3.1243 \(\int \frac{A+B x}{\sqrt{d+e x} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=195 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-A b e-4 A c d+2 b B d)}{b^3 d^{3/2}}+\frac{c \sqrt{d+e x} (A b e-2 A c d+b B d)}{b^2 d (b+c x) (c d-b e)}+\frac{\sqrt{c} \left (5 A b c e-4 A c^2 d-3 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}-\frac{A \sqrt{d+e x}}{b d x (b+c x)} \]

[Out]

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

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Rubi [A]  time = 0.82325, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-A b e-4 A c d+2 b B d)}{b^3 d^{3/2}}+\frac{c \sqrt{d+e x} (A b e-2 A c d+b B d)}{b^2 d (b+c x) (c d-b e)}-\frac{\sqrt{c} \left (-b c (5 A e+2 B d)+4 A c^2 d+3 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}-\frac{A \sqrt{d+e x}}{b d x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 103.691, size = 189, normalized size = 0.97 \[ - \frac{\sqrt{d + e x} \left (A c - B b\right )}{b x \left (b + c x\right ) \left (b e - c d\right )} - \frac{\sqrt{d + e x} \left (A b e - 2 A c d + B b d\right )}{b^{2} d x \left (b e - c d\right )} - \frac{\sqrt{c} \left (- 5 A b c e + 4 A c^{2} d + 3 B b^{2} e - 2 B b c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \left (b e - c d\right )^{\frac{3}{2}}} + \frac{\left (A b e + 4 A c d - 2 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d + e*x)*(A*c - B*b)/(b*x*(b + c*x)*(b*e - c*d)) - sqrt(d + e*x)*(A*b*e -
2*A*c*d + B*b*d)/(b**2*d*x*(b*e - c*d)) - sqrt(c)*(-5*A*b*c*e + 4*A*c**2*d + 3*B
*b**2*e - 2*B*b*c*d)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b**3*(b*e - c*
d)**(3/2)) + (A*b*e + 4*A*c*d - 2*B*b*d)*atanh(sqrt(d + e*x)/sqrt(d))/(b**3*d**(
3/2))

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Mathematica [A]  time = 0.638564, size = 167, normalized size = 0.86 \[ \frac{-\frac{\sqrt{c} \left (-b c (5 A e+2 B d)+4 A c^2 d+3 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-A b e-4 A c d+2 b B d)}{d^{3/2}}+b \sqrt{d+e x} \left (\frac{c (b B-A c)}{(b+c x) (c d-b e)}-\frac{A}{d x}\right )}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 370, normalized size = 1.9 \[{\frac{e{c}^{2}A}{{b}^{2} \left ( be-cd \right ) \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{Bce}{b \left ( be-cd \right ) \left ( cex+be \right ) }\sqrt{ex+d}}+5\,{\frac{e{c}^{2}A}{{b}^{2} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-4\,{\frac{Ad{c}^{3}}{{b}^{3} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-3\,{\frac{Bce}{b \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+2\,{\frac{B{c}^{2}d}{{b}^{2} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{A}{{b}^{2}dx}\sqrt{ex+d}}+{\frac{Ae}{{b}^{2}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{3}{2}}}}+4\,{\frac{Ac}{{b}^{3}\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{B}{{b}^{2}\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*c^2/b^2/(b*e-c*d)*(e*x+d)^(1/2)/(c*e*x+b*e)*A-e*c/b/(b*e-c*d)*(e*x+d)^(1/2)/(c
*e*x+b*e)*B+5*e*c^2/b^2/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b
*e-c*d)*c)^(1/2))*A-4*c^3/b^3/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/
2)/((b*e-c*d)*c)^(1/2))*A*d-3*e*c/b/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+
d)^(1/2)/((b*e-c*d)*c)^(1/2))*B+2*c^2/b^2/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan(c
*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d-1/b^2*A/d*(e*x+d)^(1/2)/x+e/b^2/d^(3/2)*
arctanh((e*x+d)^(1/2)/d^(1/2))*A+4/b^3/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*
c-2/b^2/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B

