\(\int \frac {(d+e x^2)^{9/2}}{(a d+(b d+a e) x^2+b e x^4)^{3/2}} \, dx\) [8]

Optimal result

Integrand size = 37, antiderivative size = 237 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {(b d-a e)^3 x \sqrt {d+e x^2}}{a b^3 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {e^2 (12 b d-7 a e) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{8 b^3 \sqrt {d+e x^2}}+\frac {e^3 x^3 \sqrt {a d+(b d+a e) x^2+b e x^4}}{4 b^2 \sqrt {d+e x^2}}+\frac {3 e \left (8 b^2 d^2-12 a b d e+5 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{8 b^{7/2}} \] Output:

(-a*e+b*d)^3*x*(e*x^2+d)^(1/2)/a/b^3/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)+1/8 
*e^2*(-7*a*e+12*b*d)*x*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/b^3/(e*x^2+d)^(1/ 
2)+1/4*e^3*x^3*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/b^2/(e*x^2+d)^(1/2)+3/8*e 
*(5*a^2*e^2-12*a*b*d*e+8*b^2*d^2)*arctanh(b^(1/2)*x*(e*x^2+d)^(1/2)/(a*d+( 
a*e+b*d)*x^2+b*e*x^4)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {\sqrt {d+e x^2} \left (\sqrt {b} x \left (8 b^3 d^3-15 a^3 e^3+a^2 b e^2 \left (36 d-5 e x^2\right )+2 a b^2 e \left (-12 d^2+6 d e x^2+e^2 x^4\right )\right )-3 a e \left (8 b^2 d^2-12 a b d e+5 a^2 e^2\right ) \sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{8 a b^{7/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[(d + e*x^2)^(9/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(3/2),x]
 

Output:

(Sqrt[d + e*x^2]*(Sqrt[b]*x*(8*b^3*d^3 - 15*a^3*e^3 + a^2*b*e^2*(36*d - 5* 
e*x^2) + 2*a*b^2*e*(-12*d^2 + 6*d*e*x^2 + e^2*x^4)) - 3*a*e*(8*b^2*d^2 - 1 
2*a*b*d*e + 5*a^2*e^2)*Sqrt[a + b*x^2]*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]] 
))/(8*a*b^(7/2)*Sqrt[(a + b*x^2)*(d + e*x^2)])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1395, 315, 27, 403, 299, 224, 219}

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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{9/2}}{\left (x^2 (a e+b d)+a d+b e x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {e \left (\frac {\int \frac {a d (8 b d-5 a e)-(2 b d-5 a e) (4 b d-3 a e) x^2}{\sqrt {b x^2+a}}dx}{4 b}-\frac {x \sqrt {a+b x^2} \left (d+e x^2\right ) (4 b d-5 a e)}{4 b}\right )}{a b}+\frac {x \left (d+e x^2\right )^2 (b d-a e)}{a b \sqrt {a+b x^2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {e \left (\frac {\frac {3 a \left (5 a^2 e^2-12 a b d e+8 b^2 d^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}-\frac {x \sqrt {a+b x^2} (2 b d-5 a e) (4 b d-3 a e)}{2 b}}{4 b}-\frac {x \sqrt {a+b x^2} \left (d+e x^2\right ) (4 b d-5 a e)}{4 b}\right )}{a b}+\frac {x \left (d+e x^2\right )^2 (b d-a e)}{a b \sqrt {a+b x^2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {e \left (\frac {\frac {3 a \left (5 a^2 e^2-12 a b d e+8 b^2 d^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}-\frac {x \sqrt {a+b x^2} (2 b d-5 a e) (4 b d-3 a e)}{2 b}}{4 b}-\frac {x \sqrt {a+b x^2} \left (d+e x^2\right ) (4 b d-5 a e)}{4 b}\right )}{a b}+\frac {x \left (d+e x^2\right )^2 (b d-a e)}{a b \sqrt {a+b x^2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {e \left (\frac {\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (5 a^2 e^2-12 a b d e+8 b^2 d^2\right )}{2 b^{3/2}}-\frac {x \sqrt {a+b x^2} (2 b d-5 a e) (4 b d-3 a e)}{2 b}}{4 b}-\frac {x \sqrt {a+b x^2} \left (d+e x^2\right ) (4 b d-5 a e)}{4 b}\right )}{a b}+\frac {x \left (d+e x^2\right )^2 (b d-a e)}{a b \sqrt {a+b x^2}}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[(d + e*x^2)^(9/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(3/2),x]
 

