a*d*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)
+ a*d*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))/b - a*d*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))/b - 4*a*e**2*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)/s
qrt(-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 - d**2*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**2*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**2*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) + 4*d*e*Pi
ecewise((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*sqrt(-a*e/
b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b + 2*e*sqrt(d + e*x)/b**2

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GIAC/XCAS [A]  time = 0.212652, size = 165, normalized size = 1.94 \[ \frac{3 \,{\left (b d e - a e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{2 \, \sqrt{x e + d} e}{b^{2}} - \frac{\sqrt{x e + d} b d e - \sqrt{x e + d} a e^{2}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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