\(\int \frac {(e x)^{3/2} (c+d x)^{3/2}}{a+b x^2} \, dx\) [860]

Optimal result

Integrand size = 26, antiderivative size = 323 \[ \int \frac {(e x)^{3/2} (c+d x)^{3/2}}{a+b x^2} \, dx=\frac {5 c e \sqrt {e x} \sqrt {c+d x}}{4 b}+\frac {d (e x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \left (\sqrt [4]{-a} \sqrt {b} c-(-a)^{3/4} d\right ) e^{3/2} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{b^2}+\frac {\left (3 b c^2-8 a d^2\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{4 b^2 \sqrt {d}}-\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \left (\sqrt [4]{-a} \sqrt {b} c+(-a)^{3/4} d\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{b^2} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.69 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.98 \[ \int \frac {(e x)^{3/2} (c+d x)^{3/2}}{a+b x^2} \, dx=\frac {(e x)^{3/2} \left (16 a d^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )+b \left (5 c \sqrt {d} \sqrt {x} \sqrt {c+d x}+2 d^{3/2} x^{3/2} \sqrt {c+d x}+6 c^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )\right )-a \sqrt {d} \text {RootSum}\left [a d^4-4 a d^3 \text {$\#$1}^2+16 b c^2 \text {$\#$1}^4+6 a d^2 \text {$\#$1}^4-4 a d \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-b c^2 d^3 \log (x)+a d^5 \log (x)+2 b c^2 d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )-2 a d^5 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right )-5 b c^2 d^2 \log (x) \text {$\#$1}^2-3 a d^4 \log (x) \text {$\#$1}^2+10 b c^2 d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+6 a d^4 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^2+5 b c^2 d \log (x) \text {$\#$1}^4+3 a d^3 \log (x) \text {$\#$1}^4-10 b c^2 d \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-6 a d^3 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+b c^2 \log (x) \text {$\#$1}^6-a d^2 \log (x) \text {$\#$1}^6-2 b c^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6+2 a d^2 \log \left (-\sqrt {c}+\sqrt {c+d x}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^6}{-a d^3 \text {$\#$1}+8 b c^2 \text {$\#$1}^3+3 a d^2 \text {$\#$1}^3-3 a d \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]\right )}{4 b^2 \sqrt {d} x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*(16*a*d^2*ArcTanh[(Sqrt[d]*Sqrt[x])/(Sqrt[c] - Sqrt[c + d*x]) 
] + b*(5*c*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] + 2*d^(3/2)*x^(3/2)*Sqrt[c + d*x] 
 + 6*c^2*ArcTanh[(Sqrt[d]*Sqrt[x])/(-Sqrt[c] + Sqrt[c + d*x])]) - a*Sqrt[d 
]*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 & , (-(b*c^2*d^3*Log[x]) + a*d^5*Log[x] + 2*b*c^2*d^3*Log[-Sqrt[ 
c] + Sqrt[c + d*x] - Sqrt[x]*#1] - 2*a*d^5*Log[-Sqrt[c] + Sqrt[c + d*x] - 
Sqrt[x]*#1] - 5*b*c^2*d^2*Log[x]*#1^2 - 3*a*d^4*Log[x]*#1^2 + 10*b*c^2*d^2 
*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 + 6*a*d^4*Log[-Sqrt[c] + 
Sqrt[c + d*x] - Sqrt[x]*#1]*#1^2 + 5*b*c^2*d*Log[x]*#1^4 + 3*a*d^3*Log[x]* 
#1^4 - 10*b*c^2*d*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 - 6*a*d^ 
3*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^4 + b*c^2*Log[x]*#1^6 - a* 
d^2*Log[x]*#1^6 - 2*b*c^2*Log[-Sqrt[c] + Sqrt[c + d*x] - Sqrt[x]*#1]*#1^6 
+ 2*a*d^2*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) & ]))/(4*b^2*Sqrt[d]*x^ 
(3/2))
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.49, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {607, 90, 60, 65, 221, 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.

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

\(\Big \downarrow \) 607

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {d e \left (\frac {5}{4} c \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )}{b}-\frac {e \int \frac {\sqrt {e x} \left (2 a c d-\left (b c^2-a d^2\right ) x\right )}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {d e \left (\frac {5}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )}{b}-\frac {e \int \frac {\sqrt {e x} \left (2 a c d-\left (b c^2-a d^2\right ) x\right )}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{b}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {d e \left (\frac {5}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )}{b}-\frac {e \int \frac {\sqrt {e x} \left (2 a c d-\left (b c^2-a d^2\right ) x\right )}{\sqrt {c+d x} \left (b x^2+a\right )}dx}{b}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2353

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 607

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S 
ymbol] :> Simp[e*(d/b)   Int[(e*x)^(m - 1)*(c + d*x)^(n - 2)*(2*c + d*x), x 
], x] - Simp[e/b   Int[(e*x)^(m - 1)*(c + d*x)^(n - 2)*(Simp[2*a*c*d - (b*c 
^2 - a*d^2)*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[ 
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. \(539\) vs. \(2(241)=482\).

