\(\int \frac {1}{x (c+d x) (a+b x^2)^{5/2}} \, dx\) [1283]

Optimal result

Integrand size = 22, antiderivative size = 191 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\frac {b (c-d x)}{3 a \left (b c^2+a d^2\right ) \left (a+b x^2\right )^{3/2}}+\frac {b \left (3 c \left (b c^2+2 a d^2\right )-d \left (2 b c^2+5 a d^2\right ) x\right )}{3 a^2 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}+\frac {d^5 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c \left (b c^2+a d^2\right )^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2} c} \] Output:

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

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\frac {b \left (a^2 d^2 (7 c-6 d x)+b^2 c^2 x^2 (3 c-2 d x)+a b \left (4 c^3-3 c^2 d x+6 c d^2 x^2-5 d^3 x^3\right )\right )}{3 a^2 \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^{3/2}}+\frac {2 d^5 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{c \left (-b c^2-a d^2\right )^{5/2}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2} c} \] Input:

Integrate[1/(x*(c + d*x)*(a + b*x^2)^(5/2)),x]
 

Output:

(b*(a^2*d^2*(7*c - 6*d*x) + b^2*c^2*x^2*(3*c - 2*d*x) + a*b*(4*c^3 - 3*c^2 
*d*x + 6*c*d^2*x^2 - 5*d^3*x^3)))/(3*a^2*(b*c^2 + a*d^2)^2*(a + b*x^2)^(3/ 
2)) + (2*d^5*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) 
- a*d^2]])/(c*(-(b*c^2) - a*d^2)^(5/2)) + (2*ArcTanh[(Sqrt[b]*x - Sqrt[a + 
 b*x^2])/Sqrt[a]])/(a^(5/2)*c)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.21, 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.

\(\displaystyle \int \frac {1}{x \left (a+b x^2\right )^{5/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (\frac {1}{c x \left (a+b x^2\right )^{5/2}}-\frac {d}{c \left (a+b x^2\right )^{5/2} (c+d x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2} c}-\frac {d \left (3 a^2 d^3+b c x \left (5 a d^2+2 b c^2\right )\right )}{3 a^2 c \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}+\frac {1}{a^2 c \sqrt {a+b x^2}}+\frac {d^5 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c \left (a d^2+b c^2\right )^{5/2}}-\frac {d (a d+b c x)}{3 a c \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )}+\frac {1}{3 a c \left (a+b x^2\right )^{3/2}}\)

Input:

Int[1/(x*(c + d*x)*(a + b*x^2)^(5/2)),x]
 

Output:

1/(3*a*c*(a + b*x^2)^(3/2)) - (d*(a*d + b*c*x))/(3*a*c*(b*c^2 + a*d^2)*(a 
+ b*x^2)^(3/2)) + 1/(a^2*c*Sqrt[a + b*x^2]) - (d*(3*a^2*d^3 + b*c*(2*b*c^2 
 + 5*a*d^2)*x))/(3*a^2*c*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^2]) + (d^5*ArcTanh 
[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c*(b*c^2 + a*d^2)^ 
(5/2)) - ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]/(a^(5/2)*c)
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(667\) vs. \(2(171)=342\).

Time = 0.40 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.50

method	result	size
default	\(\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{c}-\frac {\frac {d^{2}}{3 \left (a \,d^{2}+b \,c^{2}\right ) \left (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}}\right )^{\frac {3}{2}}}+\frac {b c d \left (\frac {\frac {4 b \left (x +\frac {c}{d}\right )}{3}-\frac {4 b c}{3 d}}{\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \left (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}}\right )^{\frac {3}{2}}}+\frac {16 b \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{3 {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right )}^{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}}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {d^{2} \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \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}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \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}}}}-\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 )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}}{c}\)	\(668\)

Input:

int(1/x/(d*x+c)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/c*(1/3/a/(b*x^2+a)^(3/2)+1/a*(1/a/(b*x^2+a)^(1/2)-1/a^(3/2)*ln((2*a+2*a^ 
(1/2)*(b*x^2+a)^(1/2))/x)))-1/c*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/ 
d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2 
*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(3/2)+16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*( 
2*b*(x+c/d)-2*b*c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2) 
)+1/(a*d^2+b*c^2)*d^2*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a 
*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a 
*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/ 
d^2)^(1/2)-1/(a*d^2+b*c^2)*d^2/((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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (172) = 344\).