Output:

(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*(((b*d - a*e)*x*(d + e*x^2)^2)/(a*b*Sqrt[ 
a + b*x^2]) + (e*(-1/4*((4*b*d - 5*a*e)*x*Sqrt[a + b*x^2]*(d + e*x^2))/b + 
 (-1/2*((2*b*d - 5*a*e)*(4*b*d - 3*a*e)*x*Sqrt[a + b*x^2])/b + (3*a*(8*b^2 
*d^2 - 12*a*b*d*e + 5*a^2*e^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^ 
(3/2)))/(4*b)))/(a*b)))/Sqrt[a*d + (b*d + a*e)*x^2 + b*e*x^4]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 403

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

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {x \,e^{2} \left (-2 b e \,x^{2}+7 a e -12 b d \right ) \left (b \,x^{2}+a \right ) \sqrt {e \,x^{2}+d}}{8 b^{3} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}+\frac {\left (3 b e \left (5 a^{2} e^{2}-12 a b d e +8 b^{2} d^{2}\right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {7 a^{2} e^{3} x}{\sqrt {b \,x^{2}+a}}+\frac {8 b^{3} d^{3} x}{a \sqrt {b \,x^{2}+a}}-\frac {12 a b d \,e^{2} x}{\sqrt {b \,x^{2}+a}}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {e \,x^{2}+d}}{8 b^{3} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}\)	\(221\)
default	\(\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \left (2 a \,b^{\frac {5}{2}} e^{3} x^{5}-5 a^{2} b^{\frac {3}{2}} e^{3} x^{3}+12 a \,b^{\frac {5}{2}} d \,e^{2} x^{3}+15 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{3} e^{3} \sqrt {b \,x^{2}+a}-36 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} b d \,e^{2} \sqrt {b \,x^{2}+a}+24 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a \,b^{2} d^{2} e \sqrt {b \,x^{2}+a}-15 a^{3} e^{3} x \sqrt {b}+36 a^{2} b^{\frac {3}{2}} d \,e^{2} x -24 a \,b^{\frac {5}{2}} d^{2} e x +8 b^{\frac {7}{2}} d^{3} x \right )}{8 b^{\frac {7}{2}} \sqrt {e \,x^{2}+d}\, \left (b \,x^{2}+a \right ) a}\)	\(232\)

Input:

int((e*x^2+d)^(9/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x,method=_RETURNVERB 
OSE)
 

Output:

-1/8*x*e^2*(-2*b*e*x^2+7*a*e-12*b*d)*(b*x^2+a)/b^3/((e*x^2+d)*(b*x^2+a))^( 
1/2)*(e*x^2+d)^(1/2)+1/8/b^3*(3*b*e*(5*a^2*e^2-12*a*b*d*e+8*b^2*d^2)*(-x/b 
/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))+7*a^2*e^3*x/(b*x 
^2+a)^(1/2)+8*b^3*d^3*x/a/(b*x^2+a)^(1/2)-12*a*b*d*e^2*x/(b*x^2+a)^(1/2))* 
(b*x^2+a)^(1/2)/((e*x^2+d)*(b*x^2+a))^(1/2)*(e*x^2+d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 677, normalized size of antiderivative = 2.86 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (8 \, a^{2} b^{2} d^{3} e - 12 \, a^{3} b d^{2} e^{2} + 5 \, a^{4} d e^{3} + {\left (8 \, a b^{3} d^{2} e^{2} - 12 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x^{4} + {\left (8 \, a b^{3} d^{3} e - 4 \, a^{2} b^{2} d^{2} e^{2} - 7 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (\frac {2 \, b e x^{4} + {\left (2 \, b d + a e\right )} x^{2} + 2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d} \sqrt {b} x + a d}{e x^{2} + d}\right ) + 2 \, {\left (2 \, a b^{3} e^{3} x^{5} + {\left (12 \, a b^{3} d e^{2} - 5 \, a^{2} b^{2} e^{3}\right )} x^{3} + {\left (8 \, b^{4} d^{3} - 24 \, a b^{3} d^{2} e + 36 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3}\right )} x\right )} \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d}}{16 \, {\left (a b^{5} e x^{4} + a^{2} b^{4} d + {\left (a b^{5} d + a^{2} b^{4} e\right )} x^{2}\right )}}, -\frac {3 \, {\left (8 \, a^{2} b^{2} d^{3} e - 12 \, a^{3} b d^{2} e^{2} + 5 \, a^{4} d e^{3} + {\left (8 \, a b^{3} d^{2} e^{2} - 12 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x^{4} + {\left (8 \, a b^{3} d^{3} e - 4 \, a^{2} b^{2} d^{2} e^{2} - 7 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {e x^{2} + d} \sqrt {-b} x}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d}}\right ) - {\left (2 \, a b^{3} e^{3} x^{5} + {\left (12 \, a b^{3} d e^{2} - 5 \, a^{2} b^{2} e^{3}\right )} x^{3} + {\left (8 \, b^{4} d^{3} - 24 \, a b^{3} d^{2} e + 36 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3}\right )} x\right )} \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d}}{8 \, {\left (a b^{5} e x^{4} + a^{2} b^{4} d + {\left (a b^{5} d + a^{2} b^{4} e\right )} x^{2}\right )}}\right ] \] Input:

