\(\int \frac {(d+e x^2)^{3/2} (A+B x^2+C x^4)}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx\) [147]

Optimal result

Integrand size = 49, antiderivative size = 284 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\left (5 a^2 C e+8 b^2 (B d+A e)-6 a b (C d+B e)\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{16 b^3 \sqrt {d+e x^2}}-\frac {(5 a C e-6 b (C d+B e)) x^3 \sqrt {a d+(b d+a e) x^2+b e x^4}}{24 b^2 \sqrt {d+e x^2}}+\frac {C e x^5 \sqrt {a d+(b d+a e) x^2+b e x^4}}{6 b \sqrt {d+e x^2}}+\frac {\left (8 A b^2 (2 b d-a e)-a \left (8 b^2 B d+5 a^2 C e-6 a b (C d+B e)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{16 b^{7/2}} \] Output:

1/16*(5*a^2*C*e+8*b^2*(A*e+B*d)-6*a*b*(B*e+C*d))*x*(a*d+(a*e+b*d)*x^2+b*e* 
x^4)^(1/2)/b^3/(e*x^2+d)^(1/2)-1/24*(5*C*a*e-6*b*(B*e+C*d))*x^3*(a*d+(a*e+ 
b*d)*x^2+b*e*x^4)^(1/2)/b^2/(e*x^2+d)^(1/2)+1/6*C*e*x^5*(a*d+(a*e+b*d)*x^2 
+b*e*x^4)^(1/2)/b/(e*x^2+d)^(1/2)+1/16*(8*A*b^2*(-a*e+2*b*d)-a*(8*b^2*B*d+ 
5*a^2*C*e-6*a*b*(B*e+C*d)))*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 = 0.46 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\sqrt {d+e x^2} \left (\sqrt {b} x \left (a+b x^2\right ) \left (15 a^2 C e-2 a b \left (9 C d+9 B e+5 C e x^2\right )+4 b^2 \left (6 B d+6 A e+3 C d x^2+3 B e x^2+2 C e x^4\right )\right )-3 \left (8 A b^2 (2 b d-a e)+a \left (-8 b^2 B d-5 a^2 C e+6 a b (C d+B e)\right )\right ) \sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{48 b^{7/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/Sqrt[a*d + (b*d + a*e)*x 
^2 + b*e*x^4],x]
 

Output:

(Sqrt[d + e*x^2]*(Sqrt[b]*x*(a + b*x^2)*(15*a^2*C*e - 2*a*b*(9*C*d + 9*B*e 
 + 5*C*e*x^2) + 4*b^2*(6*B*d + 6*A*e + 3*C*d*x^2 + 3*B*e*x^2 + 2*C*e*x^4)) 
 - 3*(8*A*b^2*(2*b*d - a*e) + a*(-8*b^2*B*d - 5*a^2*C*e + 6*a*b*(C*d + B*e 
)))*Sqrt[a + b*x^2]*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]))/(48*b^(7/2)*Sqrt 
[(a + b*x^2)*(d + e*x^2)])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1395, 2256, 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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 2256

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \int \left (\frac {C e x^6}{\sqrt {b x^2+a}}+\frac {(C d+B e) x^4}{\sqrt {b x^2+a}}+\frac {(B d+A e) x^2}{\sqrt {b x^2+a}}+\frac {A d}{\sqrt {b x^2+a}}\right )dx}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (-\frac {5 a^3 C e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{7/2}}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (B e+C d)}{8 b^{5/2}}+\frac {5 a^2 C e x \sqrt {a+b x^2}}{16 b^3}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (A e+B d)}{2 b^{3/2}}+\frac {A d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}+\frac {x \sqrt {a+b x^2} (A e+B d)}{2 b}-\frac {3 a x \sqrt {a+b x^2} (B e+C d)}{8 b^2}-\frac {5 a C e x^3 \sqrt {a+b x^2}}{24 b^2}+\frac {x^3 \sqrt {a+b x^2} (B e+C d)}{4 b}+\frac {C e x^5 \sqrt {a+b x^2}}{6 b}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

Int[((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/Sqrt[a*d + (b*d + a*e)*x^2 + b 
*e*x^4],x]
 

Output:

(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*((5*a^2*C*e*x*Sqrt[a + b*x^2])/(16*b^3) + 
 ((B*d + A*e)*x*Sqrt[a + b*x^2])/(2*b) - (3*a*(C*d + B*e)*x*Sqrt[a + b*x^2 
])/(8*b^2) - (5*a*C*e*x^3*Sqrt[a + b*x^2])/(24*b^2) + ((C*d + B*e)*x^3*Sqr 
t[a + b*x^2])/(4*b) + (C*e*x^5*Sqrt[a + b*x^2])/(6*b) + (A*d*ArcTanh[(Sqrt 
[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - (5*a^3*C*e*ArcTanh[(Sqrt[b]*x)/Sqrt[a + 
 b*x^2]])/(16*b^(7/2)) - (a*(B*d + A*e)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2 
]])/(2*b^(3/2)) + (3*a^2*(C*d + B*e)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]) 
/(8*b^(5/2))))/Sqrt[a*d + (b*d + a*e)*x^2 + b*e*x^4]
 

