\(\int \frac {1}{\sqrt {e x} (c+d x)^{3/2} (a+b x^2)^2} \, dx\) [930]

Optimal result

Integrand size = 26, antiderivative size = 392 \[ \int \frac {1}{\sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=-\frac {d^2 \left (b c^2-2 a d^2\right ) \sqrt {e x}}{a c \left (b c^2+a d^2\right )^2 e \sqrt {c+d x}}+\frac {b \sqrt {e x} (c-d x)}{2 a \left (b c^2+a d^2\right ) e \sqrt {c+d x} \left (a+b x^2\right )}+\frac {3 \sqrt {b} \left (b^{3/2} c^3+3 a \sqrt {b} c d^2+2 \sqrt {-a} a d^3\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} \left (b c^2+a d^2\right )^2 \sqrt {e}}+\frac {3 \sqrt {b} \left (2 (-a)^{3/2} d^3+\sqrt {b} c \left (b c^2+3 a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{4 (-a)^{7/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} \left (b c^2+a d^2\right )^2 \sqrt {e}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.89 (sec) , antiderivative size = 712, normalized size of antiderivative = 1.82 \[ \int \frac {1}{\sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\frac {-x \left (-4 a^2 d^4+a b d^2 \left (c^2+c d x-4 d^2 x^2\right )+b^2 c^2 \left (-c^2+c d x+2 d^2 x^2\right )\right )-2 a c d^{7/2} \sqrt {x} \sqrt {c+d x} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {17 b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+16 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-6 b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]-c d^{3/2} \sqrt {x} \sqrt {c+d x} \left (a+b x^2\right ) \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {31 a b c^2 d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+32 a^2 d^4 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )+6 b^2 c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+12 a b c d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-a b d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b c^3+3 b c^2 \text {$\#$1}+8 a d^2 \text {$\#$1}-3 b c \text {$\#$1}^2+b \text {$\#$1}^3}\&\right ]}{2 a c \left (b c^2+a d^2\right )^2 \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \] Input:

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

Output:

(-(x*(-4*a^2*d^4 + a*b*d^2*(c^2 + c*d*x - 4*d^2*x^2) + b^2*c^2*(-c^2 + c*d 
*x + 2*d^2*x^2))) - 2*a*c*d^(7/2)*Sqrt[x]*Sqrt[c + d*x]*(a + b*x^2)*RootSu 
m[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 
& , (17*b*c^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 16*a 
*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] - 6*b*c*Log[c + 
 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + b*Log[c + 2*d*x - 2*Sq 
rt[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) & ] - c*d^(3/2)*Sqrt[x]*Sqrt[c + d*x]*(a + b*x^2)*Ro 
otSum[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 & , (31*a*b*c^2*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - 
#1] + 32*a^2*d^4*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1] + 6 
*b^2*c^3*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 + 12*a*b 
*c*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1 - a*b*d^2* 
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) & ])/(2*a*c*(b*c^2 + a*d^2)^2 
*Sqrt[e*x]*Sqrt[c + d*x]*(a + b*x^2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(795\) vs. \(2(392)=784\).

Time = 2.59 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.03, 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.

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

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b}{2 a \sqrt {e x} \left (-a b-b^2 x^2\right ) (c+d x)^{3/2}}-\frac {b}{4 a \sqrt {e x} \left (\sqrt {-a} \sqrt {b}-b x\right )^2 (c+d x)^{3/2}}-\frac {b}{4 a \sqrt {e x} \left (\sqrt {-a} \sqrt {b}+b x\right )^2 (c+d x)^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. \(7254\) vs. \(2(316)=632\).

Time = 0.58 (sec) , antiderivative size = 7255, normalized size of antiderivative = 18.51

method	result	size
default	\(\text {Expression too large to display}\)	\(7255\)

Input:

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

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6355 vs. \(2 (317) = 634\).

Time = 5.09 (sec) , antiderivative size = 6355, normalized size of antiderivative = 16.21 \[ \int \frac {1}{\sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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