integrate((e*x^2+d)^(9/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
fricas")
 

Output:

[1/16*(3*(8*a^2*b^2*d^3*e - 12*a^3*b*d^2*e^2 + 5*a^4*d*e^3 + (8*a*b^3*d^2* 
e^2 - 12*a^2*b^2*d*e^3 + 5*a^3*b*e^4)*x^4 + (8*a*b^3*d^3*e - 4*a^2*b^2*d^2 
*e^2 - 7*a^3*b*d*e^3 + 5*a^4*e^4)*x^2)*sqrt(b)*log((2*b*e*x^4 + (2*b*d + a 
*e)*x^2 + 2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d)*sqrt(b)* 
x + a*d)/(e*x^2 + d)) + 2*(2*a*b^3*e^3*x^5 + (12*a*b^3*d*e^2 - 5*a^2*b^2*e 
^3)*x^3 + (8*b^4*d^3 - 24*a*b^3*d^2*e + 36*a^2*b^2*d*e^2 - 15*a^3*b*e^3)*x 
)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d))/(a*b^5*e*x^4 + a^ 
2*b^4*d + (a*b^5*d + a^2*b^4*e)*x^2), -1/8*(3*(8*a^2*b^2*d^3*e - 12*a^3*b* 
d^2*e^2 + 5*a^4*d*e^3 + (8*a*b^3*d^2*e^2 - 12*a^2*b^2*d*e^3 + 5*a^3*b*e^4) 
*x^4 + (8*a*b^3*d^3*e - 4*a^2*b^2*d^2*e^2 - 7*a^3*b*d*e^3 + 5*a^4*e^4)*x^2 
)*sqrt(-b)*arctan(sqrt(e*x^2 + d)*sqrt(-b)*x/sqrt(b*e*x^4 + (b*d + a*e)*x^ 
2 + a*d)) - (2*a*b^3*e^3*x^5 + (12*a*b^3*d*e^2 - 5*a^2*b^2*e^3)*x^3 + (8*b 
^4*d^3 - 24*a*b^3*d^2*e + 36*a^2*b^2*d*e^2 - 15*a^3*b*e^3)*x)*sqrt(b*e*x^4 
 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d))/(a*b^5*e*x^4 + a^2*b^4*d + (a*b 
^5*d + a^2*b^4*e)*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(9/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(9/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
maxima")
 

Output:

integrate((e*x^2 + d)^(9/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, e^{3} x^{2}}{b} + \frac {12 \, a b^{4} d e^{2} - 5 \, a^{2} b^{3} e^{3}}{a b^{5}}\right )} x^{2} + \frac {8 \, b^{5} d^{3} - 24 \, a b^{4} d^{2} e + 36 \, a^{2} b^{3} d e^{2} - 15 \, a^{3} b^{2} e^{3}}{a b^{5}}\right )} x}{8 \, \sqrt {b x^{2} + a}} - \frac {3 \, {\left (8 \, b^{2} d^{2} e - 12 \, a b d e^{2} + 5 \, a^{2} e^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \] Input:

integrate((e*x^2+d)^(9/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
giac")
 

