\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx\) [151]

Optimal result

Integrand size = 49, antiderivative size = 297 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=-\frac {\left (C d^2-B d e+A e^2\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{4 d e (b d-a e) \left (d+e x^2\right )^{5/2}}+\frac {\left (2 b d \left (C d^2+e (B d-3 A e)\right )-a e \left (5 C d^2-e (B d+3 A e)\right )\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{8 d^2 e (b d-a e)^2 \left (d+e x^2\right )^{3/2}}-\frac {\left (a d (4 b B d-3 a C d-a B e)-A \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{8 d^{5/2} (b d-a e)^{5/2}} \] Output:

-1/4*(A*e^2-B*d*e+C*d^2)*x*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/d/e/(-a*e+b*d 
)/(e*x^2+d)^(5/2)+1/8*(2*b*d*(C*d^2+e*(-3*A*e+B*d))-a*e*(5*C*d^2-e*(3*A*e+ 
B*d)))*x*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/d^2/e/(-a*e+b*d)^2/(e*x^2+d)^(3 
/2)-1/8*(a*d*(-B*a*e+4*B*b*d-3*C*a*d)-A*(3*a^2*e^2-8*a*b*d*e+8*b^2*d^2))*a 
rctanh((-a*e+b*d)^(1/2)*x*(e*x^2+d)^(1/2)/d^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x 
^4)^(1/2))/d^(5/2)/(-a*e+b*d)^(5/2)
 

Mathematica [A] (verified)

Time = 12.03 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\frac {4 d^2 (2 C d-B e) x \left (d+e x^2\right ) \left (e \left (a+b x^2\right )-\frac {(2 b d-a e) \left (d+e x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}\right )}{d \sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}}\right )}{b d-a e}-\frac {\left (C d^2+e (-B d+A e)\right ) x \left (d e \left (a+b x^2\right ) \left (2 b d \left (4 d+3 e x^2\right )-a e \left (5 d+3 e x^2\right )\right )-\frac {\left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \left (d+e x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b d-a e) x^2}{d \left (a+b x^2\right )}}}\right )}{(b d-a e)^2}+\frac {8 C d^{5/2} \sqrt {a+b x^2} \left (d+e x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {b d-a e} x}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {b d-a e}}}{8 d^3 e^2 \left (d+e x^2\right )^{3/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)^(5/2)*Sqrt[a*d + (b*d + a*e)*x^ 
2 + b*e*x^4]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.49, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1395, 7293, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \sqrt {x^2 (a e+b d)+a d+b e x^4}} \, dx\)

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x^2 + C*x^4)/((d + e*x^2)^(5/2)*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]*(-1/4*((C*d^2 - B*d*e + A*e^2)*x*Sqrt[a + 
 b*x^2])/(d*e*(b*d - a*e)*(d + e*x^2)^2) + ((2*C*d - B*e)*x*Sqrt[a + b*x^2 
])/(2*d*e*(b*d - a*e)*(d + e*x^2)) - (3*(2*b*d - a*e)*(C*d^2 - B*d*e + A*e 
^2)*x*Sqrt[a + b*x^2])/(8*d^2*e*(b*d - a*e)^2*(d + e*x^2)) + (C*ArcTanh[(S 
qrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/(Sqrt[d]*e^2*Sqrt[b*d - a*e] 
) - ((2*b*d - a*e)*(2*C*d - B*e)*ArcTanh[(Sqrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt 
[a + b*x^2])])/(2*d^(3/2)*e^2*(b*d - a*e)^(3/2)) + ((8*b^2*d^2 - 8*a*b*d*e 
 + 3*a^2*e^2)*(C*d^2 - B*d*e + A*e^2)*ArcTanh[(Sqrt[b*d - a*e]*x)/(Sqrt[d] 
*Sqrt[a + b*x^2])])/(8*d^(5/2)*e^2*(b*d - a*e)^(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 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5778\) vs. \(2(271)=542\).

Time = 0.21 (sec) , antiderivative size = 5779, normalized size of antiderivative = 19.46

method	result	size
default	\(\text {Expression too large to display}\)	\(5779\)

Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x,me 
thod=_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. 707 vs. \(2 (269) = 538\).

