3.1697 \(\int \frac{1}{(d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=119 \[ \frac{2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.191186, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.0864799, size = 96, normalized size = 0.81 \[ \frac{2 (a+b x) \left (\sqrt{b d-a e}-\sqrt{b} \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{\sqrt{(a+b x)^2} \sqrt{d+e x} (b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 90, normalized size = 0.8 \[ -2\,{\frac{bx+a}{\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ae-bd \right ) \sqrt{ex+d}\sqrt{b \left ( ae-bd \right ) }} \left ( b\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}+\sqrt{b \left ( ae-bd \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218439, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) - 2}{{\left (b d - a e\right )} \sqrt{e x + d}}, -\frac{2 \,{\left (\sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) - 1\right )}}{{\left (b d - a e\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-(sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sq
rt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2)/((b*d - a*e)*sqrt(e*x + d)), -2
*(sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(s
qrt(e*x + d)*b)) - 1)/((b*d - a*e)*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)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214503, size = 109, normalized size = 0.92 \[ 2 \,{\left (\frac{b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} + \frac{1}{{\left (b d - a e\right )} \sqrt{x e + d}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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