\(\int \frac {1}{(e x)^{5/2} \sqrt {c+d x} (a+b x^2)} \, dx\) [880]

Optimal result

Integrand size = 26, antiderivative size = 236 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2}}+\frac {4 d \sqrt {c+d x}}{3 a c^2 e^2 \sqrt {e x}}-\frac {b \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} e^{5/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} e^{5/2}} \] Output:

-2/3*(d*x+c)^(1/2)/a/c/e/(e*x)^(3/2)+4/3*d*(d*x+c)^(1/2)/a/c^2/e^2/(e*x)^( 
1/2)-b*arctan((b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2 
)/(d*x+c)^(1/2))/(-a)^(7/4)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/e^(5/2)-b*arcta 
nh((b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^( 
1/2))/(-a)^(7/4)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)/e^(5/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.20 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {x \left (-2 (c-2 d x) \sqrt {c+d x}+12 b c^2 d^{3/2} x^{3/2} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {\log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}}{-b c^3+3 b c^2 \text {$\#$1}+8 a d^2 \text {$\#$1}-3 b c \text {$\#$1}^2+b \text {$\#$1}^3}\&\right ]\right )}{3 a c^2 (e x)^{5/2}} \] Input:

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

Output:

(x*(-2*(c - 2*d*x)*Sqrt[c + d*x] + 12*b*c^2*d^(3/2)*x^(3/2)*RootSum[b*c^4 
- 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*#1^2 - 4*b*c*#1^3 + b*#1^4 & , (Log 
[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1)/(-(b*c^3) + 3*b*c^2 
*#1 + 8*a*d^2*#1 - 3*b*c*#1^2 + b*#1^3) & ]))/(3*a*c^2*(e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {615, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {c+d x} \left (3 \sqrt {-a} \sqrt {b} c-2 a d\right )}{3 a^2 c^2 e^2 \sqrt {e x}}+\frac {\sqrt {c+d x} \left (3 \sqrt {-a} \sqrt {b} c+2 a d\right )}{3 a^2 c^2 e^2 \sqrt {e x}}-\frac {b \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} e^{5/2} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {b \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} e^{5/2} \sqrt {\sqrt {-a} d+\sqrt {b} c}}-\frac {2 \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

Input:

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

Output:

(-2*Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)) - ((3*Sqrt[-a]*Sqrt[b]*c - 2*a*d) 
*Sqrt[c + d*x])/(3*a^2*c^2*e^2*Sqrt[e*x]) + ((3*Sqrt[-a]*Sqrt[b]*c + 2*a*d 
)*Sqrt[c + d*x])/(3*a^2*c^2*e^2*Sqrt[e*x]) - (b*ArcTan[(Sqrt[Sqrt[b]*c - S 
qrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(7/4)*Sqr 
t[Sqrt[b]*c - Sqrt[-a]*d]*e^(5/2)) - (b*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a] 
*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(7/4)*Sqrt[Sqrt[ 
b]*c + Sqrt[-a]*d]*e^(5/2))
 

Defintions of rubi rules used

rule 615

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

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. \(442\) vs. \(2(176)=352\).

Time = 0.50 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.88

method	result	size
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-2 d x +c \right )}{3 c^{2} a x \,e^{2} \sqrt {e x}}+\frac {\left (\frac {b \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a \sqrt {-a b}\, \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}-\frac {b \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{e^{2} \sqrt {e x}\, \sqrt {d x +c}}\)	\(443\)
default	\(\frac {\sqrt {d x +c}\, b \left (3 \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) a \,c^{2} d^{2} e \,x^{2} \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}+3 \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) b \,c^{4} e \,x^{2} \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}-3 \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) a \,c^{2} d^{2} e \,x^{2} \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}-3 \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) b \,c^{4} e \,x^{2} \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}+8 x \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a^{2} d^{3}+8 x \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a b \,c^{2} d -4 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a^{2} c \,d^{2}-4 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a b \,c^{3}\right )}{6 e^{2} x \,c^{2} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} \left (\sqrt {-a b}\, d +b c \right ) \left (b c -\sqrt {-a b}\, d \right ) \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\)	\(790\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1265 vs. \(2 (176) = 352\).

