\(\int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx\) [1189]

Optimal result

Integrand size = 22, antiderivative size = 148 \[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {a+b x^2}}{2 a c x^2}+\frac {d \sqrt {a+b x^2}}{a c^2 x}+\frac {d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^3 \sqrt {b c^2+a d^2}}+\frac {\left (b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2} c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a} \left (c \left (b c^2+a d^2\right ) (-c+2 d x) \sqrt {a+b x^2}-4 a d^3 \sqrt {-b c^2-a d^2} x^2 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )\right )-2 \left (b^2 c^4-a b c^2 d^2-2 a^2 d^4\right ) x^2 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2} c^3 \left (b c^2+a d^2\right ) x^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^3 \sqrt {a+b x^2} (c+d x)} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (-\frac {d^3}{c^3 \sqrt {a+b x^2} (c+d x)}+\frac {d^2}{c^3 x \sqrt {a+b x^2}}-\frac {d}{c^2 x^2 \sqrt {a+b x^2}}+\frac {1}{c x^3 \sqrt {a+b x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2} c}-\frac {d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a} c^3}+\frac {d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^3 \sqrt {a d^2+b c^2}}+\frac {d \sqrt {a+b x^2}}{a c^2 x}-\frac {\sqrt {a+b x^2}}{2 a c x^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.43

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 d x +c \right )}{2 a \,c^{2} x^{2}}+\frac {-\frac {\left (2 a \,d^{2}-b \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {2 a \,d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 a \,c^{2}}\)	\(211\)
default	\(\frac {-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}}{c}-\frac {d^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{3} \sqrt {a}}+\frac {d \sqrt {b \,x^{2}+a}}{a \,c^{2} x}+\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{3} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(235\)

Input:

int(1/x^3/(d*x+c)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(b*x^2+a)^(1/2)*(-2*d*x+c)/a/c^2/x^2+1/2/a/c^2*(-(2*a*d^2-b*c^2)/c/a^ 
(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+2*a*d^2/c/((a*d^2+b*c^2)/d^2)^ 
(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)* 
(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 962, normalized size of antiderivative = 6.50 \[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

integrate(1/x^3/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/4*(2*sqrt(b*c^2 + a*d^2)*a^2*d^3*x^2*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2 
*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqr 
t(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - (b^2*c^4 - a*b*c^2*d^2 - 2*a^2* 
d^4)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*( 
a*b*c^4 + a^2*c^2*d^2 - 2*(a*b*c^3*d + a^2*c*d^3)*x)*sqrt(b*x^2 + a))/((a^ 
2*b*c^5 + a^3*c^3*d^2)*x^2), 1/4*(4*sqrt(-b*c^2 - a*d^2)*a^2*d^3*x^2*arcta 
n(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + 
(b^2*c^2 + a*b*d^2)*x^2)) - (b^2*c^4 - a*b*c^2*d^2 - 2*a^2*d^4)*sqrt(a)*x^ 
2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a*b*c^4 + a^2*c 
^2*d^2 - 2*(a*b*c^3*d + a^2*c*d^3)*x)*sqrt(b*x^2 + a))/((a^2*b*c^5 + a^3*c 
^3*d^2)*x^2), 1/2*(sqrt(b*c^2 + a*d^2)*a^2*d^3*x^2*log((2*a*b*c*d*x - a*b* 
c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a*d^2)*(b*c*x 
 - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - (b^2*c^4 - a*b*c^2*d 
^2 - 2*a^2*d^4)*sqrt(-a)*x^2*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (a*b*c^4 
 + a^2*c^2*d^2 - 2*(a*b*c^3*d + a^2*c*d^3)*x)*sqrt(b*x^2 + a))/((a^2*b*c^5 
 + a^3*c^3*d^2)*x^2), 1/2*(2*sqrt(-b*c^2 - a*d^2)*a^2*d^3*x^2*arctan(sqrt( 
-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^ 
2 + a*b*d^2)*x^2)) - (b^2*c^4 - a*b*c^2*d^2 - 2*a^2*d^4)*sqrt(-a)*x^2*arct 
an(sqrt(b*x^2 + a)*sqrt(-a)/a) - (a*b*c^4 + a^2*c^2*d^2 - 2*(a*b*c^3*d + a 
^2*c*d^3)*x)*sqrt(b*x^2 + a))/((a^2*b*c^5 + a^3*c^3*d^2)*x^2)]
 

Sympy [F]

\[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x^{2}} \left (c + d x\right )}\, dx \] Input:

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

Output:

Integral(1/(x**3*sqrt(a + b*x**2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x + c\right )} x^{3}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(b*x^2 + a)*(d*x + c)*x^3), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.62 \[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx=-b^{\frac {3}{2}} {\left (\frac {2 \, d^{3} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} b^{\frac {3}{2}} c^{3}} + \frac {{\left (b c^{2} - 2 \, a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a b^{\frac {3}{2}} c^{3}} - \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} \sqrt {b} c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a \sqrt {b} c + 2 \, a^{2} d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a b c^{2}}\right )} \] Input:

integrate(1/x^3/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {1}{x^3\,\sqrt {b\,x^2+a}\,\left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.27 \[ \int \frac {1}{x^3 (c+d x) \sqrt {a+b x^2}} \, dx=\frac {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a^{2} d^{3} x^{2}-4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a^{2} d^{3} x^{2}-2 \sqrt {b \,x^{2}+a}\, a^{2} c^{2} d^{2}+4 \sqrt {b \,x^{2}+a}\, a^{2} c \,d^{3} x -2 \sqrt {b \,x^{2}+a}\, a b \,c^{4}+4 \sqrt {b \,x^{2}+a}\, a b \,c^{3} d x +2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) a^{2} d^{4} x^{2}+\sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) a b \,c^{2} d^{2} x^{2}-\sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) b^{2} c^{4} x^{2}-2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) a^{2} d^{4} x^{2}-\sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) a b \,c^{2} d^{2} x^{2}+\sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) b^{2} c^{4} x^{2}}{4 a^{2} c^{3} x^{2} \left (a \,d^{2}+b \,c^{2}\right )} \] Input:

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

Output:

(4*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a 
*d + b*c*x)*a**2*d**3*x**2 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*d** 
3*x**2 - 2*sqrt(a + b*x**2)*a**2*c**2*d**2 + 4*sqrt(a + b*x**2)*a**2*c*d** 
3*x - 2*sqrt(a + b*x**2)*a*b*c**4 + 4*sqrt(a + b*x**2)*a*b*c**3*d*x + 2*sq 
rt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*d**4*x**2 + sqrt(a)*log(sqrt(a 
+ b*x**2) - sqrt(a))*a*b*c**2*d**2*x**2 - sqrt(a)*log(sqrt(a + b*x**2) - s 
qrt(a))*b**2*c**4*x**2 - 2*sqrt(a)*log(sqrt(a + b*x**2) + sqrt(a))*a**2*d* 
*4*x**2 - sqrt(a)*log(sqrt(a + b*x**2) + sqrt(a))*a*b*c**2*d**2*x**2 + sqr 
t(a)*log(sqrt(a + b*x**2) + sqrt(a))*b**2*c**4*x**2)/(4*a**2*c**3*x**2*(a* 
d**2 + b*c**2))