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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)/((c*x^2 + b*x)^2*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.75702, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(((2*(B*b*c^2 - 2*A*c^3)*d^2 - (3*B*b^2*c - 5*A*b*c^2)*d*e)*x^2 + (2*(B*b^
2*c - 2*A*b*c^2)*d^2 - (3*B*b^3 - 5*A*b^2*c)*d*e)*x)*sqrt(d)*sqrt(c/(c*d - b*e))
*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*
x + b)) + 2*(A*b^2*c*d - A*b^3*e - (A*b^2*c*e + (B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt
(e*x + d)*sqrt(d) + ((A*b^2*c*e^2 + 2*(B*b*c^2 - 2*A*c^3)*d^2 - (2*B*b^2*c - 3*A
*b*c^2)*d*e)*x^2 + (A*b^3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)*d^2 - (2*B*b^3 - 3*A*b^2
*c)*d*e)*x)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x))/(((b^3*c^2*d^2 - b
^4*c*d*e)*x^2 + (b^4*c*d^2 - b^5*d*e)*x)*sqrt(d)), 1/2*(2*((2*(B*b*c^2 - 2*A*c^3
)*d^2 - (3*B*b^2*c - 5*A*b*c^2)*d*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d^2 - (3*B*b
^3 - 5*A*b^2*c)*d*e)*x)*sqrt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c
/(c*d - b*e))/(sqrt(e*x + d)*c)) - 2*(A*b^2*c*d - A*b^3*e - (A*b^2*c*e + (B*b^2*
c - 2*A*b*c^2)*d)*x)*sqrt(e*x + d)*sqrt(d) - ((A*b^2*c*e^2 + 2*(B*b*c^2 - 2*A*c^
3)*d^2 - (2*B*b^2*c - 3*A*b*c^2)*d*e)*x^2 + (A*b^3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)
*d^2 - (2*B*b^3 - 3*A*b^2*c)*d*e)*x)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*
d)/x))/(((b^3*c^2*d^2 - b^4*c*d*e)*x^2 + (b^4*c*d^2 - b^5*d*e)*x)*sqrt(d)), -1/2
*(((2*(B*b*c^2 - 2*A*c^3)*d^2 - (3*B*b^2*c - 5*A*b*c^2)*d*e)*x^2 + (2*(B*b^2*c -
2*A*b*c^2)*d^2 - (3*B*b^3 - 5*A*b^2*c)*d*e)*x)*sqrt(-d)*sqrt(c/(c*d - b*e))*log
((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x +
b)) + 2*(A*b^2*c*d - A*b^3*e - (A*b^2*c*e + (B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(e*x
+ d)*sqrt(-d) - 2*((A*b^2*c*e^2 + 2*(B*b*c^2 - 2*A*c^3)*d^2 - (2*B*b^2*c - 3*A*
b*c^2)*d*e)*x^2 + (A*b^3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)*d^2 - (2*B*b^3 - 3*A*b^2*
c)*d*e)*x)*arctan(d/(sqrt(e*x + d)*sqrt(-d))))/(((b^3*c^2*d^2 - b^4*c*d*e)*x^2 +
(b^4*c*d^2 - b^5*d*e)*x)*sqrt(-d)), (((2*(B*b*c^2 - 2*A*c^3)*d^2 - (3*B*b^2*c -
5*A*b*c^2)*d*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d^2 - (3*B*b^3 - 5*A*b^2*c)*d*e)
*x)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt
(e*x + d)*c)) - (A*b^2*c*d - A*b^3*e - (A*b^2*c*e + (B*b^2*c - 2*A*b*c^2)*d)*x)*
sqrt(e*x + d)*sqrt(-d) + ((A*b^2*c*e^2 + 2*(B*b*c^2 - 2*A*c^3)*d^2 - (2*B*b^2*c
- 3*A*b*c^2)*d*e)*x^2 + (A*b^3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)*d^2 - (2*B*b^3 - 3*
A*b^2*c)*d*e)*x)*arctan(d/(sqrt(e*x + d)*sqrt(-d))))/(((b^3*c^2*d^2 - b^4*c*d*e)
*x^2 + (b^4*c*d^2 - b^5*d*e)*x)*sqrt(-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)/(c*x**2+b*x)**2/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.290374, size = 425, normalized size = 2.18 \[ -\frac{{\left (2 \, B b c^{2} d - 4 \, A c^{3} d - 3 \, B b^{2} c e + 5 \, A b c^{2} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c d - b^{4} e\right )} \sqrt{-c^{2} d + b c e}} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b c d e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d e - \sqrt{x e + d} B b c d^{2} e + 2 \, \sqrt{x e + d} A c^{2} d^{2} e +{\left (x e + d\right )}^{\frac{3}{2}} A b c e^{2} - 2 \, \sqrt{x e + d} A b c d e^{2} + \sqrt{x e + d} A b^{2} e^{3}}{{\left (b^{2} c d^{2} - b^{3} d e\right )}{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}} + \frac{{\left (2 \, B b d - 4 \, A c d - A b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*B*b*c^2*d - 4*A*c^3*d - 3*B*b^2*c*e + 5*A*b*c^2*e)*arctan(sqrt(x*e + d)*c/sq
rt(-c^2*d + b*c*e))/((b^3*c*d - b^4*e)*sqrt(-c^2*d + b*c*e)) + ((x*e + d)^(3/2)*
B*b*c*d*e - 2*(x*e + d)^(3/2)*A*c^2*d*e - sqrt(x*e + d)*B*b*c*d^2*e + 2*sqrt(x*e
+ d)*A*c^2*d^2*e + (x*e + d)^(3/2)*A*b*c*e^2 - 2*sqrt(x*e + d)*A*b*c*d*e^2 + sq
rt(x*e + d)*A*b^2*e^3)/((b^2*c*d^2 - b^3*d*e)*((x*e + d)^2*c - 2*(x*e + d)*c*d +
c*d^2 + (x*e + d)*b*e - b*d*e)) + (2*B*b*d - 4*A*c*d - A*b*e)*arctan(sqrt(x*e +
d)/sqrt(-d))/(b^3*sqrt(-d)*d)