Defintions of rubi rules used

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])
 

rule 2009

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

rule 2256

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 
)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {x \left (8 e C \,x^{4} b^{2}+12 B \,b^{2} e \,x^{2}-10 C a b e \,x^{2}+12 C \,b^{2} d \,x^{2}+24 A e \,b^{2}-18 B a b e +24 b^{2} B d +15 a^{2} C e -18 C a b d \right ) \left (b \,x^{2}+a \right ) \sqrt {e \,x^{2}+d}}{48 b^{3} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}-\frac {\left (8 A a \,b^{2} e -16 A d \,b^{3}-6 B \,a^{2} b e +8 B a \,b^{2} d +5 C \,a^{3} e -6 C \,a^{2} b d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {e \,x^{2}+d}}{16 b^{\frac {7}{2}} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}\)	\(217\)
default	\(-\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \left (-8 C \,b^{\frac {5}{2}} e \,x^{5} \sqrt {b \,x^{2}+a}-12 B \,b^{\frac {5}{2}} e \,x^{3} \sqrt {b \,x^{2}+a}+10 C a \,b^{\frac {3}{2}} e \,x^{3} \sqrt {b \,x^{2}+a}-12 C \,b^{\frac {5}{2}} d \,x^{3} \sqrt {b \,x^{2}+a}-24 A \,b^{\frac {5}{2}} e x \sqrt {b \,x^{2}+a}+18 B a \,b^{\frac {3}{2}} e x \sqrt {b \,x^{2}+a}-24 B \,b^{\frac {5}{2}} d x \sqrt {b \,x^{2}+a}-15 C \,a^{2} e x \sqrt {b}\, \sqrt {b \,x^{2}+a}+18 C a \,b^{\frac {3}{2}} d x \sqrt {b \,x^{2}+a}+24 A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a \,b^{2} e -48 A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) b^{3} d -18 B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} b e +24 B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a \,b^{2} d +15 C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{3} e -18 C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} b d \right )}{48 b^{\frac {7}{2}} \sqrt {e \,x^{2}+d}\, \sqrt {b \,x^{2}+a}}\)	\(351\)

Input:

int((e*x^2+d)^(3/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

1/48*x*(8*C*b^2*e*x^4+12*B*b^2*e*x^2-10*C*a*b*e*x^2+12*C*b^2*d*x^2+24*A*b^ 
2*e-18*B*a*b*e+24*B*b^2*d+15*C*a^2*e-18*C*a*b*d)*(b*x^2+a)/b^3/((e*x^2+d)* 
(b*x^2+a))^(1/2)*(e*x^2+d)^(1/2)-1/16*(8*A*a*b^2*e-16*A*b^3*d-6*B*a^2*b*e+ 
8*B*a*b^2*d+5*C*a^3*e-6*C*a^2*b*d)/b^(7/2)*ln(b^(1/2)*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 = 0.11 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.18 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\left [-\frac {3 \, {\left (2 \, {\left (3 \, C a^{2} b - 4 \, B a b^{2} + 8 \, A b^{3}\right )} d^{2} - {\left (5 \, C a^{3} - 6 \, B a^{2} b + 8 \, A a b^{2}\right )} d e + {\left (2 \, {\left (3 \, C a^{2} b - 4 \, B a b^{2} + 8 \, A b^{3}\right )} d e - {\left (5 \, C a^{3} - 6 \, B a^{2} b + 8 \, A a b^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (\frac {2 \, b e x^{4} + {\left (2 \, b d + a e\right )} x^{2} - 2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d} \sqrt {b} x + a d}{e x^{2} + d}\right ) - 2 \, {\left (8 \, C b^{3} e x^{5} + 2 \, {\left (6 \, C b^{3} d - {\left (5 \, C a b^{2} - 6 \, B b^{3}\right )} e\right )} x^{3} - 3 \, {\left (2 \, {\left (3 \, C a b^{2} - 4 \, B b^{3}\right )} d - {\left (5 \, C a^{2} b - 6 \, B a b^{2} + 8 \, A b^{3}\right )} e\right )} x\right )} \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d}}{96 \, {\left (b^{4} e x^{2} + b^{4} d\right )}}, -\frac {3 \, {\left (2 \, {\left (3 \, C a^{2} b - 4 \, B a b^{2} + 8 \, A b^{3}\right )} d^{2} - {\left (5 \, C a^{3} - 6 \, B a^{2} b + 8 \, A a b^{2}\right )} d e + {\left (2 \, {\left (3 \, C a^{2} b - 4 \, B a b^{2} + 8 \, A b^{3}\right )} d e - {\left (5 \, C a^{3} - 6 \, B a^{2} b + 8 \, A a b^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {e x^{2} + d} \sqrt {-b} x}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d}}\right ) - {\left (8 \, C b^{3} e x^{5} + 2 \, {\left (6 \, C b^{3} d - {\left (5 \, C a b^{2} - 6 \, B b^{3}\right )} e\right )} x^{3} - 3 \, {\left (2 \, {\left (3 \, C a b^{2} - 4 \, B b^{3}\right )} d - {\left (5 \, C a^{2} b - 6 \, B a b^{2} + 8 \, A b^{3}\right )} e\right )} x\right )} \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d}}{48 \, {\left (b^{4} e x^{2} + b^{4} d\right )}}\right ] \] Input:

integrate((e*x^2+d)^(3/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="fricas")
 