Output:

1/8*((2*e^3*x^2/b + (12*a*b^4*d*e^2 - 5*a^2*b^3*e^3)/(a*b^5))*x^2 + (8*b^5 
*d^3 - 24*a*b^4*d^2*e + 36*a^2*b^3*d*e^2 - 15*a^3*b^2*e^3)/(a*b^5))*x/sqrt 
(b*x^2 + a) - 3/8*(8*b^2*d^2*e - 12*a*b*d*e^2 + 5*a^2*e^3)*log(abs(-sqrt(b 
)*x + sqrt(b*x^2 + a)))/b^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{9/2}}{{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{3/2}} \,d x \] Input:

int((d + e*x^2)^(9/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2),x)
 

Output:

int((d + e*x^2)^(9/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.93 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {-15 \sqrt {b \,x^{2}+a}\, a^{3} b \,e^{3} x +36 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,e^{2} x -5 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} e^{3} x^{3}-24 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{2} e x +12 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,e^{2} x^{3}+2 \sqrt {b \,x^{2}+a}\, a \,b^{3} e^{3} x^{5}+8 \sqrt {b \,x^{2}+a}\, b^{4} d^{3} x +15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} e^{3}-36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d \,e^{2}+15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b \,e^{3} x^{2}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d^{2} e -36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d \,e^{2} x^{2}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d^{2} e \,x^{2}-10 \sqrt {b}\, a^{4} e^{3}+27 \sqrt {b}\, a^{3} b d \,e^{2}-10 \sqrt {b}\, a^{3} b \,e^{3} x^{2}-24 \sqrt {b}\, a^{2} b^{2} d^{2} e +27 \sqrt {b}\, a^{2} b^{2} d \,e^{2} x^{2}+8 \sqrt {b}\, a \,b^{3} d^{3}-24 \sqrt {b}\, a \,b^{3} d^{2} e \,x^{2}+8 \sqrt {b}\, b^{4} d^{3} x^{2}}{8 a \,b^{4} \left (b \,x^{2}+a \right )} \] Input:

int((e*x^2+d)^(9/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x)
 

Output:

( - 15*sqrt(a + b*x**2)*a**3*b*e**3*x + 36*sqrt(a + b*x**2)*a**2*b**2*d*e* 
*2*x - 5*sqrt(a + b*x**2)*a**2*b**2*e**3*x**3 - 24*sqrt(a + b*x**2)*a*b**3 
*d**2*e*x + 12*sqrt(a + b*x**2)*a*b**3*d*e**2*x**3 + 2*sqrt(a + b*x**2)*a* 
b**3*e**3*x**5 + 8*sqrt(a + b*x**2)*b**4*d**3*x + 15*sqrt(b)*log((sqrt(a + 
 b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*e**3 - 36*sqrt(b)*log((sqrt(a + b*x**2 
) + sqrt(b)*x)/sqrt(a))*a**3*b*d*e**2 + 15*sqrt(b)*log((sqrt(a + b*x**2) + 
 sqrt(b)*x)/sqrt(a))*a**3*b*e**3*x**2 + 24*sqrt(b)*log((sqrt(a + b*x**2) + 
 sqrt(b)*x)/sqrt(a))*a**2*b**2*d**2*e - 36*sqrt(b)*log((sqrt(a + b*x**2) + 
 sqrt(b)*x)/sqrt(a))*a**2*b**2*d*e**2*x**2 + 24*sqrt(b)*log((sqrt(a + b*x* 
*2) + sqrt(b)*x)/sqrt(a))*a*b**3*d**2*e*x**2 - 10*sqrt(b)*a**4*e**3 + 27*s 
qrt(b)*a**3*b*d*e**2 - 10*sqrt(b)*a**3*b*e**3*x**2 - 24*sqrt(b)*a**2*b**2* 
d**2*e + 27*sqrt(b)*a**2*b**2*d*e**2*x**2 + 8*sqrt(b)*a*b**3*d**3 - 24*sqr 
t(b)*a*b**3*d**2*e*x**2 + 8*sqrt(b)*b**4*d**3*x**2)/(8*a*b**4*(a + b*x**2) 
)