Time = 0.14 (sec) , antiderivative size = 1439, normalized size of antiderivative = 4.85 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="fricas")
 

Output:

[1/16*((3*A*a^2*d^3*e^2 + (3*A*a^2*e^5 + (3*C*a^2 - 4*B*a*b + 8*A*b^2)*d^2 
*e^3 + (B*a^2 - 8*A*a*b)*d*e^4)*x^6 + (3*C*a^2 - 4*B*a*b + 8*A*b^2)*d^5 + 
(B*a^2 - 8*A*a*b)*d^4*e + 3*(3*A*a^2*d*e^4 + (3*C*a^2 - 4*B*a*b + 8*A*b^2) 
*d^3*e^2 + (B*a^2 - 8*A*a*b)*d^2*e^3)*x^4 + 3*(3*A*a^2*d^2*e^3 + (3*C*a^2 
- 4*B*a*b + 8*A*b^2)*d^4*e + (B*a^2 - 8*A*a*b)*d^3*e^2)*x^2)*sqrt(b*d^2 - 
a*d*e)*log((2*b*d^2*x^2 + (2*b*d*e - a*e^2)*x^4 + a*d^2 + 2*sqrt(b*e*x^4 + 
 (b*d + a*e)*x^2 + a*d)*sqrt(b*d^2 - a*d*e)*sqrt(e*x^2 + d)*x)/(e^2*x^4 + 
2*d*e*x^2 + d^2)) + 2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*((2*C*b^2*d^5 
- 3*A*a^2*d*e^4 - (7*C*a*b - 2*B*b^2)*d^4*e + (5*C*a^2 - B*a*b - 6*A*b^2)* 
d^3*e^2 - (B*a^2 - 9*A*a*b)*d^2*e^3)*x^3 - (5*A*a^2*d^2*e^3 + (3*C*a*b - 4 
*B*b^2)*d^5 - (3*C*a^2 - 5*B*a*b - 8*A*b^2)*d^4*e - (B*a^2 + 13*A*a*b)*d^3 
*e^2)*x)*sqrt(e*x^2 + d))/(b^3*d^9 - 3*a*b^2*d^8*e + 3*a^2*b*d^7*e^2 - a^3 
*d^6*e^3 + (b^3*d^6*e^3 - 3*a*b^2*d^5*e^4 + 3*a^2*b*d^4*e^5 - a^3*d^3*e^6) 
*x^6 + 3*(b^3*d^7*e^2 - 3*a*b^2*d^6*e^3 + 3*a^2*b*d^5*e^4 - a^3*d^4*e^5)*x 
^4 + 3*(b^3*d^8*e - 3*a*b^2*d^7*e^2 + 3*a^2*b*d^6*e^3 - a^3*d^5*e^4)*x^2), 
 -1/8*((3*A*a^2*d^3*e^2 + (3*A*a^2*e^5 + (3*C*a^2 - 4*B*a*b + 8*A*b^2)*d^2 
*e^3 + (B*a^2 - 8*A*a*b)*d*e^4)*x^6 + (3*C*a^2 - 4*B*a*b + 8*A*b^2)*d^5 + 
(B*a^2 - 8*A*a*b)*d^4*e + 3*(3*A*a^2*d*e^4 + (3*C*a^2 - 4*B*a*b + 8*A*b^2) 
*d^3*e^2 + (B*a^2 - 8*A*a*b)*d^2*e^3)*x^4 + 3*(3*A*a^2*d^2*e^3 + (3*C*a^2 
- 4*B*a*b + 8*A*b^2)*d^4*e + (B*a^2 - 8*A*a*b)*d^3*e^2)*x^2)*sqrt(-b*d^...
 

Sympy [F]

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

integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(5/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4 
)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4)/(sqrt((a + b*x**2)*(d + e*x**2))*(d + e*x** 
2)**(5/2)), x)
 

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(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)/(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(e*x^ 
2 + d)^(5/2)), x)
 

Giac [F]

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

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(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)/(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*(e*x^ 
2 + d)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (e\,x^2+d\right )}^{5/2}\,\sqrt {b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/((d + e*x^2)^(5/2)*(a*d + x^2*(a*e + b*d) + b*e*x^ 
4)^(1/2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/((d + e*x^2)^(5/2)*(a*d + x^2*(a*e + b*d) + b*e*x^ 
4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 3209, normalized size of antiderivative = 10.80 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx =\text {Too large to display} \] Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x)
 

Output:

( - 6*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x 
**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*d**2*e**4 - 12*sqrt(d)*s 
qrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)* 
sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**4*d*e**5*x**2 - 6*sqrt(d)*sqrt(a*e - b*d) 
*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sq 
rt(d)*sqrt(b)))*a**4*e**6*x**4 + 26*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e 
 - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b))) 
*a**3*b*d**3*e**3 + 52*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqr 
t(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**3*b*d**2* 
e**4*x**2 + 26*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqr 
t(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**3*b*d*e**5*x**4 - 
 6*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2 
) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a**3*c*d**4*e**2 - 12*sqrt(d)*sq 
rt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*s 
qrt(b)*x)/(sqrt(d)*sqrt(b)))*a**3*c*d**3*e**3*x**2 - 6*sqrt(d)*sqrt(a*e - 
b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x) 
/(sqrt(d)*sqrt(b)))*a**3*c*d**2*e**4*x**4 - 36*sqrt(d)*sqrt(a*e - b*d)*ata 
n((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d 
)*sqrt(b)))*a**2*b**2*d**4*e**2 - 72*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a* 
e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(...