\(\int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} (a+b x^2)} \, dx\) [887]

Optimal result

Integrand size = 26, antiderivative size = 302 \[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2}{3 a c e (e x)^{3/2} \sqrt {c+d x}}+\frac {8 d}{3 a c^2 e^2 \sqrt {e x} \sqrt {c+d x}}+\frac {2 d^2 \left (5 b c^2+8 a d^2\right ) \sqrt {e x}}{3 a c^3 \left (b c^2+a d^2\right ) e^3 \sqrt {c+d x}}-\frac {b^{3/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2} e^{5/2}}-\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2} e^{5/2}} \] Output:

-2/3/a/c/e/(e*x)^(3/2)/(d*x+c)^(1/2)+8/3*d/a/c^2/e^2/(e*x)^(1/2)/(d*x+c)^( 
1/2)+2/3*d^2*(8*a*d^2+5*b*c^2)*(e*x)^(1/2)/a/c^3/(a*d^2+b*c^2)/e^3/(d*x+c) 
^(1/2)-b^(3/2)*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)^(7/4)/(b^(1/2)*c-(-a)^(1/2)*d)^(3/2)/e^(5/2) 
-b^(3/2)*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)^(7/4)/(b^(1/2)*c+(-a)^(1/2)*d)^(3/2)/e^(5/2)
 

Mathematica [C] (verified)

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

Time = 0.40 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-2 b c^2 \left (c^2-4 c d x-5 d^2 x^2\right )+2 a d^2 \left (-c^2+4 c d x+8 d^2 x^2\right )+3 b^2 c^3 d^{3/2} x^{3/2} \sqrt {c+d x} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-6 c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{3 a c^3 \left (b c^2+a d^2\right ) (e x)^{5/2} \sqrt {c+d x}} \] Input:

Integrate[1/((e*x)^(5/2)*(c + d*x)^(3/2)*(a + b*x^2)),x]
 

Output:

(x*(-2*b*c^2*(c^2 - 4*c*d*x - 5*d^2*x^2) + 2*a*d^2*(-c^2 + 4*c*d*x + 8*d^2 
*x^2) + 3*b^2*c^3*d^(3/2)*x^(3/2)*Sqrt[c + d*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] - 6*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 
) & ]))/(3*a*c^3*(b*c^2 + a*d^2)*(e*x)^(5/2)*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.58, 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.

\(\displaystyle \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right ) (c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {\sqrt {-a}}{2 a (e x)^{5/2} \left (\sqrt {-a}-\sqrt {b} x\right ) (c+d x)^{3/2}}+\frac {\sqrt {-a}}{2 a (e x)^{5/2} \left (\sqrt {-a}+\sqrt {b} x\right ) (c+d x)^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {-a} \sqrt {b} c-4 a d}{3 a^2 c^2 e^2 \sqrt {e x} \sqrt {c+d x}}+\frac {3 \sqrt {-a} \sqrt {b} c+4 a d}{3 a^2 c^2 e^2 \sqrt {e x} \sqrt {c+d x}}-\frac {b^{3/2} \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} e^{5/2} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2}}-\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{7/4} e^{5/2} \left (\sqrt {-a} d+\sqrt {b} c\right )^{3/2}}+\frac {d \sqrt {e x} \left (-2 \sqrt {-a} \sqrt {b} c d+8 a d^2+3 b c^2\right )}{3 (-a)^{3/2} c^3 e^3 \sqrt {c+d x} \left (\sqrt {-a} d+\sqrt {b} c\right )}-\frac {d \sqrt {e x} \left (2 \sqrt {-a} \sqrt {b} c d+8 a d^2+3 b c^2\right )}{3 (-a)^{3/2} c^3 e^3 \sqrt {c+d x} \left (\sqrt {b} c-\sqrt {-a} d\right )}-\frac {2}{3 a c e (e x)^{3/2} \sqrt {c+d x}}\)

Input:

Int[1/((e*x)^(5/2)*(c + d*x)^(3/2)*(a + b*x^2)),x]
 

Output:

