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

Optimal. Leaf size=202 \[ \frac{4 a b^2 d}{\left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )}-\frac{a-b \sqrt{c+d x}}{x \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}+\frac{b^2 d \log (x) \left (3 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^3}-\frac{2 b^2 d \left (3 a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^3}+\frac{2 a b d \left (a^2+3 b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2-b^2 c\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.518407, antiderivative size = 202, 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 \[ \frac{4 a b^2 d}{\left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )}-\frac{a-b \sqrt{c+d x}}{x \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}+\frac{b^2 d \log (x) \left (3 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^3}-\frac{2 b^2 d \left (3 a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^3}+\frac{2 a b d \left (a^2+3 b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2-b^2 c\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.3978, size = 184, normalized size = 0.91 \[ \frac{4 a b^{2} d}{\left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )^{2}} + \frac{2 a b d \left (a^{2} + 3 b^{2} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c} \left (a^{2} - b^{2} c\right )^{3}} + \frac{b^{2} d \left (3 a^{2} + b^{2} c\right ) \log{\left (- d x \right )}}{\left (a^{2} - b^{2} c\right )^{3}} - \frac{2 b^{2} d \left (3 a^{2} + b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{\left (a^{2} - b^{2} c\right )^{3}} - \frac{a - b \sqrt{c + d x}}{x \left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.72961, size = 292, normalized size = 1.45 \[ \frac{2 a^2 b^2 d}{\left (a^2-b^2 c\right )^2 \left (a^2-b^2 (c+d x)\right )}+\frac{b^2 d \log (x) \left (3 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^3}-\frac{b^2 d \left (3 a^2+b^2 c\right ) \log \left (a^2-b^2 (c+d x)\right )}{\left (a^2-b^2 c\right )^3}+\frac{2 b^2 d \left (3 a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )}{\left (b^2 c-a^2\right )^3}+\frac{2 a b d \left (a^2+3 b^2 c\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2-b^2 c\right )^3}-\frac{a^2+b^2 c}{x \left (a^2-b^2 c\right )^2}+\frac{2 a \sqrt{c+d x} \left (b^3 (c+2 d x)-a^2 b\right )}{x \left (a^2-b^2 c\right )^2 \left (b^2 (c+d x)-a^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 312, normalized size = 1.5 \[ -2\,{\frac{a\sqrt{dx+c}{b}^{3}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}x}}+2\,{\frac{\sqrt{dx+c}{a}^{3}b}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}x}}+{\frac{{b}^{4}{c}^{2}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}x}}-{\frac{{a}^{4}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}x}}+{\frac{d\ln \left ( dx \right ){b}^{4}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}}+3\,{\frac{d\ln \left ( dx \right ){a}^{2}{b}^{2}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}}+6\,{\frac{{b}^{3}d\sqrt{c}a}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{bd{a}^{3}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+2\,{\frac{a{b}^{2}d}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{2} \left ( a+b\sqrt{dx+c} \right ) }}-2\,{\frac{{b}^{4}d\ln \left ( a+b\sqrt{dx+c} \right ) c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}}-6\,{\frac{{b}^{2}d\ln \left ( a+b\sqrt{dx+c} \right ){a}^{2}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(-b^2*c+a^2)^3/x*(d*x+c)^(1/2)*a*b^3*c+2/(-b^2*c+a^2)^3/x*(d*x+c)^(1/2)*a^3*b
+1/(-b^2*c+a^2)^3/x*b^4*c^2-1/(-b^2*c+a^2)^3/x*a^4+d/(-b^2*c+a^2)^3*ln(d*x)*b^4*