Output:

[-1/96*(3*(2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d^2 - (5*C*a^3 - 6*B*a^2*b 
+ 8*A*a*b^2)*d*e + (2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d*e - (5*C*a^3 - 6 
*B*a^2*b + 8*A*a*b^2)*e^2)*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*(8*C*b^3*e*x^5 + 2*(6*C*b^3*d - (5*C*a*b^2 - 6*B*b^3)*e 
)*x^3 - 3*(2*(3*C*a*b^2 - 4*B*b^3)*d - (5*C*a^2*b - 6*B*a*b^2 + 8*A*b^3)*e 
)*x)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d))/(b^4*e*x^2 + b 
^4*d), -1/48*(3*(2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d^2 - (5*C*a^3 - 6*B* 
a^2*b + 8*A*a*b^2)*d*e + (2*(3*C*a^2*b - 4*B*a*b^2 + 8*A*b^3)*d*e - (5*C*a 
^3 - 6*B*a^2*b + 8*A*a*b^2)*e^2)*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)) - (8*C*b^3*e*x^5 + 2*(6*C*b^ 
3*d - (5*C*a*b^2 - 6*B*b^3)*e)*x^3 - 3*(2*(3*C*a*b^2 - 4*B*b^3)*d - (5*C*a 
^2*b - 6*B*a*b^2 + 8*A*b^3)*e)*x)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sq 
rt(e*x^2 + d))/(b^4*e*x^2 + b^4*d)]
 

Sympy [F]

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

integrate((e*x**2+d)**(3/2)*(C*x**4+B*x**2+A)/(a*d+(a*e+b*d)*x**2+b*e*x**4 
)**(1/2),x)
 

Output:

Integral((d + e*x**2)**(3/2)*(A + B*x**2 + C*x**4)/sqrt((a + b*x**2)*(d + 
e*x**2)), x)
 

Maxima [F]

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

integrate((e*x^2+d)^(3/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^(3/2)/sqrt(b*e*x^4 + (b*d + a*e) 
*x^2 + a*d), x)
 

Giac [F]

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

integrate((e*x^2+d)^(3/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^(3/2)/sqrt(b*e*x^4 + (b*d + a*e) 
*x^2 + a*d), x)
 

Mupad [F(-1)]

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

int(((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/(a*d + x^2*(a*e + b*d) + b*e*x 
^4)^(1/2),x)
 

Output:

int(((d + e*x^2)^(3/2)*(A + B*x^2 + C*x^4))/(a*d + x^2*(a*e + b*d) + b*e*x 
^4)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (A+B x^2+C x^4\right )}{\sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {15 \sqrt {b \,x^{2}+a}\, a^{2} b c e x +6 \sqrt {b \,x^{2}+a}\, a \,b^{3} e x -18 \sqrt {b \,x^{2}+a}\, a \,b^{2} c d x -10 \sqrt {b \,x^{2}+a}\, a \,b^{2} c e \,x^{3}+24 \sqrt {b \,x^{2}+a}\, b^{4} d x +12 \sqrt {b \,x^{2}+a}\, b^{4} e \,x^{3}+12 \sqrt {b \,x^{2}+a}\, b^{3} c d \,x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{3} c e \,x^{5}-15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} c e -6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} e +18 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b c d +24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d}{48 b^{4}} \] Input:

int((e*x^2+d)^(3/2)*(C*x^4+B*x^2+A)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x)
 

Output:

(15*sqrt(a + b*x**2)*a**2*b*c*e*x + 6*sqrt(a + b*x**2)*a*b**3*e*x - 18*sqr 
t(a + b*x**2)*a*b**2*c*d*x - 10*sqrt(a + b*x**2)*a*b**2*c*e*x**3 + 24*sqrt 
(a + b*x**2)*b**4*d*x + 12*sqrt(a + b*x**2)*b**4*e*x**3 + 12*sqrt(a + b*x* 
*2)*b**3*c*d*x**3 + 8*sqrt(a + b*x**2)*b**3*c*e*x**5 - 15*sqrt(b)*log((sqr 
t(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*c*e - 6*sqrt(b)*log((sqrt(a + b*x 
**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*e + 18*sqrt(b)*log((sqrt(a + b*x**2) 
+ sqrt(b)*x)/sqrt(a))*a**2*b*c*d + 24*sqrt(b)*log((sqrt(a + b*x**2) + sqrt 
(b)*x)/sqrt(a))*a*b**3*d)/(48*b**4)