3.477 \(\int \frac{1}{x^3 \left (a+b \sqrt{c+d x}\right )} \, dx\)

Optimal. Leaf size=204 \[ -\frac{a-b \sqrt{c+d x}}{2 x^2 \left (a^2-b^2 c\right )}-\frac{b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt{c+d x}\right )}{4 c x \left (a^2-b^2 c\right )^2}+\frac{a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac{2 a b^4 d^2 \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^3}-\frac{b d^2 \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3} \]

[Out]

-(a - b*Sqrt[c + d*x])/(2*(a^2 - b^2*c)*x^2) - (b*d*(4*a*b*c - (a^2 + 3*b^2*c)*S
qrt[c + d*x]))/(4*c*(a^2 - b^2*c)^2*x) - (b*(a^4 - 6*a^2*b^2*c - 3*b^4*c^2)*d^2*
ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(4*c^(3/2)*(a^2 - b^2*c)^3) + (a*b^4*d^2*Log[x])
/(a^2 - b^2*c)^3 - (2*a*b^4*d^2*Log[a + b*Sqrt[c + d*x]])/(a^2 - b^2*c)^3

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Rubi [A]  time = 0.59139, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{a-b \sqrt{c+d x}}{2 x^2 \left (a^2-b^2 c\right )}-\frac{b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt{c+d x}\right )}{4 c x \left (a^2-b^2 c\right )^2}+\frac{a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac{2 a b^4 d^2 \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^3}-\frac{b d^2 \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*Sqrt[c + d*x])),x]

[Out]

-(a - b*Sqrt[c + d*x])/(2*(a^2 - b^2*c)*x^2) - (b*d*(4*a*b*c - (a^2 + 3*b^2*c)*S
qrt[c + d*x]))/(4*c*(a^2 - b^2*c)^2*x) - (b*(a^4 - 6*a^2*b^2*c - 3*b^4*c^2)*d^2*
ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(4*c^(3/2)*(a^2 - b^2*c)^3) + (a*b^4*d^2*Log[x])
/(a^2 - b^2*c)^3 - (2*a*b^4*d^2*Log[a + b*Sqrt[c + d*x]])/(a^2 - b^2*c)^3

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Rubi in Sympy [A]  time = 47.5834, size = 187, normalized size = 0.92 \[ \frac{a b^{4} d^{2} \log{\left (- d x \right )}}{\left (a^{2} - b^{2} c\right )^{3}} - \frac{2 a b^{4} d^{2} \log{\left (a + b \sqrt{c + d x} \right )}}{\left (a^{2} - b^{2} c\right )^{3}} + \frac{b d \left (- 4 a b c + \left (a^{2} + 3 b^{2} c\right ) \sqrt{c + d x}\right )}{4 c x \left (a^{2} - b^{2} c\right )^{2}} - \frac{b d^{2} \left (a^{4} - 6 a^{2} b^{2} c - 3 b^{4} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{4 c^{\frac{3}{2}} \left (a^{2} - b^{2} c\right )^{3}} - \frac{a - b \sqrt{c + d x}}{2 x^{2} \left (a^{2} - b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b*(d*x+c)**(1/2)),x)

[Out]

a*b**4*d**2*log(-d*x)/(a**2 - b**2*c)**3 - 2*a*b**4*d**2*log(a + b*sqrt(c + d*x)
)/(a**2 - b**2*c)**3 + b*d*(-4*a*b*c + (a**2 + 3*b**2*c)*sqrt(c + d*x))/(4*c*x*(
a**2 - b**2*c)**2) - b*d**2*(a**4 - 6*a**2*b**2*c - 3*b**4*c**2)*atanh(sqrt(c +
d*x)/sqrt(c))/(4*c**(3/2)*(a**2 - b**2*c)**3) - (a - b*sqrt(c + d*x))/(2*x**2*(a
**2 - b**2*c))