Time = 1.05 (sec) , antiderivative size = 2530, normalized size of antiderivative = 13.25 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(a^3*b^2*d^5*x^4 + 2*a^4*b*d^5*x^2 + a^5*d^5)*sqrt(b*c^2 + a*d^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*sqr 
t(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)) 
 + 3*(a^2*b^3*c^6 + 3*a^3*b^2*c^4*d^2 + 3*a^4*b*c^2*d^4 + a^5*d^6 + (b^5*c 
^6 + 3*a*b^4*c^4*d^2 + 3*a^2*b^3*c^2*d^4 + a^3*b^2*d^6)*x^4 + 2*(a*b^4*c^6 
 + 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 + a^4*b*d^6)*x^2)*sqrt(a)*log(-(b 
*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(4*a^2*b^3*c^6 + 11*a^3*b 
^2*c^4*d^2 + 7*a^4*b*c^2*d^4 - (2*a*b^4*c^5*d + 7*a^2*b^3*c^3*d^3 + 5*a^3* 
b^2*c*d^5)*x^3 + 3*(a*b^4*c^6 + 3*a^2*b^3*c^4*d^2 + 2*a^3*b^2*c^2*d^4)*x^2 
 - 3*(a^2*b^3*c^5*d + 3*a^3*b^2*c^3*d^3 + 2*a^4*b*c*d^5)*x)*sqrt(b*x^2 + a 
))/(a^5*b^3*c^7 + 3*a^6*b^2*c^5*d^2 + 3*a^7*b*c^3*d^4 + a^8*c*d^6 + (a^3*b 
^5*c^7 + 3*a^4*b^4*c^5*d^2 + 3*a^5*b^3*c^3*d^4 + a^6*b^2*c*d^6)*x^4 + 2*(a 
^4*b^4*c^7 + 3*a^5*b^3*c^5*d^2 + 3*a^6*b^2*c^3*d^4 + a^7*b*c*d^6)*x^2), 1/ 
6*(6*(a^3*b^2*d^5*x^4 + 2*a^4*b*d^5*x^2 + a^5*d^5)*sqrt(-b*c^2 - a*d^2)*ar 
ctan(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)) + 3*(a^2*b^3*c^6 + 3*a^3*b^2*c^4*d^2 + 3*a^4* 
b*c^2*d^4 + a^5*d^6 + (b^5*c^6 + 3*a*b^4*c^4*d^2 + 3*a^2*b^3*c^2*d^4 + a^3 
*b^2*d^6)*x^4 + 2*(a*b^4*c^6 + 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 + a^4 
*b*d^6)*x^2)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 
 2*(4*a^2*b^3*c^6 + 11*a^3*b^2*c^4*d^2 + 7*a^4*b*c^2*d^4 - (2*a*b^4*c^5...
 

Sympy [F]

\[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{x \left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x\right )}\, dx \] Input:

integrate(1/x/(d*x+c)/(b*x**2+a)**(5/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/(x*(a + b*x**2)**(5/2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x + c\right )} x} \,d x } \] Input:

integrate(1/x/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(5/2)*(d*x + c)*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/x/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{x\,{\left (b\,x^2+a\right )}^{5/2}\,\left (c+d\,x\right )} \,d x \] Input:

int(1/(x*(a + b*x^2)^(5/2)*(c + d*x)),x)
 

Output:

int(1/(x*(a + b*x^2)^(5/2)*(c + d*x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 1365, normalized size of antiderivative = 7.15 \[ \int \frac {1}{x (c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/x/(d*x+c)/(b*x^2+a)^(5/2),x)
 

Output:

(6*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**5*d**5 + 12*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b*d**5*x**2 + 6*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**3*b** 
2*d**5*x**4 - 6*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**5*d**5 - 12*sqrt(a*d 
**2 + b*c**2)*log(c + d*x)*a**4*b*d**5*x**2 - 6*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*a**3*b**2*d**5*x**4 + 14*sqrt(a + b*x**2)*a**4*b*c**2*d**4 - 12*s 
qrt(a + b*x**2)*a**4*b*c*d**5*x + 22*sqrt(a + b*x**2)*a**3*b**2*c**4*d**2 
- 18*sqrt(a + b*x**2)*a**3*b**2*c**3*d**3*x + 12*sqrt(a + b*x**2)*a**3*b** 
2*c**2*d**4*x**2 - 10*sqrt(a + b*x**2)*a**3*b**2*c*d**5*x**3 + 8*sqrt(a + 
b*x**2)*a**2*b**3*c**6 - 6*sqrt(a + b*x**2)*a**2*b**3*c**5*d*x + 18*sqrt(a 
 + b*x**2)*a**2*b**3*c**4*d**2*x**2 - 14*sqrt(a + b*x**2)*a**2*b**3*c**3*d 
**3*x**3 + 6*sqrt(a + b*x**2)*a*b**4*c**6*x**2 - 4*sqrt(a + b*x**2)*a*b**4 
*c**5*d*x**3 + 3*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**5*d**6 + 9*sqr 
t(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**4*b*c**2*d**4 + 6*sqrt(a)*log(sqrt 
(a + b*x**2) - sqrt(a))*a**4*b*d**6*x**2 + 9*sqrt(a)*log(sqrt(a + b*x**2) 
- sqrt(a))*a**3*b**2*c**4*d**2 + 18*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a) 
)*a**3*b**2*c**2*d**4*x**2 + 3*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a** 
3*b**2*d**6*x**4 + 3*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*b**3*c** 
6 + 18*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*b**3*c**4*d**2*x**2...