\(\int \frac {(c+d x^2) (e+f x^2)}{\sqrt {a+b x^2}} \, dx\) [318]

Optimal result

Integrand size = 26, antiderivative size = 113 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=-\frac {(3 a d f-4 b (d e+c f)) x \sqrt {a+b x^2}}{8 b^2}+\frac {d f x^3 \sqrt {a+b x^2}}{4 b}+\frac {\left (8 b^2 c e+3 a^2 d f-4 a b (d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {x \sqrt {a+b x^2} \left (4 b d e+4 b c f-3 a d f+2 b d f x^2\right )}{8 b^2}+\frac {\left (-8 b^2 c e+4 a b d e+4 a b c f-3 a^2 d f\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{5/2}} \] Input:

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

Output:

(x*Sqrt[a + b*x^2]*(4*b*d*e + 4*b*c*f - 3*a*d*f + 2*b*d*f*x^2))/(8*b^2) + 
((-8*b^2*c*e + 4*a*b*d*e + 4*a*b*c*f - 3*a^2*d*f)*Log[-(Sqrt[b]*x) + Sqrt[ 
a + b*x^2]])/(8*b^(5/2))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {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.

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(f*x*Sqrt[a + b*x^2]*(c + d*x^2))/(4*b) + (((4*b*d*e + 2*b*c*f - 3*a*d*f)* 
x*Sqrt[a + b*x^2])/(2*b) + ((8*b^2*c*e + 3*a^2*d*f - 4*a*b*(d*e + c*f))*Ar 
cTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2)))/(4*b)
 

Defintions of rubi rules used

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 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]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {x \left (-2 b d f \,x^{2}+3 a d f -4 b c f -4 b d e \right ) \sqrt {b \,x^{2}+a}}{8 b^{2}}+\frac {\left (3 a^{2} d f -4 a b c f -4 a b d e +8 c e \,b^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}\)	\(89\)
pseudoelliptic	\(\frac {\frac {3 \left (a \left (a f -\frac {4 b e}{3}\right ) d -\frac {4 b c \left (a f -2 b e \right )}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}-\frac {3 \sqrt {b \,x^{2}+a}\, x \left (\frac {2 \left (\left (-f \,x^{2}-2 e \right ) d -2 c f \right ) b^{\frac {3}{2}}}{3}+a d f \sqrt {b}\right )}{8}}{b^{\frac {5}{2}}}\)	\(91\)
default	\(\frac {c e \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\left (c f +d e \right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+d f \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )\)	\(135\)

Input:

int((d*x^2+c)*(f*x^2+e)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*x*(-2*b*d*f*x^2+3*a*d*f-4*b*c*f-4*b*d*e)/b^2*(b*x^2+a)^(1/2)+1/8*(3*a 
^2*d*f-4*a*b*c*f-4*a*b*d*e+8*b^2*c*e)/b^(5/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2) 
)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.95 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\left [\frac {{\left (4 \, {\left (2 \, b^{2} c - a b d\right )} e - {\left (4 \, a b c - 3 \, a^{2} d\right )} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, b^{2} d f x^{3} + {\left (4 \, b^{2} d e + {\left (4 \, b^{2} c - 3 \, a b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, b^{3}}, -\frac {{\left (4 \, {\left (2 \, b^{2} c - a b d\right )} e - {\left (4 \, a b c - 3 \, a^{2} d\right )} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, b^{2} d f x^{3} + {\left (4 \, b^{2} d e + {\left (4 \, b^{2} c - 3 \, a b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, b^{3}}\right ] \] Input:

integrate((d*x^2+c)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/16*((4*(2*b^2*c - a*b*d)*e - (4*a*b*c - 3*a^2*d)*f)*sqrt(b)*log(-2*b*x^ 
2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(2*b^2*d*f*x^3 + (4*b^2*d*e + (4* 
b^2*c - 3*a*b*d)*f)*x)*sqrt(b*x^2 + a))/b^3, -1/8*((4*(2*b^2*c - a*b*d)*e 
- (4*a*b*c - 3*a^2*d)*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (2* 
b^2*d*f*x^3 + (4*b^2*d*e + (4*b^2*c - 3*a*b*d)*f)*x)*sqrt(b*x^2 + a))/b^3]
 

