3.374 \(\int \frac{1}{\sqrt{d+e x} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=164 \[ -\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{3/2}}-\frac{c \sqrt{d+e x} (2 c d-b e)}{b^2 d (b+c x) (c d-b e)}-\frac{\sqrt{d+e x}}{b d x (b+c x)} \]

[Out]

-((c*(2*c*d - b*e)*Sqrt[d + e*x])/(b^2*d*(c*d - b*e)*(b + c*x))) - Sqrt[d + e*x]
/(b*d*x*(b + c*x)) + ((4*c*d + b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(3/2)
) - (c^(3/2)*(4*c*d - 5*b*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 = 0.595075, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{3/2}}-\frac{c \sqrt{d+e x} (2 c d-b e)}{b^2 d (b+c x) (c d-b e)}-\frac{\sqrt{d+e x}}{b d x (b+c x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*(b*x + c*x^2)^2),x]

[Out]

-((c*(2*c*d - b*e)*Sqrt[d + e*x])/(b^2*d*(c*d - b*e)*(b + c*x))) - Sqrt[d + e*x]
/(b*d*x*(b + c*x)) + ((4*c*d + b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(3/2)
) - (c^(3/2)*(4*c*d - 5*b*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 = 68.132, size = 141, normalized size = 0.86 \[ - \frac{c \sqrt{d + e x}}{b x \left (b + c x\right ) \left (b e - c d\right )} - \frac{\sqrt{d + e x} \left (b e - 2 c d\right )}{b^{2} d x \left (b e - c d\right )} + \frac{c^{\frac{3}{2}} \left (5 b e - 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \left (b e - c d\right )^{\frac{3}{2}}} + \frac{\left (b e + 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.521659, size = 132, normalized size = 0.8 \[ \frac{-\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{3/2}}+b \sqrt{d+e x} \left (\frac{c^2}{(b+c x) (b e-c d)}-\frac{1}{d x}\right )+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{3/2}}}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^2),x]

[Out]

(b*Sqrt[d + e*x]*(-(1/(d*x)) + c^2/((-(c*d) + b*e)*(b + c*x))) + ((4*c*d + b*e)*
ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(3/2) - (c^(3/2)*(4*c*d - 5*b*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 [A]  time = 0.026, size = 202, normalized size = 1.2 \[{\frac{e{c}^{2}}{{b}^{2} \left ( be-cd \right ) \left ( cex+be \right ) }\sqrt{ex+d}}+5\,{\frac{e{c}^{2}}{{b}^{2} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-4\,{\frac{{c}^{3}d}{{b}^{3} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{b}^{2}dx}\sqrt{ex+d}}+{\frac{e}{{b}^{2}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{3}{2}}}}+4\,{\frac{c}{{b}^{3}\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(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)+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))-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))*d-1/b^2/d*(e*x+d)^(1/2)/
x+e/b^2/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+4/b^3/d^(1/2)*arctanh((e*x+d)^(1/
2)/d^(1/2))*c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.856362, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/2*(((4*c^3*d^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*b^2*c*d*e)*x)*sqrt(d)*sq
rt(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*(b^2*c*d - b^3*e + (2*b*c^2*d - b^2*c*e)*x)*sqrt(e*
x + d)*sqrt(d) + ((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b
^2*c*d*e - b^3*e^2)*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*((4*c^3*d^
2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*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*(b^2*c*d - b^
3*e + (2*b*c^2*d - b^2*c*e)*x)*sqrt(e*x + d)*sqrt(d) - ((4*c^3*d^2 - 3*b*c^2*d*e
- b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*x)*log(((e*x + 2*d)*sq
rt(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*(((4*c^3*d^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*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)*s
qrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 2*(b^2*c*d - b^3*e + (2*b*c^2*d -
b^2*c*e)*x)*sqrt(e*x + d)*sqrt(-d) - 2*((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x
^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*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)), -(((4*c^
3*d^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*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)) + (b^2*c*d -
b^3*e + (2*b*c^2*d - b^2*c*e)*x)*sqrt(e*x + d)*sqrt(-d) + ((4*c^3*d^2 - 3*b*c^2
*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (b + c x\right )^{2} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(b + c*x)**2*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.211389, size = 342, normalized size = 2.09 \[ \frac{{\left (4 \, c^{3} d - 5 \, b c^{2} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c d - b^{4} e\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d e - 2 \, \sqrt{x e + d} c^{2} d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} b c e^{2} + 2 \, \sqrt{x e + d} b c d e^{2} - \sqrt{x e + d} b^{2} e^{3}}{{\left (b^{2} c d^{2} - b^{3} d e\right )}{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}} - \frac{{\left (4 \, c d + b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*sqrt(e*x + d)),x, algorithm="giac")

[Out]

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