3.2071 \(\int \frac{a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=69 \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.107827, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.0643, size = 60, normalized size = 0.87 \[ - \frac{2 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{3}{2}}} - \frac{2}{\sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0968502, size = 69, normalized size = 1. \[ \frac{2}{\sqrt{d+e x} (b d-a e)}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 68, normalized size = 1. \[ -2\,{\frac{1}{ \left ( ae-bd \right ) \sqrt{ex+d}}}-2\,{\frac{b}{ \left ( ae-bd \right ) \sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*e-b*d)/(e*x+d)^(1/2)-2*b/(a*e-b*d)/(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((b*x + a)/((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 = 0.296398, 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((b*x + a)/((b^2*x^2 + 2*a*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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.293962, size = 101, normalized size = 1.46 \[ \frac{2 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} + \frac{2}{{\left (b d - a e\right )} \sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((b^2*x^2 + 2*a*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)) + 2/((b*d - a*e)*sqrt(x*e + d))