Time = 0.11 (sec) , antiderivative size = 1265, normalized size of antiderivative = 5.36 \[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/6*(3*a*c^2*e^3*x^2*sqrt(((a^3*b*c^2 + a^4*d^2)*e^5*sqrt(-b^5*c^2/((a^7*b 
^2*c^4 + 2*a^8*b*c^2*d^2 + a^9*d^4)*e^10)) + b^2*d)/((a^3*b*c^2 + a^4*d^2) 
*e^5))*log((sqrt(d*x + c)*sqrt(e*x)*b^3 + ((a^5*b*c^2 + a^6*d^2)*e^8*x*sqr 
t(-b^5*c^2/((a^7*b^2*c^4 + 2*a^8*b*c^2*d^2 + a^9*d^4)*e^10)) - a^2*b^2*d*e 
^3*x)*sqrt(((a^3*b*c^2 + a^4*d^2)*e^5*sqrt(-b^5*c^2/((a^7*b^2*c^4 + 2*a^8* 
b*c^2*d^2 + a^9*d^4)*e^10)) + b^2*d)/((a^3*b*c^2 + a^4*d^2)*e^5)))/x) - 3* 
a*c^2*e^3*x^2*sqrt(((a^3*b*c^2 + a^4*d^2)*e^5*sqrt(-b^5*c^2/((a^7*b^2*c^4 
+ 2*a^8*b*c^2*d^2 + a^9*d^4)*e^10)) + b^2*d)/((a^3*b*c^2 + a^4*d^2)*e^5))* 
log((sqrt(d*x + c)*sqrt(e*x)*b^3 - ((a^5*b*c^2 + a^6*d^2)*e^8*x*sqrt(-b^5* 
c^2/((a^7*b^2*c^4 + 2*a^8*b*c^2*d^2 + a^9*d^4)*e^10)) - a^2*b^2*d*e^3*x)*s 
qrt(((a^3*b*c^2 + a^4*d^2)*e^5*sqrt(-b^5*c^2/((a^7*b^2*c^4 + 2*a^8*b*c^2*d 
^2 + a^9*d^4)*e^10)) + b^2*d)/((a^3*b*c^2 + a^4*d^2)*e^5)))/x) - 3*a*c^2*e 
^3*x^2*sqrt(-((a^3*b*c^2 + a^4*d^2)*e^5*sqrt(-b^5*c^2/((a^7*b^2*c^4 + 2*a^ 
8*b*c^2*d^2 + a^9*d^4)*e^10)) - b^2*d)/((a^3*b*c^2 + a^4*d^2)*e^5))*log((s 
qrt(d*x + c)*sqrt(e*x)*b^3 + ((a^5*b*c^2 + a^6*d^2)*e^8*x*sqrt(-b^5*c^2/(( 
a^7*b^2*c^4 + 2*a^8*b*c^2*d^2 + a^9*d^4)*e^10)) + a^2*b^2*d*e^3*x)*sqrt(-( 
(a^3*b*c^2 + a^4*d^2)*e^5*sqrt(-b^5*c^2/((a^7*b^2*c^4 + 2*a^8*b*c^2*d^2 + 
a^9*d^4)*e^10)) - b^2*d)/((a^3*b*c^2 + a^4*d^2)*e^5)))/x) + 3*a*c^2*e^3*x^ 
2*sqrt(-((a^3*b*c^2 + a^4*d^2)*e^5*sqrt(-b^5*c^2/((a^7*b^2*c^4 + 2*a^8*b*c 
^2*d^2 + a^9*d^4)*e^10)) - b^2*d)/((a^3*b*c^2 + a^4*d^2)*e^5))*log((sqr...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[1,0]%%%},[0,2,0]%%%}+%%%{%%%{1,[0,1]%%%},[0,0,2]%%% 
} / %%%{%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\int \frac {1}{\sqrt {x}\, \sqrt {d x +c}\, a \,x^{2}+\sqrt {x}\, \sqrt {d x +c}\, b \,x^{4}}d x}{\sqrt {e}\, e^{2}} \] Input:

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

Output:

int(1/(sqrt(x)*sqrt(c + d*x)*a*x**2 + sqrt(x)*sqrt(c + d*x)*b*x**4),x)/(sq 
rt(e)*e**2)