3.1703 \(\int \frac{(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{3 e \sqrt{d+e x}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]

[Out]

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

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Rubi [A]  time = 0.225301, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{3 e \sqrt{d+e x}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{3/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.185392, size = 110, normalized size = 0.7 \[ \frac{-\frac{3 e^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b d-a e}}-\sqrt{b} \sqrt{d+e x} (3 a e+2 b d+5 b e x)}{4 b^{5/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 194, normalized size = 1.2 \[ -{\frac{bx+a}{4\,{b}^{2}} \left ( -3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{b}^{2}{e}^{2}-6\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xab{e}^{2}+5\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}b-3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{e}^{2}+3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(-3*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^2*e^2-6*arctan((e*x+d
)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a*b*e^2+5*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b-3
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*e^2+3*(b*(a*e-b*d))^(1/2)*(e*x+
d)^(1/2)*a*e-3*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b*d)*(b*x+a)/(b*(a*e-b*d))^(1/2
)/b^2/((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)^(3/2)/(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 = 0.21846, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} d - a b e}{\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} \sqrt{e x + d} - 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{b^{2} d - a b e}}, -\frac{\sqrt{-b^{2} d + a b e}{\left (5 \, b e x + 2 \, b d + 3 \, a e\right )} \sqrt{e x + d} + 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*(2*sqrt(b^2*d - a*b*e)*(5*b*e*x + 2*b*d + 3*a*e)*sqrt(e*x + d) - 3*(b^2*e^
2*x^2 + 2*a*b*e^2*x + a^2*e^2)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) -
2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((b^4*x^2 + 2*a*b^3*x + a^2*b^2)*sq
rt(b^2*d - a*b*e)), -1/4*(sqrt(-b^2*d + a*b*e)*(5*b*e*x + 2*b*d + 3*a*e)*sqrt(e*
x + d) + 3*(b^2*e^2*x^2 + 2*a*b*e^2*x + a^2*e^2)*arctan(-(b*d - a*e)/(sqrt(-b^2*
d + a*b*e)*sqrt(e*x + d))))/((b^4*x^2 + 2*a*b^3*x + a^2*b^2)*sqrt(-b^2*d + a*b*e
))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22631, size = 219, normalized size = 1.39 \[ -\frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \, \sqrt{-b^{2} d + a b e} b^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{5 \,{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} - 3 \, \sqrt{x e + d} b d e^{2} + 3 \, \sqrt{x e + d} a e^{3}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/(sqrt(-b^2*d + a*b*e)*b^2*
sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/4*(5*(x*e + d)^(3/2)*b*e^2 - 3*sqrt(x*
e + d)*b*d*e^2 + 3*sqrt(x*e + d)*a*e^3)/(((x*e + d)*b - b*d + a*e)^2*b^2*sign(-(
x*e + d)*b*e + b*d*e - a*e^2))