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

Optimal. Leaf size=129 \[ \frac{2 a}{\left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}-\frac{2 \left (a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac{4 a b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\left (a^2-b^2 c\right )^2}+\frac{\log (x) \left (a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.261699, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 a}{\left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}-\frac{2 \left (a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac{4 a b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\left (a^2-b^2 c\right )^2}+\frac{\log (x) \left (a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.4796, size = 117, normalized size = 0.91 \[ \frac{4 a b \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\left (a^{2} - b^{2} c\right )^{2}} + \frac{2 a}{\left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )} + \frac{\left (a^{2} + b^{2} c\right ) \log{\left (- d x \right )}}{\left (a^{2} - b^{2} c\right )^{2}} - \frac{2 \left (a^{2} + b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{\left (a^{2} - b^{2} c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a*b*sqrt(c)*atanh(sqrt(c + d*x)/sqrt(c))/(a**2 - b**2*c)**2 + 2*a/((a + b*sqrt
(c + d*x))*(a**2 - b**2*c)) + (a**2 + b**2*c)*log(-d*x)/(a**2 - b**2*c)**2 - 2*(
a**2 + b**2*c)*log(a + b*sqrt(c + d*x))/(a**2 - b**2*c)**2

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Mathematica [A]  time = 0.281962, size = 106, normalized size = 0.82 \[ \frac{\frac{2 a \left (a^2-b^2 c\right )}{a+b \sqrt{c+d x}}+\left (a^2+b^2 c\right ) \log (d x)-2 \left (a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )+4 a b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\left (a^2-b^2 c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 161, normalized size = 1.3 \[{\frac{\ln \left ( dx \right ){b}^{2}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}}+{\frac{\ln \left ( dx \right ){a}^{2}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}}+4\,{\frac{\sqrt{c}ab}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{a}{ \left ( -{b}^{2}c+{a}^{2} \right ) \left ( a+b\sqrt{dx+c} \right ) }}-2\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){b}^{2}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}}-2\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){a}^{2}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304268, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, a b^{2} c - 2 \, a^{3} -{\left (b^{3} c + a^{2} b\right )} \sqrt{d x + c} \log \left (x\right ) + 2 \,{\left (a b^{2} c + a^{3} +{\left (b^{3} c + a^{2} b\right )} \sqrt{d x + c}\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left (a b^{2} c + a^{3}\right )} \log \left (x\right ) - 2 \,{\left (\sqrt{d x + c} a b^{2} \sqrt{c} + a^{2} b \sqrt{c}\right )} \log \left (\frac{d x + 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right )}{a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5} +{\left (b^{5} c^{2} - 2 \, a^{2} b^{3} c + a^{4} b\right )} \sqrt{d x + c}}, -\frac{2 \, a b^{2} c - 2 \, a^{3} -{\left (b^{3} c + a^{2} b\right )} \sqrt{d x + c} \log \left (x\right ) - 4 \,{\left (\sqrt{d x + c} a b^{2} \sqrt{-c} + a^{2} b \sqrt{-c}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) + 2 \,{\left (a b^{2} c + a^{3} +{\left (b^{3} c + a^{2} b\right )} \sqrt{d x + c}\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left (a b^{2} c + a^{3}\right )} \log \left (x\right )}{a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5} +{\left (b^{5} c^{2} - 2 \, a^{2} b^{3} c + a^{4} b\right )} \sqrt{d x + c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b \sqrt{c + d x}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef