3.1694 \(\int \frac{(d+e x)^{3/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=161 \[ \frac{2 (a+b x) \sqrt{d+e x} (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.236634, antiderivative size = 161, 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 \[ \frac{2 (a+b x) \sqrt{d+e x} (b d-a e)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2
*(a + b*x)*(d + e*x)^(3/2))/(3*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*(b*d - a*e)
^(3/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(5/2)*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)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.213193, size = 96, normalized size = 0.6 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{3 b^{5/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 188, normalized size = 1.2 \[{\frac{2\,bx+2\,a}{3\,{b}^{2}} \left ( \sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}b+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{e}^{2}-6\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) abde+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){b}^{2}{d}^{2}-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae+3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/((b*x+a)^2)^(1/2),x)

[Out]

2/3*(b*x+a)*((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^2-6*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*b*d*e+3*a
rctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*b^2*d^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)^2)^(1/2)/b^2/(b*(a*
e-b*d))^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216723, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b d - a e\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}}{3 \, b^{2}}, -\frac{2 \,{\left (3 \,{\left (b d - a e\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}\right )}}{3 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215997, size = 200, normalized size = 1.24 \[ \frac{2 \,{\left (b^{2} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b d e{\rm sign}\left (b x + a\right ) + a^{2} e^{2}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2}{\rm sign}\left (b x + a\right ) + 3 \, \sqrt{x e + d} b^{2} d{\rm sign}\left (b x + a\right ) - 3 \, \sqrt{x e + d} a b e{\rm sign}\left (b x + a\right )\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^2*d^2*sign(b*x + a) - 2*a*b*d*e*sign(b*x + a) + a^2*e^2*sign(b*x + a))*arct
an(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^2) + 2/3*((x*e
+ d)^(3/2)*b^2*sign(b*x + a) + 3*sqrt(x*e + d)*b^2*d*sign(b*x + a) - 3*sqrt(x*e
+ d)*a*b*e*sign(b*x + a))/b^3