-2/(3*a*c*e*(e*x)^(3/2)*Sqrt[c + d*x]) - (3*Sqrt[-a]*Sqrt[b]*c - 4*a*d)/(3 
*a^2*c^2*e^2*Sqrt[e*x]*Sqrt[c + d*x]) + (3*Sqrt[-a]*Sqrt[b]*c + 4*a*d)/(3* 
a^2*c^2*e^2*Sqrt[e*x]*Sqrt[c + d*x]) + (d*(3*b*c^2 - 2*Sqrt[-a]*Sqrt[b]*c* 
d + 8*a*d^2)*Sqrt[e*x])/(3*(-a)^(3/2)*c^3*(Sqrt[b]*c + Sqrt[-a]*d)*e^3*Sqr 
t[c + d*x]) - (d*(3*b*c^2 + 2*Sqrt[-a]*Sqrt[b]*c*d + 8*a*d^2)*Sqrt[e*x])/( 
3*(-a)^(3/2)*c^3*(Sqrt[b]*c - Sqrt[-a]*d)*e^3*Sqrt[c + d*x]) - (b^(3/2)*Ar 
cTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + 
 d*x])])/((-a)^(7/4)*(Sqrt[b]*c - Sqrt[-a]*d)^(3/2)*e^(5/2)) - (b^(3/2)*Ar 
cTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c 
+ d*x])])/((-a)^(7/4)*(Sqrt[b]*c + Sqrt[-a]*d)^(3/2)*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. \(551\) vs. \(2(232)=464\).

Time = 0.53 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.83

method	result	size
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-5 d x +c \right )}{3 c^{3} a x \,e^{2} \sqrt {e x}}+\frac {\left (\frac {b^{2} \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a \left (\sqrt {-a b}\, d +b c \right ) \sqrt {-a b}\, \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}+\frac {b^{2} \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a \left (\sqrt {-a b}\, d -b c \right ) \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}-\frac {2 b \,d^{3} \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{c^{3} \left (\sqrt {-a b}\, d +b c \right ) \left (\sqrt {-a b}\, d -b c \right ) e \left (x +\frac {c}{d}\right )}\right ) \sqrt {\left (d x +c \right ) e x}}{e^{2} \sqrt {e x}\, \sqrt {d x +c}}\)	\(552\)
default	\(\text {Expression too large to display}\)	\(2457\)

Input:

int(1/(e*x)^(5/2)/(d*x+c)^(3/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-2/3*(d*x+c)^(1/2)*(-5*d*x+c)/c^3/a/x/e^2/(e*x)^(1/2)+(1/2/a*b^2/((-a*b)^( 
1/2)*d+b*c)/(-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/a*b^2/((-a*b)^(1/2)*d-b*c)/(-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))-2/c^3*b*d^3/((-a*b)^(1/2)*d+b*c)/((-a*b)^(1/2)*d- 
b*c)/e/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2))/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. 3322 vs. \(2 (232) = 464\).

Time = 0.50 (sec) , antiderivative size = 3322, normalized size of antiderivative = 11.00 \[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \left (a + b x^{2}\right ) \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x)**(5/2)/(d*x+c)**(3/2)/(b*x**2+a),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/((e*x)**(5/2)*(a + b*x**2)*(c + d*x)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)*(d*x + c)^(3/2)*(e*x)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)^(5/2)/(d*x+c)^(3/2)/(b*x^2+a),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,\left (b\,x^2+a\right )\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int(1/((e*x)^(5/2)*(a + b*x^2)*(c + d*x)^(3/2)),x)
 

Output:

int(1/((e*x)^(5/2)*(a + b*x^2)*(c + d*x)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(e x)^{5/2} (c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {\int \frac {1}{\sqrt {x}\, \sqrt {d x +c}\, a c \,x^{2}+\sqrt {x}\, \sqrt {d x +c}\, a d \,x^{3}+\sqrt {x}\, \sqrt {d x +c}\, b c \,x^{4}+\sqrt {x}\, \sqrt {d x +c}\, b d \,x^{5}}d x}{\sqrt {e}\, e^{2}} \] Input:

int(1/(e*x)^(5/2)/(d*x+c)^(3/2)/(b*x^2+a),x)
 

Output:

int(1/(sqrt(x)*sqrt(c + d*x)*a*c*x**2 + sqrt(x)*sqrt(c + d*x)*a*d*x**3 + s 
qrt(x)*sqrt(c + d*x)*b*c*x**4 + sqrt(x)*sqrt(c + d*x)*b*d*x**5),x)/(sqrt(e 
)*e**2)