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Mathematica [A]  time = 0.732582, size = 228, normalized size = 1.12 \[ \frac{b d^2 x^2 \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\sqrt{c} \left (4 a b^4 c d^2 x^2 \log \left (a^2-b^2 (c+d x)\right )+\left (a^2-b^2 c\right ) \left (2 a^3 c-a^2 b \sqrt{c+d x} (2 c+d x)-2 a b^2 c (c-2 d x)+b^3 c (2 c-3 d x) \sqrt{c+d x}\right )-4 a b^4 c d^2 x^2 \log (x)\right )+8 a b^4 c^{3/2} d^2 x^2 \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )}{4 c^{3/2} x^2 \left (b^2 c-a^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*Sqrt[c + d*x])),x]

[Out]

(8*a*b^4*c^(3/2)*d^2*x^2*ArcTanh[(b*Sqrt[c + d*x])/a] + b*(a^4 - 6*a^2*b^2*c - 3
*b^4*c^2)*d^2*x^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] + Sqrt[c]*((a^2 - b^2*c)*(2*a^3
*c - 2*a*b^2*c*(c - 2*d*x) + b^3*c*(2*c - 3*d*x)*Sqrt[c + d*x] - a^2*b*Sqrt[c +
d*x]*(2*c + d*x)) - 4*a*b^4*c*d^2*x^2*Log[x] + 4*a*b^4*c*d^2*x^2*Log[a^2 - b^2*(
c + d*x)]))/(4*c^(3/2)*(-a^2 + b^2*c)^3*x^2)

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Maple [B]  time = 0.021, size = 460, normalized size = 2.3 \[ -{\frac{3\,{b}^{5}c}{4\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{3}{a}^{2}}{2\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{b{a}^{4}}{4\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}c} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{a{b}^{4}cd}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}x}}-{\frac{a{b}^{4}{c}^{2}}{2\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}}}-{\frac{{a}^{3}d{b}^{2}}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}x}}+{\frac{{a}^{3}{b}^{2}c}{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}}}-{\frac{3\,{a}^{2}{b}^{3}c}{2\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}}\sqrt{dx+c}}+{\frac{b{a}^{4}}{4\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}}\sqrt{dx+c}}+{\frac{5\,{b}^{5}{c}^{2}}{4\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}}\sqrt{dx+c}}-{\frac{{a}^{5}}{2\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}{x}^{2}}}+{\frac{a{b}^{4}{d}^{2}\ln \left ( cdx \right ) }{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}}+{\frac{3\,{d}^{2}{b}^{5}}{4\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}\sqrt{c}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ) }+{\frac{3\,{a}^{2}{b}^{3}{d}^{2}}{2\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{b{d}^{2}{a}^{4}}{4\, \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}{\it Artanh} \left ({1\sqrt{dx+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{a{b}^{4}{d}^{2}\ln \left ( a+b\sqrt{dx+c} \right ) }{ \left ( -{b}^{2}c+{a}^{2} \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*(d*x+c)^(1/2)),x)

[Out]

-3/4/(-b^2*c+a^2)^3/x^2*b^5*c*(d*x+c)^(3/2)+1/2/(-b^2*c+a^2)^3/x^2*b^3*(d*x+c)^(
3/2)*a^2+1/4/(-b^2*c+a^2)^3/x^2*b/c*(d*x+c)^(3/2)*a^4+d/(-b^2*c+a^2)^3/x*a*b^4*c
-1/2/(-b^2*c+a^2)^3/x^2*a*b^4*c^2-d/(-b^2*c+a^2)^3/x*a^3*b^2+1/(-b^2*c+a^2)^3/x^
2*a^3*b^2*c-3/2/(-b^2*c+a^2)^3/x^2*(d*x+c)^(1/2)*a^2*b^3*c+1/4/(-b^2*c+a^2)^3/x^
2*(d*x+c)^(1/2)*b*a^4+5/4/(-b^2*c+a^2)^3/x^2*(d*x+c)^(1/2)*b^5*c^2-1/2/(-b^2*c+a
^2)^3/x^2*a^5+d^2/(-b^2*c+a^2)^3*a*b^4*ln(c*d*x)+3/4*d^2/(-b^2*c+a^2)^3*b^5*c^(1
/2)*arctanh((d*x+c)^(1/2)/c^(1/2))+3/2*d^2/(-b^2*c+a^2)^3*b^3/c^(1/2)*arctanh((d
*x+c)^(1/2)/c^(1/2))*a^2-1/4*d^2/(-b^2*c+a^2)^3*b/c^(3/2)*arctanh((d*x+c)^(1/2)/
c^(1/2))*a^4-2*a*b^4*d^2*ln(a+b*(d*x+c)^(1/2))/(-b^2*c+a^2)^3