Time = 0.48 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.67

method	result	size
risch	\(\frac {\left (2 d x +5 c \right ) x \sqrt {d x +c}\, e^{2}}{4 b \sqrt {e x}}-\frac {\left (\frac {\left (8 a \,d^{2}-3 b \,c^{2}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{b \sqrt {d e}}-\frac {4 a \left (-a \,d^{2}+b \,c^{2}+2 c d \sqrt {-a b}\right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}-\frac {4 a \left (a \,d^{2}-b \,c^{2}+2 c d \sqrt {-a b}\right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\right ) e^{2} \sqrt {\left (d x +c \right ) e x}}{8 b \sqrt {e x}\, \sqrt {d x +c}}\)	\(540\)
default	\(\text {Expression too large to display}\)	\(1041\)

Input:

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

Output:

1/4*(2*d*x+5*c)*x*(d*x+c)^(1/2)/b*e^2/(e*x)^(1/2)-1/8/b*(1/b*(8*a*d^2-3*b* 
c^2)*ln((1/2*c*e+d*e*x)/(d*e)^(1/2)+(d*e*x^2+c*e*x)^(1/2))/(d*e)^(1/2)-4*a 
*(-a*d^2+b*c^2+2*c*d*(-a*b)^(1/2))/(-a*b)^(1/2)/b/(-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))-4*a*(a*d^2-b*c^2+2*c*d*(-a*b)^(1/2))/(-a*b)^(1/ 
2)/b/(-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. 1220 vs. \(2 (241) = 482\).

Time = 1.13 (sec) , antiderivative size = 2439, normalized size of antiderivative = 7.55 \[ \int \frac {(e x)^{3/2} (c+d x)^{3/2}}{a+b x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*((3*b*c^2 - 8*a*d^2)*e*sqrt(e/d)*log(2*d*e*x - 2*sqrt(d*x + c)*sqrt( 
e*x)*d*sqrt(e/d) + c*e) + 4*b^2*sqrt(-(b^4*sqrt(-(a*b^2*c^6 - 6*a^2*b*c^4* 
d^2 + 9*a^3*c^2*d^4)*e^6/b^7) + (3*a*b*c^2*d - a^2*d^3)*e^3)/b^4)*log(-((a 
*b^2*c^4 - 2*a^2*b*c^2*d^2 - 3*a^3*d^4)*sqrt(d*x + c)*sqrt(e*x)*e^4 + (b^6 
*x*sqrt(-(a*b^2*c^6 - 6*a^2*b*c^4*d^2 + 9*a^3*c^2*d^4)*e^6/b^7) - (a*b^3*c 
^2*d - 3*a^2*b^2*d^3)*e^3*x)*sqrt(-(b^4*sqrt(-(a*b^2*c^6 - 6*a^2*b*c^4*d^2 
 + 9*a^3*c^2*d^4)*e^6/b^7) + (3*a*b*c^2*d - a^2*d^3)*e^3)/b^4))/x) - 4*b^2 
*sqrt(-(b^4*sqrt(-(a*b^2*c^6 - 6*a^2*b*c^4*d^2 + 9*a^3*c^2*d^4)*e^6/b^7) + 
 (3*a*b*c^2*d - a^2*d^3)*e^3)/b^4)*log(-((a*b^2*c^4 - 2*a^2*b*c^2*d^2 - 3* 
a^3*d^4)*sqrt(d*x + c)*sqrt(e*x)*e^4 - (b^6*x*sqrt(-(a*b^2*c^6 - 6*a^2*b*c 
^4*d^2 + 9*a^3*c^2*d^4)*e^6/b^7) - (a*b^3*c^2*d - 3*a^2*b^2*d^3)*e^3*x)*sq 
rt(-(b^4*sqrt(-(a*b^2*c^6 - 6*a^2*b*c^4*d^2 + 9*a^3*c^2*d^4)*e^6/b^7) + (3 
*a*b*c^2*d - a^2*d^3)*e^3)/b^4))/x) - 4*b^2*sqrt((b^4*sqrt(-(a*b^2*c^6 - 6 
*a^2*b*c^4*d^2 + 9*a^3*c^2*d^4)*e^6/b^7) - (3*a*b*c^2*d - a^2*d^3)*e^3)/b^ 
4)*log(-((a*b^2*c^4 - 2*a^2*b*c^2*d^2 - 3*a^3*d^4)*sqrt(d*x + c)*sqrt(e*x) 
*e^4 + (b^6*x*sqrt(-(a*b^2*c^6 - 6*a^2*b*c^4*d^2 + 9*a^3*c^2*d^4)*e^6/b^7) 
 + (a*b^3*c^2*d - 3*a^2*b^2*d^3)*e^3*x)*sqrt((b^4*sqrt(-(a*b^2*c^6 - 6*a^2 
*b*c^4*d^2 + 9*a^3*c^2*d^4)*e^6/b^7) - (3*a*b*c^2*d - a^2*d^3)*e^3)/b^4))/ 
x) + 4*b^2*sqrt((b^4*sqrt(-(a*b^2*c^6 - 6*a^2*b*c^4*d^2 + 9*a^3*c^2*d^4)*e 
^6/b^7) - (3*a*b*c^2*d - a^2*d^3)*e^3)/b^4)*log(-((a*b^2*c^4 - 2*a^2*b*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(e x)^{3/2} (c+d x)^{3/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(3/2)/(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:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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