\(\int \frac {(d+e x^2)^{7/2}}{(a d+(b d+a e) x^2+b e x^4)^{3/2}} \, dx\) [9]

Optimal result

Integrand size = 37, antiderivative size = 165 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {(b d-a e)^2 x \sqrt {d+e x^2}}{a b^2 \sqrt {a d+(b d+a e) x^2+b e x^4}}+\frac {e^2 x \sqrt {a d+(b d+a e) x^2+b e x^4}}{2 b^2 \sqrt {d+e x^2}}+\frac {e (4 b d-3 a e) \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{2 b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {\sqrt {d+e x^2} \left (\sqrt {b} x \left (2 b^2 d^2+3 a^2 e^2+a b e \left (-4 d+e x^2\right )\right )+a e (-4 b d+3 a e) \sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{2 a b^{5/2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[(d + e*x^2)^(7/2)/(a*d + (b*d + a*e)*x^2 + b*e*x^4)^(3/2),x]
 

Output:

(Sqrt[d + e*x^2]*(Sqrt[b]*x*(2*b^2*d^2 + 3*a^2*e^2 + a*b*e*(-4*d + e*x^2)) 
 + a*e*(-4*b*d + 3*a*e)*Sqrt[a + b*x^2]*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2] 
]))/(2*a*b^(5/2)*Sqrt[(a + b*x^2)*(d + e*x^2)])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1395, 315, 27, 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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (x^2 (a e+b d)+a d+b e x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

Int[(d + e*x^2)^(7/2)/(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))/(a*b*Sqrt[a 
+ b*x^2]) + (e*(-1/2*((2*b*d - 3*a*e)*x*Sqrt[a + b*x^2])/b + (a*(4*b*d - 3 
*a*e)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/(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 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 = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {\sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \left (-a \,b^{\frac {3}{2}} e^{2} x^{3}+3 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a^{2} e^{2} \sqrt {b \,x^{2}+a}-4 \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) a b d e \sqrt {b \,x^{2}+a}-3 a^{2} e^{2} x \sqrt {b}+4 a \,b^{\frac {3}{2}} d e x -2 b^{\frac {5}{2}} d^{2} x \right )}{2 b^{\frac {5}{2}} \sqrt {e \,x^{2}+d}\, \left (b \,x^{2}+a \right ) a}\)	\(151\)
risch	\(\frac {e^{2} x \left (b \,x^{2}+a \right ) \sqrt {e \,x^{2}+d}}{2 b^{2} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}-\frac {\left (\frac {a \,e^{2} x}{\sqrt {b \,x^{2}+a}}+b e \left (3 a e -4 b d \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )-\frac {2 b^{2} d^{2} x}{a \sqrt {b \,x^{2}+a}}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {e \,x^{2}+d}}{2 b^{2} \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}\)	\(169\)

Input:

int((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x,method=_RETURNVERB 
OSE)
 

Output:

-1/2*((e*x^2+d)*(b*x^2+a))^(1/2)/b^(5/2)*(-a*b^(3/2)*e^2*x^3+3*ln(b^(1/2)* 
x+(b*x^2+a)^(1/2))*a^2*e^2*(b*x^2+a)^(1/2)-4*ln(b^(1/2)*x+(b*x^2+a)^(1/2)) 
*a*b*d*e*(b*x^2+a)^(1/2)-3*a^2*e^2*x*b^(1/2)+4*a*b^(3/2)*d*e*x-2*b^(5/2)*d 
^2*x)/(e*x^2+d)^(1/2)/(b*x^2+a)/a
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.07 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\left [-\frac {{\left (4 \, a^{2} b d^{2} e - 3 \, a^{3} d e^{2} + {\left (4 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x^{4} + {\left (4 \, a b^{2} d^{2} e + a^{2} b d e^{2} - 3 \, a^{3} e^{3}\right )} x^{2}\right )} \sqrt {b} \log \left (\frac {2 \, b e x^{4} + {\left (2 \, b d + a e\right )} x^{2} - 2 \, \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d} \sqrt {b} x + a d}{e x^{2} + d}\right ) - 2 \, {\left (a b^{2} e^{2} x^{3} + {\left (2 \, b^{3} d^{2} - 4 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x\right )} \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d}}{4 \, {\left (a b^{4} e x^{4} + a^{2} b^{3} d + {\left (a b^{4} d + a^{2} b^{3} e\right )} x^{2}\right )}}, -\frac {{\left (4 \, a^{2} b d^{2} e - 3 \, a^{3} d e^{2} + {\left (4 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x^{4} + {\left (4 \, a b^{2} d^{2} e + a^{2} b d e^{2} - 3 \, a^{3} e^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {e x^{2} + d} \sqrt {-b} x}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d}}\right ) - {\left (a b^{2} e^{2} x^{3} + {\left (2 \, b^{3} d^{2} - 4 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x\right )} \sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} \sqrt {e x^{2} + d}}{2 \, {\left (a b^{4} e x^{4} + a^{2} b^{3} d + {\left (a b^{4} d + a^{2} b^{3} e\right )} x^{2}\right )}}\right ] \] Input:

integrate((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
fricas")
 