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.873207, size = 1, normalized size = 0. \[ \left [\frac{16 \, a b^{4} c^{\frac{3}{2}} d^{2} x^{2} \log \left (\sqrt{d x + c} b + a\right ) +{\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} d^{2} x^{2} \log \left (\frac{{\left (d x + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x + c} c}{x}\right ) - 2 \,{\left (2 \, b^{5} c^{3} - 4 \, a^{2} b^{3} c^{2} + 2 \, a^{4} b c -{\left (3 \, b^{5} c^{2} - 2 \, a^{2} b^{3} c - a^{4} b\right )} d x\right )} \sqrt{d x + c} \sqrt{c} - 4 \,{\left (2 \, a b^{4} c d^{2} x^{2} \log \left (x\right ) - a b^{4} c^{3} + 2 \, a^{3} b^{2} c^{2} - a^{5} c + 2 \,{\left (a b^{4} c^{2} - a^{3} b^{2} c\right )} d x\right )} \sqrt{c}}{8 \,{\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} \sqrt{c} x^{2}}, \frac{8 \, a b^{4} \sqrt{-c} c d^{2} x^{2} \log \left (\sqrt{d x + c} b + a\right ) +{\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} d^{2} x^{2} \arctan \left (\frac{c}{\sqrt{d x + c} \sqrt{-c}}\right ) -{\left (2 \, b^{5} c^{3} - 4 \, a^{2} b^{3} c^{2} + 2 \, a^{4} b c -{\left (3 \, b^{5} c^{2} - 2 \, a^{2} b^{3} c - a^{4} b\right )} d x\right )} \sqrt{d x + c} \sqrt{-c} - 2 \,{\left (2 \, a b^{4} c d^{2} x^{2} \log \left (x\right ) - a b^{4} c^{3} + 2 \, a^{3} b^{2} c^{2} - a^{5} c + 2 \,{\left (a b^{4} c^{2} - a^{3} b^{2} c\right )} d x\right )} \sqrt{-c}}{4 \,{\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} \sqrt{-c} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((sqrt(d*x + c)*b + a)*x^3),x, algorithm="fricas")

[Out]

[1/8*(16*a*b^4*c^(3/2)*d^2*x^2*log(sqrt(d*x + c)*b + a) + (3*b^5*c^2 + 6*a^2*b^3
*c - a^4*b)*d^2*x^2*log(((d*x + 2*c)*sqrt(c) - 2*sqrt(d*x + c)*c)/x) - 2*(2*b^5*
c^3 - 4*a^2*b^3*c^2 + 2*a^4*b*c - (3*b^5*c^2 - 2*a^2*b^3*c - a^4*b)*d*x)*sqrt(d*
x + c)*sqrt(c) - 4*(2*a*b^4*c*d^2*x^2*log(x) - a*b^4*c^3 + 2*a^3*b^2*c^2 - a^5*c
+ 2*(a*b^4*c^2 - a^3*b^2*c)*d*x)*sqrt(c))/((b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2
*c^2 - a^6*c)*sqrt(c)*x^2), 1/4*(8*a*b^4*sqrt(-c)*c*d^2*x^2*log(sqrt(d*x + c)*b
+ a) + (3*b^5*c^2 + 6*a^2*b^3*c - a^4*b)*d^2*x^2*arctan(c/(sqrt(d*x + c)*sqrt(-c
))) - (2*b^5*c^3 - 4*a^2*b^3*c^2 + 2*a^4*b*c - (3*b^5*c^2 - 2*a^2*b^3*c - a^4*b)
*d*x)*sqrt(d*x + c)*sqrt(-c) - 2*(2*a*b^4*c*d^2*x^2*log(x) - a*b^4*c^3 + 2*a^3*b
^2*c^2 - a^5*c + 2*(a*b^4*c^2 - a^3*b^2*c)*d*x)*sqrt(-c))/((b^6*c^4 - 3*a^2*b^4*
c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt(-c)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b \sqrt{c + d x}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a+b*(d*x+c)**(1/2)),x)

