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

Optimal. Leaf size=130 \[ -\frac{a-b \sqrt{c+d x}}{x \left (a^2-b^2 c\right )}+\frac{a b^2 d \log (x)}{\left (a^2-b^2 c\right )^2}-\frac{2 a b^2 d \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac{b d \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2-b^2 c\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.363361, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{a-b \sqrt{c+d x}}{x \left (a^2-b^2 c\right )}+\frac{a b^2 d \log (x)}{\left (a^2-b^2 c\right )^2}-\frac{2 a b^2 d \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^2}+\frac{b d \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2-b^2 c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.3431, size = 119, normalized size = 0.92 \[ \frac{a b^{2} d \log{\left (- d x \right )}}{\left (a^{2} - b^{2} c\right )^{2}} - \frac{2 a b^{2} d \log{\left (a + b \sqrt{c + d x} \right )}}{\left (a^{2} - b^{2} c\right )^{2}} + \frac{b d \left (a^{2} + b^{2} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c} \left (a^{2} - b^{2} c\right )^{2}} - \frac{a - b \sqrt{c + d x}}{x \left (a^{2} - b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.325795, size = 144, normalized size = 1.11 \[ \frac{\sqrt{c} \left (-\left (a^2-b^2 c\right ) \left (a-b \sqrt{c+d x}\right )-a b^2 d x \log \left (a^2-b^2 (c+d x)\right )+a b^2 d x \log (x)\right )+b d x \left (a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )-2 a b^2 \sqrt{c} d x \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )}{\sqrt{c} x \left (a^2-b^2 c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.027, size = 216, normalized size = 1.7 \[ -{\frac{{b}^{3}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}\sqrt{dx+c}}+{\frac{{a}^{2}b}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}\sqrt{dx+c}}+{\frac{a{b}^{2}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}}-{\frac{{a}^{3}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}x}}+{\frac{a{b}^{2}d\ln \left ( dx \right ) }{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}}+{\frac{{b}^{3}d}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}\sqrt{c}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }+{\frac{{a}^{2}bd}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-2\,{\frac{a{b}^{2}d\ln \left ( a+b\sqrt{dx+c} \right ) }{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.352942, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, a b^{2} \sqrt{c} d x \log \left (\sqrt{d x + c} b + a\right ) -{\left (b^{3} c + a^{2} b\right )} d x \log \left (\frac{{\left (d x + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x + c} c}{x}\right ) + 2 \,{\left (b^{3} c - a^{2} b\right )} \sqrt{d x + c} \sqrt{c} - 2 \,{\left (a b^{2} d x \log \left (x\right ) + a b^{2} c - a^{3}\right )} \sqrt{c}}{2 \,{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt{c} x}, -\frac{2 \, a b^{2} \sqrt{-c} d x \log \left (\sqrt{d x + c} b + a\right ) +{\left (b^{3} c + a^{2} b\right )} d x \arctan \left (\frac{c}{\sqrt{d x + c} \sqrt{-c}}\right ) +{\left (b^{3} c - a^{2} b\right )} \sqrt{d x + c} \sqrt{-c} -{\left (a b^{2} d x \log \left (x\right ) + a b^{2} c - a^{3}\right )} \sqrt{-c}}{{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(4*a*b^2*sqrt(c)*d*x*log(sqrt(d*x + c)*b + a) - (b^3*c + a^2*b)*d*x*log(((
d*x + 2*c)*sqrt(c) + 2*sqrt(d*x + c)*c)/x) + 2*(b^3*c - a^2*b)*sqrt(d*x + c)*sqr
t(c) - 2*(a*b^2*d*x*log(x) + a*b^2*c - a^3)*sqrt(c))/((b^4*c^2 - 2*a^2*b^2*c + a
^4)*sqrt(c)*x), -(2*a*b^2*sqrt(-c)*d*x*log(sqrt(d*x + c)*b + a) + (b^3*c + a^2*b
)*d*x*arctan(c/(sqrt(d*x + c)*sqrt(-c))) + (b^3*c - a^2*b)*sqrt(d*x + c)*sqrt(-c
) - (a*b^2*d*x*log(x) + a*b^2*c - a^3)*sqrt(-c))/((b^4*c^2 - 2*a^2*b^2*c + a^4)*
sqrt(-c)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.286877, size = 342, normalized size = 2.63 \[ -\frac{2 \, a b^{3} d{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{5} c^{2} - 2 \, a^{2} b^{3} c + a^{4} b} + \frac{a b^{2} d{\rm ln}\left (-d x\right )}{b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}} - \frac{{\left (b^{3} c d + a^{2} b d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{{\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt{-c}} - \frac{a b^{2} c d{\rm ln}\left (c\right ) - 2 \, a b^{2} c d{\rm ln}\left ({\left | a \right |}\right ) - a b^{2} c d + a^{3} d}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c} + \frac{a b^{2} c d - a^{3} d -{\left (b^{3} c d - a^{2} b d\right )} \sqrt{d x + c}}{{\left (b^{2} c - a^{2}\right )}^{2} d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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