Output:

[-1/4*((4*a^2*b*d^2*e - 3*a^3*d*e^2 + (4*a*b^2*d*e^2 - 3*a^2*b*e^3)*x^4 + 
(4*a*b^2*d^2*e + a^2*b*d*e^2 - 3*a^3*e^3)*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*(a*b^2*e^2*x^3 + (2*b^3*d^2 - 4*a*b^2*d* 
e + 3*a^2*b*e^2)*x)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d)) 
/(a*b^4*e*x^4 + a^2*b^3*d + (a*b^4*d + a^2*b^3*e)*x^2), -1/2*((4*a^2*b*d^2 
*e - 3*a^3*d*e^2 + (4*a*b^2*d*e^2 - 3*a^2*b*e^3)*x^4 + (4*a*b^2*d^2*e + a^ 
2*b*d*e^2 - 3*a^3*e^3)*x^2)*sqrt(-b)*arctan(sqrt(e*x^2 + d)*sqrt(-b)*x/sqr 
t(b*e*x^4 + (b*d + a*e)*x^2 + a*d)) - (a*b^2*e^2*x^3 + (2*b^3*d^2 - 4*a*b^ 
2*d*e + 3*a^2*b*e^2)*x)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + 
 d))/(a*b^4*e*x^4 + a^2*b^3*d + (a*b^4*d + a^2*b^3*e)*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(7/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (b e x^{4} + {\left (b d + a e\right )} x^{2} + a d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
maxima")
 

Output:

integrate((e*x^2 + d)^(7/2)/(b*e*x^4 + (b*d + a*e)*x^2 + a*d)^(3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.56 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {{\left (\frac {e^{2} x^{2}}{b} + \frac {2 \, b^{3} d^{2} - 4 \, a b^{2} d e + 3 \, a^{2} b e^{2}}{a b^{3}}\right )} x}{2 \, \sqrt {b x^{2} + a}} - \frac {{\left (4 \, b d e - 3 \, a e^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \] Input:

integrate((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x, algorithm=" 
giac")
 

Output:

1/2*(e^2*x^2/b + (2*b^3*d^2 - 4*a*b^2*d*e + 3*a^2*b*e^2)/(a*b^3))*x/sqrt(b 
*x^2 + a) - 1/2*(4*b*d*e - 3*a*e^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) 
/b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{{\left (b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d\right )}^{3/2}} \,d x \] Input:

int((d + e*x^2)^(7/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2),x)
 

Output:

int((d + e*x^2)^(7/2)/(a*d + x^2*(a*e + b*d) + b*e*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.72 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a d+(b d+a e) x^2+b e x^4\right )^{3/2}} \, dx=\frac {12 \sqrt {b \,x^{2}+a}\, a^{2} b \,e^{2} x -16 \sqrt {b \,x^{2}+a}\, a \,b^{2} d e x +4 \sqrt {b \,x^{2}+a}\, a \,b^{2} e^{2} x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{3} d^{2} x -12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} e^{2}+16 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b d e -12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b \,e^{2} x^{2}+16 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d e \,x^{2}+9 \sqrt {b}\, a^{3} e^{2}-16 \sqrt {b}\, a^{2} b d e +9 \sqrt {b}\, a^{2} b \,e^{2} x^{2}+8 \sqrt {b}\, a \,b^{2} d^{2}-16 \sqrt {b}\, a \,b^{2} d e \,x^{2}+8 \sqrt {b}\, b^{3} d^{2} x^{2}}{8 a \,b^{3} \left (b \,x^{2}+a \right )} \] Input:

int((e*x^2+d)^(7/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(3/2),x)
 

Output:

(12*sqrt(a + b*x**2)*a**2*b*e**2*x - 16*sqrt(a + b*x**2)*a*b**2*d*e*x + 4* 
sqrt(a + b*x**2)*a*b**2*e**2*x**3 + 8*sqrt(a + b*x**2)*b**3*d**2*x - 12*sq 
rt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*e**2 + 16*sqrt(b)*l 
og((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*d*e - 12*sqrt(b)*log((sq 
rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*e**2*x**2 + 16*sqrt(b)*log((sq 
rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*d*e*x**2 + 9*sqrt(b)*a**3*e**2 
 - 16*sqrt(b)*a**2*b*d*e + 9*sqrt(b)*a**2*b*e**2*x**2 + 8*sqrt(b)*a*b**2*d 
**2 - 16*sqrt(b)*a*b**2*d*e*x**2 + 8*sqrt(b)*b**3*d**2*x**2)/(8*a*b**3*(a 
+ b*x**2))