[Out]

Integral(1/(x**3*(a + b*sqrt(c + d*x))), x)

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GIAC/XCAS [A]  time = 0.295693, size = 648, normalized size = 3.18 \[ \frac{2 \, a b^{5} d^{2}{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{7} c^{3} - 3 \, a^{2} b^{5} c^{2} + 3 \, a^{4} b^{3} c - a^{6} b} - \frac{a b^{4} d^{2}{\rm ln}\left (-d x\right )}{b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}} + \frac{{\left (3 \, b^{5} c^{2} d^{2} + 6 \, a^{2} b^{3} c d^{2} - a^{4} b d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{4 \,{\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} \sqrt{-c}} + \frac{2 \, a b^{4} c^{2} d^{2}{\rm ln}\left (c\right ) - 4 \, a b^{4} c^{2} d^{2}{\rm ln}\left ({\left | a \right |}\right ) - 3 \, a b^{4} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{2} - a^{5} d^{2}}{2 \,{\left (b^{6} c^{5} - 3 \, a^{2} b^{4} c^{4} + 3 \, a^{4} b^{2} c^{3} - a^{6} c^{2}\right )}} + \frac{6 \, a b^{4} c^{3} d^{2} - 8 \, a^{3} b^{2} c^{2} d^{2} + 2 \, a^{5} c d^{2} +{\left (3 \, b^{5} c^{2} d^{2} - 2 \, a^{2} b^{3} c d^{2} - a^{4} b d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 4 \,{\left (a b^{4} c^{2} d^{2} - a^{3} b^{2} c d^{2}\right )}{\left (d x + c\right )} -{\left (5 \, b^{5} c^{3} d^{2} - 6 \, a^{2} b^{3} c^{2} d^{2} + a^{4} b c d^{2}\right )} \sqrt{d x + c}}{4 \,{\left (b^{2} c - a^{2}\right )}^{3} c d^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((sqrt(d*x + c)*b + a)*x^3),x, algorithm="giac")

[Out]

2*a*b^5*d^2*ln(abs(sqrt(d*x + c)*b + a))/(b^7*c^3 - 3*a^2*b^5*c^2 + 3*a^4*b^3*c
- a^6*b) - a*b^4*d^2*ln(-d*x)/(b^6*c^3 - 3*a^2*b^4*c^2 + 3*a^4*b^2*c - a^6) + 1/
4*(3*b^5*c^2*d^2 + 6*a^2*b^3*c*d^2 - a^4*b*d^2)*arctan(sqrt(d*x + c)/sqrt(-c))/(
(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt(-c)) + 1/2*(2*a*b^4*c^2*d
^2*ln(c) - 4*a*b^4*c^2*d^2*ln(abs(a)) - 3*a*b^4*c^2*d^2 + 4*a^3*b^2*c*d^2 - a^5*
d^2)/(b^6*c^5 - 3*a^2*b^4*c^4 + 3*a^4*b^2*c^3 - a^6*c^2) + 1/4*(6*a*b^4*c^3*d^2
- 8*a^3*b^2*c^2*d^2 + 2*a^5*c*d^2 + (3*b^5*c^2*d^2 - 2*a^2*b^3*c*d^2 - a^4*b*d^2
)*(d*x + c)^(3/2) - 4*(a*b^4*c^2*d^2 - a^3*b^2*c*d^2)*(d*x + c) - (5*b^5*c^3*d^2
- 6*a^2*b^3*c^2*d^2 + a^4*b*c*d^2)*sqrt(d*x + c))/((b^2*c - a^2)^3*c*d^2*x^2)