3.1705 \(\int \frac{1}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac{3 e \sqrt{d+e x}}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(3*e*Sqrt[d + e*x])/(4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e
*x]/(2*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e^2*(a + b*x)*A
rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*Sqrt[b]*(b*d - a*e)^(5/2)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.301718, antiderivative size = 172, 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 \[ -\frac{3 e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac{3 e \sqrt{d+e x}}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x}}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*Sqrt[d + e*x])/(4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e
*x]/(2*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e^2*(a + b*x)*A
rcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*Sqrt[b]*(b*d - a*e)^(5/2)*Sq
rt[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/(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.180275, size = 121, normalized size = 0.7 \[ \frac{\sqrt{b} \sqrt{d+e x} \sqrt{b d-a e} (5 a e-2 b d+3 b e x)-3 e^2 (a+b x)^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (a+b x) \sqrt{(a+b x)^2} (b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 203, normalized size = 1.2 \[{\frac{bx+a}{4\, \left ( ae-bd \right ) ^{2}} \left ( 3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{b}^{2}{e}^{2}+6\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xab{e}^{2}+3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xbe+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{e}^{2}+5\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae-2\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225951, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} d - a b e}{\left (3 \, b e x - 2 \, b d + 5 \, a e\right )} \sqrt{e x + d} + 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e}}, \frac{\sqrt{-b^{2} d + a b e}{\left (3 \, b e x - 2 \, b d + 5 \, a e\right )} \sqrt{e x + d} - 3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} +{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(d + e*x)*((a + b*x)**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.226034, size = 392, normalized size = 2.28 \[ -\frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{2} d^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 2 \, a b d e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{3 \,{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} - 5 \, \sqrt{x e + d} b d e^{2} + 5 \, \sqrt{x e + d} a e^{3}}{4 \,{\left (b^{2} d^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 2 \, a b d e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

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