3.1871 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.632146, 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 \[ -\frac{A b-a B}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{a B e-5 A b e+4 b B d}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{3 e (a+b x) (a B e-5 A b e+4 b B d)}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e (a+b x) (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

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]

-(4*b*B*d - 5*A*b*e + a*B*e)/(4*b*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (A*b - a*B)/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (3*e*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x))/(4*b*(b*d - a*e
)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e*(4*b*B*d - 5*A*b*e + a*B
*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*Sqrt[b]*(b*d
- a*e)^(7/2)*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 = 1.15753, size = 174, normalized size = 0.62 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (\frac{2 (a B-A b) (b d-a e)}{(a+b x)^2}+\frac{-3 a B e+7 A b e-4 b B d}{a+b x}+\frac{8 e (A e-B d)}{d+e x}\right )}{(b d-a e)^3}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{7/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

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]*((2*(-(A*b) + a*B)*(b*d - a*e))/(a + b*x)^2 + (-4*b
*B*d + 7*A*b*e - 3*a*B*e)/(a + b*x) + (8*e*(-(B*d) + A*e))/(d + e*x)))/(b*d - a*
e)^3 + (3*e*(4*b*B*d - 5*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(Sqrt[b]*(b*d - a*e)^(7/2))))/(4*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.037, size = 681, normalized size = 2.4 \[ -{\frac{bx+a}{4\, \left ( ae-bd \right ) ^{3}} \left ( 15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}{b}^{3}{e}^{2}-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}a{b}^{2}{e}^{2}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}{b}^{3}de+30\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}xa{b}^{2}{e}^{2}-6\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}x{a}^{2}b{e}^{2}-24\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}xa{b}^{2}de+15\,A\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}{e}^{2}+15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{2}b{e}^{2}-3\,B\sqrt{b \left ( ae-bd \right ) }{x}^{2}ab{e}^{2}-12\,B\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}de-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{3}{e}^{2}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{2}bde+25\,A\sqrt{b \left ( ae-bd \right ) }xab{e}^{2}+5\,A\sqrt{b \left ( ae-bd \right ) }x{b}^{2}de-5\,B\sqrt{b \left ( ae-bd \right ) }x{a}^{2}{e}^{2}-21\,B\sqrt{b \left ( ae-bd \right ) }xabde-4\,B\sqrt{b \left ( ae-bd \right ) }x{b}^{2}{d}^{2}+8\,A\sqrt{b \left ( ae-bd \right ) }{a}^{2}{e}^{2}+9\,A\sqrt{b \left ( ae-bd \right ) }abde-2\,A\sqrt{b \left ( ae-bd \right ) }{b}^{2}{d}^{2}-13\,B\sqrt{b \left ( ae-bd \right ) }{a}^{2}de-2\,B\sqrt{b \left ( ae-bd \right ) }ab{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}{\frac{1}{\sqrt{ex+d}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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*(15*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^2*b^3*e^2
-3*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^2*a*b^2*e^2-12*
B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^2*b^3*d*e+30*A*arc
tan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x*a*b^2*e^2-6*B*arctan((e
*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x*a^2*b*e^2-24*B*arctan((e*x+d)
^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x*a*b^2*d*e+15*A*(b*(a*e-b*d))^(1/2)
*x^2*b^2*e^2+15*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^2*
b*e^2-3*B*(b*(a*e-b*d))^(1/2)*x^2*a*b*e^2-12*B*(b*(a*e-b*d))^(1/2)*x^2*b^2*d*e-3
*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^3*e^2-12*B*arctan
((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^2*b*d*e+25*A*(b*(a*e-b*d))
^(1/2)*x*a*b*e^2+5*A*(b*(a*e-b*d))^(1/2)*x*b^2*d*e-5*B*(b*(a*e-b*d))^(1/2)*x*a^2
*e^2-21*B*(b*(a*e-b*d))^(1/2)*x*a*b*d*e-4*B*(b*(a*e-b*d))^(1/2)*x*b^2*d^2+8*A*(b
*(a*e-b*d))^(1/2)*a^2*e^2+9*A*(b*(a*e-b*d))^(1/2)*a*b*d*e-2*A*(b*(a*e-b*d))^(1/2
)*b^2*d^2-13*B*(b*(a*e-b*d))^(1/2)*a^2*d*e-2*B*(b*(a*e-b*d))^(1/2)*a*b*d^2)*(b*x
+a)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(1/2)/(a*e-b*d)^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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(3/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/8*(3*(4*B*a^2*b*d*e + (B*a^3 - 5*A*a^2*b)*e^2 + (4*B*b^3*d*e + (B*a*b^2 - 5*A
*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e + (B*a^2*b - 5*A*a*b^2)*e^2)*x)*sqrt(e*x + d)*
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)) + 2*(8*A*a^2*e^2 - 2*(B*a*b + A*b^2)*d^2 - (13*B*a^2 - 9*A*a*b)*d*e
- 3*(4*B*b^2*d*e + (B*a*b - 5*A*b^2)*e^2)*x^2 - (4*B*b^2*d^2 + (21*B*a*b - 5*A*
b^2)*d*e + 5*(B*a^2 - 5*A*a*b)*e^2)*x)*sqrt(b^2*d - a*b*e))/((a^2*b^3*d^3 - 3*a^
3*b^2*d^2*e + 3*a^4*b*d*e^2 - a^5*e^3 + (b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e
^2 - a^3*b^2*e^3)*x^2 + 2*(a*b^4*d^3 - 3*a^2*b^3*d^2*e + 3*a^3*b^2*d*e^2 - a^4*b
*e^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d)), 1/4*(3*(4*B*a^2*b*d*e + (B*a^3 - 5*
A*a^2*b)*e^2 + (4*B*b^3*d*e + (B*a*b^2 - 5*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e +
(B*a^2*b - 5*A*a*b^2)*e^2)*x)*sqrt(e*x + d)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a
*b*e)*sqrt(e*x + d))) + (8*A*a^2*e^2 - 2*(B*a*b + A*b^2)*d^2 - (13*B*a^2 - 9*A*a
*b)*d*e - 3*(4*B*b^2*d*e + (B*a*b - 5*A*b^2)*e^2)*x^2 - (4*B*b^2*d^2 + (21*B*a*b
- 5*A*b^2)*d*e + 5*(B*a^2 - 5*A*a*b)*e^2)*x)*sqrt(-b^2*d + a*b*e))/((a^2*b^3*d^
3 - 3*a^3*b^2*d^2*e + 3*a^4*b*d*e^2 - a^5*e^3 + (b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2
*b^3*d*e^2 - a^3*b^2*e^3)*x^2 + 2*(a*b^4*d^3 - 3*a^2*b^3*d^2*e + 3*a^3*b^2*d*e^2
- a^4*b*e^3)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))]

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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 = 0.325318, size = 848, normalized size = 3.03 \[ \frac{3 \,{\left (4 \, B b d e^{2} + B a e^{3} - 5 \, A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (B d e^{2} - A e^{3}\right )}}{{\left (b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{x e + d}} + \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{2} - 4 \, \sqrt{x e + d} B b^{2} d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{3} - 7 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{3} - \sqrt{x e + d} B a b d e^{3} + 9 \, \sqrt{x e + d} A b^{2} d e^{3} + 5 \, \sqrt{x e + d} B a^{2} e^{4} - 9 \, \sqrt{x e + d} A a b e^{4}}{4 \,{\left (b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

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)^(3/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

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