3.1640 \(\int \frac{1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=99 \[ -\frac{3 e}{\sqrt{d+e x} (b d-a e)^2}-\frac{1}{(a+b x) \sqrt{d+e x} (b d-a e)}+\frac{3 \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.16657, antiderivative size = 99, 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 \[ -\frac{3 e}{\sqrt{d+e x} (b d-a e)^2}-\frac{1}{(a+b x) \sqrt{d+e x} (b d-a e)}+\frac{3 \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.8591, size = 92, normalized size = 0.93 \[ - \frac{3 \sqrt{b} e \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{5}{2}}} - \frac{3 b \sqrt{d + e x}}{\left (a + b x\right ) \left (a e - b d\right )^{2}} - \frac{2}{\left (a + b x\right ) \sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.2313, size = 90, normalized size = 0.91 \[ \frac{3 \sqrt{b} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{5/2}}-\frac{2 a e+b (d+3 e x)}{(a+b x) \sqrt{d+e x} (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 101, normalized size = 1. \[ -2\,{\frac{e}{ \left ( ae-bd \right ) ^{2}\sqrt{ex+d}}}-{\frac{be}{ \left ( ae-bd \right ) ^{2} \left ( bex+ae \right ) }\sqrt{ex+d}}-3\,{\frac{be}{ \left ( ae-bd \right ) ^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226401, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \, b e x - 3 \,{\left (b e x + a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 2 \, b d + 4 \, a e}{2 \,{\left (a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x\right )} \sqrt{e x + d}}, -\frac{3 \, b e x - 3 \,{\left (b e x + a e\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + b d + 2 \, a e}{{\left (a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210772, size = 207, normalized size = 2.09 \[ -\frac{3 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e}} - \frac{3 \,{\left (x e + d\right )} b e - 2 \, b d e + 2 \, a e^{2}}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left ({\left (x e + d\right )}^{\frac{3}{2}} b - \sqrt{x e + d} b d + \sqrt{x e + d} a e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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