3.2132 \(\int \frac{(a+b x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ -\frac{(d+e x)^{3/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (a+b x) \sqrt{d+e x}}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.276643, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(d+e x)^{3/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (a+b x) \sqrt{d+e x}}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.154066, size = 95, normalized size = 0.64 \[ \frac{\sqrt{b} \sqrt{d+e x} (3 a e-b d+2 b e x)-3 e (a+b x) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 222, normalized size = 1.5 \[{\frac{ \left ( bx+a \right ) ^{2}}{{b}^{2}} \left ( -3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xab{e}^{2}+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{b}^{2}de+2\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xbe-3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{e}^{2}+3\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) abde+3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae-\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}bd \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.330908, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b e x + a e\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (2 \, b e x - b d + 3 \, a e\right )} \sqrt{e x + d}}{2 \,{\left (b^{3} x + a b^{2}\right )}}, -\frac{3 \,{\left (b e x + a e\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (2 \, b e x - b d + 3 \, a e\right )} \sqrt{e x + d}}{b^{3} x + a b^{2}}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

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

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

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)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.303512, size = 270, normalized size = 1.82 \[ -\frac{3 \,{\left (b d e^{2} - a e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{\sqrt{-b^{2} d + a b e} b^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \, \sqrt{x e + d} e}{b^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{{\left (\sqrt{x e + d} b d e^{2} - \sqrt{x e + d} a e^{3}\right )} e^{\left (-1\right )}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

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)^(3/2),x, algorithm="giac")

[Out]

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