\(\int \frac {1}{(e x)^{3/2} \sqrt {c+d x} (a+b x^2)} \, dx\) [879]

Optimal result

Integrand size = 26, antiderivative size = 211 \[ \int \frac {1}{(e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {2 \sqrt {c+d x}}{a c e \sqrt {e x}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} e^{3/2}}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{5/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} e^{3/2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {x \left (-2 \sqrt {c+d x}-b c \sqrt {d} \sqrt {x} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-2 c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{a c (e x)^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. \(419\) vs. \(2(155)=310\).

Time = 0.49 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.99

method	result	size
risch	\(-\frac {2 \sqrt {d x +c}}{a c e \sqrt {e x}}+\frac {\left (\frac {\ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}+\frac {\ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{e \sqrt {e x}\, \sqrt {d x +c}}\)	\(420\)
default	\(\frac {\sqrt {d x +c}\, b \left (\ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) a c \,d^{2} e x \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}+\ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) b \,c^{3} e x \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}+\ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) a c \,d^{2} e x \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}+\ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) b \,c^{3} e x \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}-4 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a \,d^{2}-4 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b \,c^{2}\right )}{2 e c \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a \left (\sqrt {-a b}\, d +b c \right ) \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \left (b c -\sqrt {-a b}\, d \right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\)	\(625\)

Input:

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

Output:

-2*(d*x+c)^(1/2)/a/c/e/(e*x)^(1/2)+(1/2/a/(-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/a/(-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*((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. 1231 vs. \(2 (155) = 310\).

Time = 0.14 (sec) , antiderivative size = 1231, normalized size of antiderivative = 5.83 \[ \int \frac {1}{(e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(a*c*e^2*x*sqrt(-((a^2*b*c^2 + a^3*d^2)*e^3*sqrt(-b^3*c^2/((a^5*b^2*c^ 
4 + 2*a^6*b*c^2*d^2 + a^7*d^4)*e^6)) + b*d)/((a^2*b*c^2 + a^3*d^2)*e^3))*l 
og((sqrt(d*x + c)*sqrt(e*x)*b^2*c + (a*b^2*c^2*e^2*x + (a^4*b*c^2*d + a^5* 
d^3)*e^5*x*sqrt(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^2*d^2 + a^7*d^4)*e^6))) 
*sqrt(-((a^2*b*c^2 + a^3*d^2)*e^3*sqrt(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^ 
2*d^2 + a^7*d^4)*e^6)) + b*d)/((a^2*b*c^2 + a^3*d^2)*e^3)))/x) - a*c*e^2*x 
*sqrt(-((a^2*b*c^2 + a^3*d^2)*e^3*sqrt(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^ 
2*d^2 + a^7*d^4)*e^6)) + b*d)/((a^2*b*c^2 + a^3*d^2)*e^3))*log((sqrt(d*x + 
 c)*sqrt(e*x)*b^2*c - (a*b^2*c^2*e^2*x + (a^4*b*c^2*d + a^5*d^3)*e^5*x*sqr 
t(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^2*d^2 + a^7*d^4)*e^6)))*sqrt(-((a^2*b 
*c^2 + a^3*d^2)*e^3*sqrt(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^2*d^2 + a^7*d^ 
4)*e^6)) + b*d)/((a^2*b*c^2 + a^3*d^2)*e^3)))/x) + a*c*e^2*x*sqrt(((a^2*b* 
c^2 + a^3*d^2)*e^3*sqrt(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^2*d^2 + a^7*d^4 
)*e^6)) - b*d)/((a^2*b*c^2 + a^3*d^2)*e^3))*log((sqrt(d*x + c)*sqrt(e*x)*b 
^2*c + (a*b^2*c^2*e^2*x - (a^4*b*c^2*d + a^5*d^3)*e^5*x*sqrt(-b^3*c^2/((a^ 
5*b^2*c^4 + 2*a^6*b*c^2*d^2 + a^7*d^4)*e^6)))*sqrt(((a^2*b*c^2 + a^3*d^2)* 
e^3*sqrt(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^2*d^2 + a^7*d^4)*e^6)) - b*d)/ 
((a^2*b*c^2 + a^3*d^2)*e^3)))/x) - a*c*e^2*x*sqrt(((a^2*b*c^2 + a^3*d^2)*e 
^3*sqrt(-b^3*c^2/((a^5*b^2*c^4 + 2*a^6*b*c^2*d^2 + a^7*d^4)*e^6)) - b*d)/( 
(a^2*b*c^2 + a^3*d^2)*e^3))*log((sqrt(d*x + c)*sqrt(e*x)*b^2*c - (a*b^2...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/(e*x)^(3/2)/(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)^(3/2)), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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