\(\int \frac {(c+d x^2) (e+f x^2)}{(a+b x^2)^{3/2}} \, dx\) [339]

Optimal result

Integrand size = 26, antiderivative size = 101 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(b c-a d) (b e-a f) x}{a b^2 \sqrt {a+b x^2}}+\frac {d f x \sqrt {a+b x^2}}{2 b^2}-\frac {(3 a d f-2 b (d e+c f)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (2 b^2 c e-2 a b d e-2 a b c f+3 a^2 d f+a b d f x^2\right )}{2 a b^2 \sqrt {a+b x^2}}+\frac {(-2 b d e-2 b c f+3 a d f) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{5/2}} \] Input:

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

Output:

(x*(2*b^2*c*e - 2*a*b*d*e - 2*a*b*c*f + 3*a^2*d*f + a*b*d*f*x^2))/(2*a*b^2 
*Sqrt[a + b*x^2]) + ((-2*b*d*e - 2*b*c*f + 3*a*d*f)*Log[-(Sqrt[b]*x) + Sqr 
t[a + b*x^2]])/(2*b^(5/2))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {401, 25, 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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

((b*e - a*f)*x*(c + d*x^2))/(a*b*Sqrt[a + b*x^2]) + (-1/2*(d*(2*b*e - 3*a* 
f)*x*Sqrt[a + b*x^2])/b - (a*(3*a*d*f - 2*b*(d*e + c*f))*ArcTanh[(Sqrt[b]* 
x)/Sqrt[a + b*x^2]])/(2*b^(3/2)))/(a*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 401

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

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {-\frac {3 a \left (a d f -\frac {2 b \left (c f +d e \right )}{3}\right ) \sqrt {b \,x^{2}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{2}+\sqrt {b}\, x \left (\frac {3 a^{2} d f}{2}-\left (-\frac {1}{2} d f \,x^{2}+c f +d e \right ) b a +c e \,b^{2}\right )}{\sqrt {b \,x^{2}+a}\, b^{\frac {5}{2}} a}\)	\(99\)
risch	\(\frac {d f x \sqrt {b \,x^{2}+a}}{2 b^{2}}-\frac {b \left (3 a d f -2 b c f -2 b d e \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {a d f x}{\sqrt {b \,x^{2}+a}}-\frac {2 c e \,b^{2} x}{a \sqrt {b \,x^{2}+a}}}{2 b^{2}}\)	\(113\)
default	\(\frac {c e x}{a \sqrt {b \,x^{2}+a}}+\left (c f +d e \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+d f \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )\)	\(125\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.21 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (2 \, a^{2} b d e + {\left (2 \, a b^{2} d e + {\left (2 \, a b^{2} c - 3 \, a^{2} b d\right )} f\right )} x^{2} + {\left (2 \, a^{2} b c - 3 \, a^{3} d\right )} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (a b^{2} d f x^{3} + {\left (2 \, {\left (b^{3} c - a b^{2} d\right )} e - {\left (2 \, a b^{2} c - 3 \, a^{2} b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, -\frac {{\left (2 \, a^{2} b d e + {\left (2 \, a b^{2} d e + {\left (2 \, a b^{2} c - 3 \, a^{2} b d\right )} f\right )} x^{2} + {\left (2 \, a^{2} b c - 3 \, a^{3} d\right )} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (a b^{2} d f x^{3} + {\left (2 \, {\left (b^{3} c - a b^{2} d\right )} e - {\left (2 \, a b^{2} c - 3 \, a^{2} b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (90) = 180\).

Time = 5.66 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.80 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=c f \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + d e \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + d f \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {c e x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

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

Output:

c*f*(asinh(sqrt(b)*x/sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) 
 + d*e*(asinh(sqrt(b)*x/sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a 
))) + d*f*(3*sqrt(a)*x/(2*b**2*sqrt(1 + b*x**2/a)) - 3*a*asinh(sqrt(b)*x/s 
qrt(a))/(2*b**(5/2)) + x**3/(2*sqrt(a)*b*sqrt(1 + b*x**2/a))) + c*e*x/(a** 
(3/2)*sqrt(1 + b*x**2/a))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d f x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {c e x}{\sqrt {b x^{2} + a} a} + \frac {3 \, a d f x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {3 \, a d f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} - \frac {{\left (d e + c f\right )} x}{\sqrt {b x^{2} + a} b} + \frac {{\left (d e + c f\right )} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \] Input:

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

Output:

1/2*d*f*x^3/(sqrt(b*x^2 + a)*b) + c*e*x/(sqrt(b*x^2 + a)*a) + 3/2*a*d*f*x/ 
(sqrt(b*x^2 + a)*b^2) - 3/2*a*d*f*arcsinh(b*x/sqrt(a*b))/b^(5/2) - (d*e + 
c*f)*x/(sqrt(b*x^2 + a)*b) + (d*e + c*f)*arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {d f x^{2}}{b} + \frac {2 \, b^{3} c e - 2 \, a b^{2} d e - 2 \, a b^{2} c f + 3 \, a^{2} b d f}{a b^{3}}\right )} x}{2 \, \sqrt {b x^{2} + a}} - \frac {{\left (2 \, b d e + 2 \, b c f - 3 \, a d f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 376, normalized size of antiderivative = 3.72 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {12 \sqrt {b \,x^{2}+a}\, a^{2} b d f x -8 \sqrt {b \,x^{2}+a}\, a \,b^{2} c f x -8 \sqrt {b \,x^{2}+a}\, a \,b^{2} d e x +4 \sqrt {b \,x^{2}+a}\, a \,b^{2} d f \,x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{3} c e x -12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} d f +8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b c f +8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b d e -12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b d f \,x^{2}+8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c f \,x^{2}+8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d e \,x^{2}+9 \sqrt {b}\, a^{3} d f -8 \sqrt {b}\, a^{2} b c f -8 \sqrt {b}\, a^{2} b d e +9 \sqrt {b}\, a^{2} b d f \,x^{2}+8 \sqrt {b}\, a \,b^{2} c e -8 \sqrt {b}\, a \,b^{2} c f \,x^{2}-8 \sqrt {b}\, a \,b^{2} d e \,x^{2}+8 \sqrt {b}\, b^{3} c e \,x^{2}}{8 a \,b^{3} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

(12*sqrt(a + b*x**2)*a**2*b*d*f*x - 8*sqrt(a + b*x**2)*a*b**2*c*f*x - 8*sq 
rt(a + b*x**2)*a*b**2*d*e*x + 4*sqrt(a + b*x**2)*a*b**2*d*f*x**3 + 8*sqrt( 
a + b*x**2)*b**3*c*e*x - 12*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqr 
t(a))*a**3*d*f + 8*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a** 
2*b*c*f + 8*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*d*e 
 - 12*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b*d*f*x**2 
+ 8*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*c*f*x**2 + 
8*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*d*e*x**2 + 9* 
sqrt(b)*a**3*d*f - 8*sqrt(b)*a**2*b*c*f - 8*sqrt(b)*a**2*b*d*e + 9*sqrt(b) 
*a**2*b*d*f*x**2 + 8*sqrt(b)*a*b**2*c*e - 8*sqrt(b)*a*b**2*c*f*x**2 - 8*sq 
rt(b)*a*b**2*d*e*x**2 + 8*sqrt(b)*b**3*c*e*x**2)/(8*a*b**3*(a + b*x**2))