c+3*d/(-b^2*c+a^2)^3*ln(d*x)*a^2*b^2+6*d/(-b^2*c+a^2)^3*b^3*c^(1/2)*arctanh((d*x
+c)^(1/2)/c^(1/2))*a+2*d/(-b^2*c+a^2)^3*b/c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))
*a^3+2*a*b^2*d/(-b^2*c+a^2)^2/(a+b*(d*x+c)^(1/2))-2*d*b^4/(-b^2*c+a^2)^3*ln(a+b*
(d*x+c)^(1/2))*c-6*d*b^2/(-b^2*c+a^2)^3*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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.605176, size = 1, normalized size = 0. \[ \left [-\frac{{\left (b^{5} c^{2} - 2 \, a^{2} b^{3} c + a^{4} b +{\left (b^{5} c + 3 \, a^{2} b^{3}\right )} d x \log \left (x\right )\right )} \sqrt{d x + c} \sqrt{c} - 2 \,{\left ({\left (b^{5} c + 3 \, a^{2} b^{3}\right )} \sqrt{d x + c} \sqrt{c} d x +{\left (a b^{4} c + 3 \, a^{3} b^{2}\right )} \sqrt{c} d x\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left ({\left (3 \, a b^{4} c + a^{3} b^{2}\right )} \sqrt{d x + c} d x +{\left (3 \, a^{2} b^{3} c + a^{4} b\right )} d x\right )} \log \left (\frac{{\left (d x + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x + c} c}{x}\right ) -{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5} -{\left (a b^{4} c + 3 \, a^{3} b^{2}\right )} d x \log \left (x\right ) + 4 \,{\left (a b^{4} c - a^{3} b^{2}\right )} d x\right )} \sqrt{c}}{{\left (b^{7} c^{3} - 3 \, a^{2} b^{5} c^{2} + 3 \, a^{4} b^{3} c - a^{6} b\right )} \sqrt{d x + c} \sqrt{c} x +{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \sqrt{c} x}, -\frac{{\left (b^{5} c^{2} - 2 \, a^{2} b^{3} c + a^{4} b +{\left (b^{5} c + 3 \, a^{2} b^{3}\right )} d x \log \left (x\right )\right )} \sqrt{d x + c} \sqrt{-c} - 2 \,{\left ({\left (3 \, a b^{4} c + a^{3} b^{2}\right )} \sqrt{d x + c} d x +{\left (3 \, a^{2} b^{3} c + a^{4} b\right )} d x\right )} \arctan \left (\frac{c}{\sqrt{d x + c} \sqrt{-c}}\right ) - 2 \,{\left ({\left (b^{5} c + 3 \, a^{2} b^{3}\right )} \sqrt{d x + c} \sqrt{-c} d x +{\left (a b^{4} c + 3 \, a^{3} b^{2}\right )} \sqrt{-c} d x\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5} -{\left (a b^{4} c + 3 \, a^{3} b^{2}\right )} d x \log \left (x\right ) + 4 \,{\left (a b^{4} c - a^{3} b^{2}\right )} d x\right )} \sqrt{-c}}{{\left (b^{7} c^{3} - 3 \, a^{2} b^{5} c^{2} + 3 \, a^{4} b^{3} c - a^{6} b\right )} \sqrt{d x + c} \sqrt{-c} x +{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-((b^5*c^2 - 2*a^2*b^3*c + a^4*b + (b^5*c + 3*a^2*b^3)*d*x*log(x))*sqrt(d*x + c
)*sqrt(c) - 2*((b^5*c + 3*a^2*b^3)*sqrt(d*x + c)*sqrt(c)*d*x + (a*b^4*c + 3*a^3*
b^2)*sqrt(c)*d*x)*log(sqrt(d*x + c)*b + a) - ((3*a*b^4*c + a^3*b^2)*sqrt(d*x + c
)*d*x + (3*a^2*b^3*c + a^4*b)*d*x)*log(((d*x + 2*c)*sqrt(c) - 2*sqrt(d*x + c)*c)
/x) - (a*b^4*c^2 - 2*a^3*b^2*c + a^5 - (a*b^4*c + 3*a^3*b^2)*d*x*log(x) + 4*(a*b
^4*c - a^3*b^2)*d*x)*sqrt(c))/((b^7*c^3 - 3*a^2*b^5*c^2 + 3*a^4*b^3*c - a^6*b)*s
qrt(d*x + c)*sqrt(c)*x + (a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*sqrt(c)
*x), -((b^5*c^2 - 2*a^2*b^3*c + a^4*b + (b^5*c + 3*a^2*b^3)*d*x*log(x))*sqrt(d*x
+ c)*sqrt(-c) - 2*((3*a*b^4*c + a^3*b^2)*sqrt(d*x + c)*d*x + (3*a^2*b^3*c + a^4
*b)*d*x)*arctan(c/(sqrt(d*x + c)*sqrt(-c))) - 2*((b^5*c + 3*a^2*b^3)*sqrt(d*x +
c)*sqrt(-c)*d*x + (a*b^4*c + 3*a^3*b^2)*sqrt(-c)*d*x)*log(sqrt(d*x + c)*b + a) -
(a*b^4*c^2 - 2*a^3*b^2*c + a^5 - (a*b^4*c + 3*a^3*b^2)*d*x*log(x) + 4*(a*b^4*c
- a^3*b^2)*d*x)*sqrt(-c))/((b^7*c^3 - 3*a^2*b^5*c^2 + 3*a^4*b^3*c - a^6*b)*sqrt(
d*x + c)*sqrt(-c)*x + (a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*sqrt(-c)*x
)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(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^2),x, algorithm="giac")

[Out]

undef