\(\int \frac {\sqrt {e x}}{(c+d x)^{3/2} (a+b x^2)} \, dx\) [884]

Optimal result

Integrand size = 26, antiderivative size = 206 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 d \sqrt {e x}}{\left (b c^2+a d^2\right ) \sqrt {c+d x}}+\frac {\sqrt {e} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} \left (\sqrt {b} c-\sqrt {-a} d\right )^{3/2}}-\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{\sqrt [4]{-a} \left (\sqrt {b} c+\sqrt {-a} d\right )^{3/2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.37 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {e x} \left (\frac {8 d}{\sqrt {c+d x}}-\frac {\text {RootSum}\left [a d^4-4 a d^3 \text {$\#$1}^2+16 b c^2 \text {$\#$1}^4+6 a d^2 \text {$\#$1}^4-4 a d \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {a d^4 \log (x)-2 a d^4 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )+4 b c^2 d \log (x) \text {$\#$1}^2-3 a d^3 \log (x) \text {$\#$1}^2-8 b c^2 d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+6 a d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2-4 b c^2 \log (x) \text {$\#$1}^4+3 a d^2 \log (x) \text {$\#$1}^4+8 b c^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-6 a d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-a d \log (x) \text {$\#$1}^6+2 a d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6}{a d^3 \text {$\#$1}-8 b c^2 \text {$\#$1}^3-3 a d^2 \text {$\#$1}^3+3 a d \text {$\#$1}^5-a \text {$\#$1}^7}\&\right ]}{\sqrt {x}}\right )}{4 \left (b c^2+a d^2\right )} \] Input:

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

Output:

-1/4*(Sqrt[e*x]*((8*d)/Sqrt[c + d*x] - RootSum[a*d^4 - 4*a*d^3*#1^2 + 16*b 
*c^2*#1^4 + 6*a*d^2*#1^4 - 4*a*d*#1^6 + a*#1^8 & , (a*d^4*Log[x] - 2*a*d^4 
*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1] + 4*b*c^2*d*Log[x]*#1^2 - 3*a* 
d^3*Log[x]*#1^2 - 8*b*c^2*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^ 
2 + 6*a*d^3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 - 4*b*c^2*Log[ 
x]*#1^4 + 3*a*d^2*Log[x]*#1^4 + 8*b*c^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqr 
t[x]*#1]*#1^4 - 6*a*d^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 - 
a*d*Log[x]*#1^6 + 2*a*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^6)/( 
a*d^3*#1 - 8*b*c^2*#1^3 - 3*a*d^2*#1^3 + 3*a*d*#1^5 - a*#1^7) & ]/Sqrt[x]) 
)/(b*c^2 + a*d^2)
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.25, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {612, 48, 2353, 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.

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

\(\Big \downarrow \) 612

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

\(\Big \downarrow \) 48

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

\(\Big \downarrow \) 2353

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*d*Sqrt[e*x])/((b*c^2 + a*d^2)*Sqrt[c + d*x]) + (e*(((Sqrt[b]*c + Sqrt[ 
-a]*d)*ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e] 
*Sqrt[c + d*x])])/((-a)^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e]) - ((Sq 
rt[b]*c - Sqrt[-a]*d)*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((- 
a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[-a]*d] 
*Sqrt[e])))/(b*c^2 + a*d^2)
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 612

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S 
ymbol] :> Simp[(-e)*c*(d/(b*c^2 + a*d^2))   Int[(e*x)^(m - 1)*(c + d*x)^n, 
x], x] + Simp[e/(b*c^2 + a*d^2)   Int[(e*x)^(m - 1)*(c + d*x)^(n + 1)*((a*d 
 + b*c*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[n, -1] & 
& GtQ[m, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2353

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1928\) vs. \(2(154)=308\).

Time = 0.49 (sec) , antiderivative size = 1929, normalized size of antiderivative = 9.36

method	result	size
default	\(\text {Expression too large to display}\)	\(1929\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2918 vs. \(2 (154) = 308\).

Time = 0.14 (sec) , antiderivative size = 2918, normalized size of antiderivative = 14.17 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*((b*c^3 + a*c*d^2 + (b*c^2*d + a*d^3)*x)*sqrt(-((3*b*c^2*d - a*d^3)*e 
 + (b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*sqrt(-(b^3*c^6 
- 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)*e^2/(a*b^6*c^12 + 6*a^2*b^5*c^10*d^2 
+ 15*a^3*b^4*c^8*d^4 + 20*a^4*b^3*c^6*d^6 + 15*a^5*b^2*c^4*d^8 + 6*a^6*b*c 
^2*d^10 + a^7*d^12)))/(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d 
^6))*log(-((b*c^2 - 3*a*d^2)*sqrt(d*x + c)*sqrt(e*x)*e + ((b^2*c^4 - 4*a*b 
*c^2*d^2 + 3*a^2*d^4)*e*x + 2*(a*b^3*c^6*d + 3*a^2*b^2*c^4*d^3 + 3*a^3*b*c 
^2*d^5 + a^4*d^7)*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)*e^2/ 
(a*b^6*c^12 + 6*a^2*b^5*c^10*d^2 + 15*a^3*b^4*c^8*d^4 + 20*a^4*b^3*c^6*d^6 
 + 15*a^5*b^2*c^4*d^8 + 6*a^6*b*c^2*d^10 + a^7*d^12))*x)*sqrt(-((3*b*c^2*d 
 - a*d^3)*e + (b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*sqrt 
(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)*e^2/(a*b^6*c^12 + 6*a^2*b^ 
5*c^10*d^2 + 15*a^3*b^4*c^8*d^4 + 20*a^4*b^3*c^6*d^6 + 15*a^5*b^2*c^4*d^8 
+ 6*a^6*b*c^2*d^10 + a^7*d^12)))/(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2* 
d^4 + a^3*d^6)))/x) - (b*c^3 + a*c*d^2 + (b*c^2*d + a*d^3)*x)*sqrt(-((3*b* 
c^2*d - a*d^3)*e + (b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6) 
*sqrt(-(b^3*c^6 - 6*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4)*e^2/(a*b^6*c^12 + 6*a 
^2*b^5*c^10*d^2 + 15*a^3*b^4*c^8*d^4 + 20*a^4*b^3*c^6*d^6 + 15*a^5*b^2*c^4 
*d^8 + 6*a^6*b*c^2*d^10 + a^7*d^12)))/(b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b 
*c^2*d^4 + a^3*d^6))*log(-((b*c^2 - 3*a*d^2)*sqrt(d*x + c)*sqrt(e*x)*e ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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