3.1698 \(\int \frac{1}{(d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=168 \[ \frac{2 b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}+\frac{2 (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{2 b^{3/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.244763, antiderivative size = 168, 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 \[ \frac{2 b (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}+\frac{2 (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{2 b^{3/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.220793, size = 117, normalized size = 0.7 \[ -\frac{2 (a+b x) \left (3 b^{3/2} (d+e x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )+\sqrt{d+e x} \sqrt{b d-a e} (a e-4 b d-3 b e x)\right )}{3 \sqrt{(a+b x)^2} (d+e x)^2 (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.017, size = 130, normalized size = 0.8 \[{\frac{2\,bx+2\,a}{3\, \left ( ae-bd \right ) ^{2}} \left ( 3\,{b}^{2}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}+3\,\sqrt{b \left ( ae-bd \right ) }xbe-\sqrt{b \left ( ae-bd \right ) }ae+4\,\sqrt{b \left ( ae-bd \right ) }bd \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219589, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, b e x + 3 \,{\left (b e x + b d\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 8 \, b d - 2 \, a e}{3 \,{\left (b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} +{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}, \frac{2 \,{\left (3 \, b e x - 3 \,{\left (b e x + b d\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 4 \, b d - a e\right )}}{3 \,{\left (b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} +{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/3*(6*b*e*x + 3*(b*e*x + b*d)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2
*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 8*b*d
- 2*a*e)/((b^2*d^3 - 2*a*b*d^2*e + a^2*d*e^2 + (b^2*d^2*e - 2*a*b*d*e^2 + a^2*e
^3)*x)*sqrt(e*x + d)), 2/3*(3*b*e*x - 3*(b*e*x + b*d)*sqrt(e*x + d)*sqrt(-b/(b*d
- a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)) + 4*b*d - a
*e)/((b^2*d^3 - 2*a*b*d^2*e + a^2*d*e^2 + (b^2*d^2*e - 2*a*b*d*e^2 + a^2*e^3)*x)
*sqrt(e*x + d))]

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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(1/(e*x+d)**(5/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216782, size = 170, normalized size = 1.01 \[ \frac{2}{3} \,{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} + \frac{3 \,{\left (x e + d\right )} b + b d - a e}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (x e + d\right )}^{\frac{3}{2}}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

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