3.464 \(\int \frac{1}{x^2 (a+b x)^2 \sqrt{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.579787, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{3/2}}-\frac{b \sqrt{c+d x} (2 b c-a d)}{a^2 c (a+b x) (b c-a d)}-\frac{\sqrt{c+d x}}{a c x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.4371, size = 141, normalized size = 0.86 \[ - \frac{b \sqrt{c + d x}}{a x \left (a + b x\right ) \left (a d - b c\right )} - \frac{\sqrt{c + d x} \left (a d - 2 b c\right )}{a^{2} c x \left (a d - b c\right )} + \frac{b^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{3} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\left (a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.512727, size = 132, normalized size = 0.8 \[ \frac{-\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}}+a \sqrt{c+d x} \left (\frac{b^2}{(a+b x) (a d-b c)}-\frac{1}{c x}\right )+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2}}}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

(a*Sqrt[c + d*x]*(-(1/(c*x)) + b^2/((-(b*c) + a*d)*(a + b*x))) + ((4*b*c + a*d)*
ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(3/2) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt
[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2))/a^3

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Maple [A]  time = 0.026, size = 202, normalized size = 1.2 \[ -{\frac{1}{{a}^{2}cx}\sqrt{dx+c}}+{\frac{d}{{a}^{2}}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}+4\,{\frac{b}{{a}^{3}\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+{\frac{{b}^{2}d}{{a}^{2} \left ( ad-bc \right ) \left ( bdx+ad \right ) }\sqrt{dx+c}}+5\,{\frac{{b}^{2}d}{{a}^{2} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-4\,{\frac{{b}^{3}c}{{a}^{3} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a^2/c*(d*x+c)^(1/2)/x+d/a^2/c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2))+4/a^3/c^(1
/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b+d*b^2/a^2/(a*d-b*c)*(d*x+c)^(1/2)/(b*d*x+a*
d)+5*d*b^2/a^2/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b
)^(1/2))-4*b^3/a^3/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*
c)*b)^(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/((b*x + a)^2*sqrt(d*x + c)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.942068, 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/((b*x + a)^2*sqrt(d*x + c)*x^2),x, algorithm="fricas")

[Out]

[1/2*(((4*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(c)*sq
rt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/
(b*c - a*d)))/(b*x + a)) - 2*(a^2*b*c - a^3*d + (2*a*b^2*c - a^2*b*d)*x)*sqrt(d*
x + c)*sqrt(c) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a
^2*b*c*d - a^3*d^2)*x)*log(((d*x + 2*c)*sqrt(c) + 2*sqrt(d*x + c)*c)/x))/(((a^3*
b^2*c^2 - a^4*b*c*d)*x^2 + (a^4*b*c^2 - a^5*c*d)*x)*sqrt(c)), -1/2*(2*((4*b^3*c^
2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(c)*sqrt(-b/(b*c - a*d
))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x + c)*b)) + 2*(a^2*b*c - a^
3*d + (2*a*b^2*c - a^2*b*d)*x)*sqrt(d*x + c)*sqrt(c) - ((4*b^3*c^2 - 3*a*b^2*c*d
- a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*log(((d*x + 2*c)*sq
rt(c) + 2*sqrt(d*x + c)*c)/x))/(((a^3*b^2*c^2 - a^4*b*c*d)*x^2 + (a^4*b*c^2 - a^
5*c*d)*x)*sqrt(c)), 1/2*(((4*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b
*c*d)*x)*sqrt(-c)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*s
qrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) - 2*(a^2*b*c - a^3*d + (2*a*b^2*c -
a^2*b*d)*x)*sqrt(d*x + c)*sqrt(-c) - 2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x
^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*arctan(c/(sqrt(d*x + c)*sqrt(-c)))
)/(((a^3*b^2*c^2 - a^4*b*c*d)*x^2 + (a^4*b*c^2 - a^5*c*d)*x)*sqrt(-c)), -(((4*b^
3*c^2 - 5*a*b^2*c*d)*x^2 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(-c)*sqrt(-b/(b*c
- a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x + c)*b)) + (a^2*b*c -
a^3*d + (2*a*b^2*c - a^2*b*d)*x)*sqrt(d*x + c)*sqrt(-c) + ((4*b^3*c^2 - 3*a*b^2
*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*arctan(c/(sqrt(
d*x + c)*sqrt(-c))))/(((a^3*b^2*c^2 - a^4*b*c*d)*x^2 + (a^4*b*c^2 - a^5*c*d)*x)*
sqrt(-c))]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.217068, size = 319, normalized size = 1.95 \[ \frac{{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b c - a^{4} d\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d - 2 \, \sqrt{d x + c} b^{2} c^{2} d -{\left (d x + c\right )}^{\frac{3}{2}} a b d^{2} + 2 \, \sqrt{d x + c} a b c d^{2} - \sqrt{d x + c} a^{2} d^{3}}{{\left (a^{2} b c^{2} - a^{3} c d\right )}{\left ({\left (d x + c\right )}^{2} b - 2 \,{\left (d x + c\right )} b c + b c^{2} +{\left (d x + c\right )} a d - a c d\right )}} - \frac{{\left (4 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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