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.23 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d f x^{3}}{4 b} + \frac {x \left (- \frac {3 a d f}{4 b} + c f + d e\right )}{2 b}\right ) + \left (- \frac {a \left (- \frac {3 a d f}{4 b} + c f + d e\right )}{2 b} + c e\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\frac {c e x + \frac {d f x^{5}}{5} + \frac {x^{3} \left (c f + d e\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:

integrate((d*x**2+c)*(f*x**2+e)/(b*x**2+a)**(1/2),x)
 

Output:

Piecewise((sqrt(a + b*x**2)*(d*f*x**3/(4*b) + x*(-3*a*d*f/(4*b) + c*f + d* 
e)/(2*b)) + (-a*(-3*a*d*f/(4*b) + c*f + d*e)/(2*b) + c*e)*Piecewise((log(2 
*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x* 
*2), True)), Ne(b, 0)), ((c*e*x + d*f*x**5/5 + x**3*(c*f + d*e)/3)/sqrt(a) 
, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} d f x^{3}}{4 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a d f x}{8 \, b^{2}} + \frac {c e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} + \frac {3 \, a^{2} d f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} + \frac {\sqrt {b x^{2} + a} {\left (d e + c f\right )} x}{2 \, b} - \frac {{\left (d e + c f\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} \] Input:

integrate((d*x^2+c)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

1/4*sqrt(b*x^2 + a)*d*f*x^3/b - 3/8*sqrt(b*x^2 + a)*a*d*f*x/b^2 + c*e*arcs 
inh(b*x/sqrt(a*b))/sqrt(b) + 3/8*a^2*d*f*arcsinh(b*x/sqrt(a*b))/b^(5/2) + 
1/2*sqrt(b*x^2 + a)*(d*e + c*f)*x/b - 1/2*(d*e + c*f)*a*arcsinh(b*x/sqrt(a 
*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {1}{8} \, \sqrt {b x^{2} + a} {\left (\frac {2 \, d f x^{2}}{b} + \frac {4 \, b^{2} d e + 4 \, b^{2} c f - 3 \, a b d f}{b^{3}}\right )} x - \frac {{\left (8 \, b^{2} c e - 4 \, a b d e - 4 \, a b c f + 3 \, a^{2} d f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {5}{2}}} \] Input:

integrate((d*x^2+c)*(f*x^2+e)/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

1/8*sqrt(b*x^2 + a)*(2*d*f*x^2/b + (4*b^2*d*e + 4*b^2*c*f - 3*a*b*d*f)/b^3 
)*x - 1/8*(8*b^2*c*e - 4*a*b*d*e - 4*a*b*c*f + 3*a^2*d*f)*log(abs(-sqrt(b) 
*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

int(((c + d*x^2)*(e + f*x^2))/(a + b*x^2)^(1/2),x)
 

Output:

int(((c + d*x^2)*(e + f*x^2))/(a + b*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.60 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, a b d f x +4 \sqrt {b \,x^{2}+a}\, b^{2} c f x +4 \sqrt {b \,x^{2}+a}\, b^{2} d e x +2 \sqrt {b \,x^{2}+a}\, b^{2} d f \,x^{3}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} d f -4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c f -4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b d e +8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c e}{8 b^{3}} \] Input:

int((d*x^2+c)*(f*x^2+e)/(b*x^2+a)^(1/2),x)
 

Output:

( - 3*sqrt(a + b*x**2)*a*b*d*f*x + 4*sqrt(a + b*x**2)*b**2*c*f*x + 4*sqrt( 
a + b*x**2)*b**2*d*e*x + 2*sqrt(a + b*x**2)*b**2*d*f*x**3 + 3*sqrt(b)*log( 
(sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*d*f - 4*sqrt(b)*log((sqrt(a + 
 b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c*f - 4*sqrt(b)*log((sqrt(a + b*x**2) + 
 sqrt(b)*x)/sqrt(a))*a*b*d*e + 8*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x 
)/sqrt(a))*b**2*c*e)/(8*b**3)