3.2068 \(\int \frac{(a+b x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=86 \[ -\frac{2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}+\frac{2 \sqrt{d+e x} (b d-a e)}{b^2}+\frac{2 (d+e x)^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.110864, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}+\frac{2 \sqrt{d+e x} (b d-a e)}{b^2}+\frac{2 (d+e x)^{3/2}}{3 b} \]

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

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Rubi in Sympy [A]  time = 34.9361, size = 75, normalized size = 0.87 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 b} - \frac{2 \sqrt{d + e x} \left (a e - b d\right )}{b^{2}} + \frac{2 \left (a e - b d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{5}{2}}} \]

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

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Mathematica [A]  time = 0.138478, size = 77, normalized size = 0.9 \[ \frac{2 \sqrt{d+e x} (-3 a e+4 b d+b e x)}{3 b^2}-\frac{2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}} \]

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

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Maple [B]  time = 0.008, size = 167, normalized size = 1.9 \[{\frac{2}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{\sqrt{ex+d}ae}{{b}^{2}}}+2\,{\frac{\sqrt{ex+d}d}{b}}+2\,{\frac{{a}^{2}{e}^{2}}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-4\,{\frac{aed}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+2\,{\frac{{d}^{2}}{\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297982, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b d - a e\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}}{3 \, b^{2}}, -\frac{2 \,{\left (3 \,{\left (b d - a e\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt{e x + d}\right )}}{3 \, b^{2}}\right ] \]

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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 69.1576, size = 201, normalized size = 2.34 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 b} + \frac{\sqrt{d + e x} \left (- 2 a e + 2 b d\right )}{b^{2}} + \frac{2 \left (a e - b d\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}}} & \text{for}\: \frac{a e - b d}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: d + e x > \frac{- a e + b d}{b} \wedge \frac{a e - b d}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: \frac{a e - b d}{b} < 0 \wedge d + e x < \frac{- a e + b d}{b} \end{cases}\right )}{b^{2}} \]

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]

2*(d + e*x)**(3/2)/(3*b) + sqrt(d + e*x)*(-2*a*e + 2*b*d)/b**2 + 2*(a*e - b*d)**
2*Piecewise((atan(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b*sqrt((a*e - b*d)/b)), (a
*e - b*d)/b > 0), (-acoth(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b*sqrt((-a*e + b*
d)/b)), ((a*e - b*d)/b < 0) & (d + e*x > (-a*e + b*d)/b)), (-atanh(sqrt(d + e*x)
/sqrt((-a*e + b*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b < 0) & (d + e*x
< (-a*e + b*d)/b)))/b**2

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GIAC/XCAS [A]  time = 0.280928, size = 151, normalized size = 1.76 \[ \frac{2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2} + 3 \, \sqrt{x e + d} b^{2} d - 3 \, \sqrt{x e + d} a b e\right )}}{3 \, b^{3}} \]

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, algorithm="giac")

[Out]

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