3.1861 \(\int \frac{(A+B x) \sqrt{d+e x}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=173 \[ \frac{2 (a+b x) \sqrt{d+e x} (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B (a+b x) (d+e x)^{3/2}}{3 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*(A*b - a*B)*(a + b*x)*Sqrt[d + e*x])/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2
*B*(a + b*x)*(d + e*x)^(3/2))/(3*b*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*(A*b -
a*B)*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 = 0.328432, antiderivative size = 173, 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 \[ \frac{2 (a+b x) \sqrt{d+e x} (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B (a+b x) (d+e x)^{3/2}}{3 b e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(a + b*x)*Sqrt[d + e*x])/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2
*B*(a + b*x)*(d + e*x)^(3/2))/(3*b*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*(A*b -
a*B)*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)**(1/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.274368, size = 114, normalized size = 0.66 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{d+e x} (-3 a B e+3 A b e+b B (d+e x))+3 e (a B-A b) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{3 b^{5/2} e \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 226, normalized size = 1.3 \[{\frac{2\,bx+2\,a}{3\,{b}^{2}e} \left ( -3\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) ab{e}^{2}+3\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){b}^{2}de+B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}b+3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}{e}^{2}-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) abde+3\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}be-3\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ae \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290826, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a - A b\right )} e \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (B b e x + B b d - 3 \,{\left (B a - A b\right )} e\right )} \sqrt{e x + d}}{3 \, b^{2} e}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} e \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) +{\left (B b e x + B b d - 3 \,{\left (B a - A b\right )} e\right )} \sqrt{e x + d}\right )}}{3 \, b^{2} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*sqrt(d + e*x)/sqrt((a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.28836, size = 227, normalized size = 1.31 \[ -\frac{2 \,{\left (B a b d{\rm sign}\left (b x + a\right ) - A b^{2} d{\rm sign}\left (b x + a\right ) - B a^{2} e{\rm sign}\left (b x + a\right ) + A a b e{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{2} e^{2}{\rm sign}\left (b x + a\right ) - 3 \, \sqrt{x e + d} B a b e^{3}{\rm sign}\left (b x + a\right ) + 3 \, \sqrt{x e + d} A b^{2} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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