3.1868 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=280 \[ -\frac{(d+e x)^{5/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (a+b x) (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}+\frac{3 e (a+b x) \sqrt{d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(3*e*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^3*(b*d - a*e)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - ((4*b*B*d + A*b*e - 5*a*B*e)*(d + e*x)^(3/2))/(4*b
^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(5/2))/(2
*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e*(4*b*B*d + A*b*e
- 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(7/2
)*Sqrt[b*d - a*e]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.589691, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(d+e x)^{5/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e+A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (a+b x) (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}+\frac{3 e (a+b x) \sqrt{d+e x} (-5 a B e+A b e+4 b B d)}{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

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*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^3*(b*d - a*e)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - ((4*b*B*d + A*b*e - 5*a*B*e)*(d + e*x)^(3/2))/(4*b
^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(5/2))/(2
*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e*(4*b*B*d + A*b*e
- 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(7/2
)*Sqrt[b*d - a*e]*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 = 0.433416, size = 157, normalized size = 0.56 \[ \frac{(a+b x)^3 \left (-\frac{\sqrt{d+e x} \left (B \left (-15 a^2 e+a b (2 d-25 e x)+4 b^2 x (d-2 e x)\right )+A b (3 a e+2 b d+5 b e x)\right )}{b^3 (a+b x)^2}-\frac{3 e (-5 a B e+A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{b d-a e}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^3*(-((Sqrt[d + e*x]*(A*b*(2*b*d + 3*a*e + 5*b*e*x) + B*(-15*a^2*e + a
*b*(2*d - 25*e*x) + 4*b^2*x*(d - 2*e*x))))/(b^3*(a + b*x)^2)) - (3*e*(4*b*B*d +
A*b*e - 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(7/2)*Sqrt
[b*d - a*e])))/(4*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.03, size = 608, normalized size = 2.2 \[ -{\frac{bx+a}{4\,{b}^{3}e} \left ( -3\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{b}^{3}{e}^{3}+15\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}a{b}^{2}{e}^{3}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){x}^{2}{b}^{3}d{e}^{2}-6\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xa{b}^{2}{e}^{3}-8\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{x}^{2}{b}^{2}{e}^{2}+30\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}b{e}^{3}-24\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xa{b}^{2}d{e}^{2}+5\,A\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{2}e-3\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}b{e}^{3}-9\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}abe+4\,B\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}{b}^{2}d-16\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xab{e}^{2}+15\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}bd{e}^{2}+3\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}ab{e}^{2}-3\,A\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}de-15\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}+11\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde-4\,B\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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]

-1/4*(-3*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^3*e^3+15*B*arctan((
e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a*b^2*e^3-12*B*arctan((e*x+d)^(1/2)*b/(b
*(a*e-b*d))^(1/2))*x^2*b^3*d*e^2-6*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*x*a*b^2*e^3-8*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*b^2*e^2+30*B*arctan((e*x+
d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^2*b*e^3-24*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-
b*d))^(1/2))*x*a*b^2*d*e^2+5*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^2*e-3*A*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b*e^3-9*B*(b*(a*e-b*d))^(1/2)*(e*x+d)
^(3/2)*a*b*e+4*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^2*d-16*B*(b*(a*e-b*d))^(1/2
)*(e*x+d)^(1/2)*x*a*b*e^2+15*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*e
^3-12*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b*d*e^2+3*A*(b*(a*e-b*d)
)^(1/2)*(e*x+d)^(1/2)*a*b*e^2-3*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^2*d*e-15*B
*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*e^2+11*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2
)*a*b*d*e-4*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^2*d^2)/e*(b*x+a)/(b*(a*e-b*d))
^(1/2)/b^3/((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 = 0.306571, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (8 \, B b^{2} e x^{2} - 2 \,{\left (B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - A a b\right )} e -{\left (4 \, B b^{2} d - 5 \,{\left (5 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{b^{2} d - a b e} \sqrt{e x + d} + 3 \,{\left (4 \, B a^{2} b d e -{\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{b^{2} d - a b e}}, \frac{{\left (8 \, B b^{2} e x^{2} - 2 \,{\left (B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - A a b\right )} e -{\left (4 \, B b^{2} d - 5 \,{\left (5 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt{-b^{2} d + a b e} \sqrt{e x + d} - 3 \,{\left (4 \, B a^{2} b d e -{\left (5 \, B a^{3} - A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (5 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (5 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]

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]

[1/8*(2*(8*B*b^2*e*x^2 - 2*(B*a*b + A*b^2)*d + 3*(5*B*a^2 - A*a*b)*e - (4*B*b^2*
d - 5*(5*B*a*b - A*b^2)*e)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 3*(4*B*a^2*b*d
*e - (5*B*a^3 - A*a^2*b)*e^2 + (4*B*b^3*d*e - (5*B*a*b^2 - A*b^3)*e^2)*x^2 + 2*(
4*B*a*b^2*d*e - (5*B*a^2*b - A*a*b^2)*e^2)*x)*log((sqrt(b^2*d - a*b*e)*(b*e*x +
2*b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((b^5*x^2 + 2*a*b^4*
x + a^2*b^3)*sqrt(b^2*d - a*b*e)), 1/4*((8*B*b^2*e*x^2 - 2*(B*a*b + A*b^2)*d + 3
*(5*B*a^2 - A*a*b)*e - (4*B*b^2*d - 5*(5*B*a*b - A*b^2)*e)*x)*sqrt(-b^2*d + a*b*
e)*sqrt(e*x + d) - 3*(4*B*a^2*b*d*e - (5*B*a^3 - A*a^2*b)*e^2 + (4*B*b^3*d*e - (
5*B*a*b^2 - A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (5*B*a^2*b - A*a*b^2)*e^2)*x)*a
rctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((b^5*x^2 + 2*a*b^4*x
+ a^2*b^3)*sqrt(-b^2*d + a*b*e))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.303709, size = 425, normalized size = 1.52 \[ -\frac{2 \, \sqrt{x e + d} B e}{b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{3 \,{\left (4 \, B b d e^{2} - 5 \, B a e^{3} + A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{4 \, \sqrt{-b^{2} d + a b e} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{{\left (4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{2} - 4 \, \sqrt{x e + d} B b^{2} d^{2} e^{2} - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{3} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{3} + 11 \, \sqrt{x e + d} B a b d e^{3} - 3 \, \sqrt{x e + d} A b^{2} d e^{3} - 7 \, \sqrt{x e + d} B a^{2} e^{4} + 3 \, \sqrt{x e + d} A a b e^{4}\right )} e^{\left (-1\right )}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

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]

-2*sqrt(x*e + d)*B*e/(b^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 3/4*(4*B*b*d*e
^2 - 5*B*a*e^3 + A*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^(-1)/(s
qrt(-b^2*d + a*b*e)*b^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/4*(4*(x*e + d)
^(3/2)*B*b^2*d*e^2 - 4*sqrt(x*e + d)*B*b^2*d^2*e^2 - 9*(x*e + d)^(3/2)*B*a*b*e^3
+ 5*(x*e + d)^(3/2)*A*b^2*e^3 + 11*sqrt(x*e + d)*B*a*b*d*e^3 - 3*sqrt(x*e + d)*
A*b^2*d*e^3 - 7*sqrt(x*e + d)*B*a^2*e^4 + 3*sqrt(x*e + d)*A*a*b*e^4)*e^(-1)/(((x
*e + d)*b - b*d + a*e)^2*b^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2))