\(\int \frac {1}{(d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}})^2} \, dx\) [11]

Optimal result

Integrand size = 28, antiderivative size = 266 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx=-\frac {2 \left (d^2 e-b d f^2+a e f^2\right )}{\left (2 d e-b f^2\right )^2 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}-\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{\left (2 d e-b f^2\right )^3}-\frac {2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}{\left (2 d e-b f^2\right )^3} \] Output:

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

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx=\frac {2 \left (b f^3 (-d+e x)+2 e f \left (a f^2-d e x\right )\right ) \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}{\left (-2 d e+b f^2\right )^2 \left (d^2+2 d e x-f^2 (a+b x)\right )}-\frac {2 \left (4 a^2 e^3 f^2 x+b d x \left (2 d^2 e^2-2 b d e f^2+b^2 f^4+2 d e^3 x-b e^2 f^2 x\right )+a \left (4 d^3 e^2+2 b e^3 f^2 x^2+d^2 \left (-4 b e f^2+4 e^3 x\right )+d \left (b^2 f^4-6 b e^2 f^2 x-4 e^4 x^2\right )\right )\right )}{(b d-2 a e) \left (-2 d e+b f^2\right )^2 \left (-d^2-2 d e x+f^2 (a+b x)\right )}-\frac {2 \left (4 a e^2 f^2-b^2 f^4\right ) \log \left (-\sqrt {a} f+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}{\left (2 d e-b f^2\right )^3}+\frac {2 \left (4 a e^2 f^2-b^2 f^4\right ) \log \left (-a f^2+d e x-b f^2 x-d f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+\sqrt {a} f \left (d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{\left (2 d e-b f^2\right )^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2541, 1195, 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.

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

\(\Big \downarrow \) 2541

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

rule 2541

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Simp[2   Subst[Int[(g + h*x^n)^p*((d 
^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2)/(-2*d*e + b*f^2 + 2*e*x 
)^2), x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e 
, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6304\) vs. \(2(259)=518\).

Time = 0.07 (sec) , antiderivative size = 6305, normalized size of antiderivative = 23.70

method	result	size
default	\(\text {Expression too large to display}\)	\(6305\)

Input:

int(1/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2))^2,x,method=_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. 826 vs. \(2 (251) = 502\).

Time = 1.36 (sec) , antiderivative size = 826, normalized size of antiderivative = 3.11 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(a*b^2*f^6 + (3*b^2*d^2 - 14*a*b*d*e + 8*a^2*e^2)*f^4 - 2*(b*d^3*e - 
4*a*d^2*e^2)*f^2 - 4*(b^2*e^2*f^4 - 4*b*d*e^3*f^2 + 4*d^2*e^4)*x^2 + (b^3* 
f^6 - 8*b^2*d*e*f^4 + 20*b*d^2*e^2*f^2 - 16*d^3*e^3)*x - 2*(a*b^2*f^6 + 4* 
a*d^2*e^2*f^2 - (b^2*d^2 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2*( 
b^2*d*e + 2*a*b*e^2)*f^4)*x)*log(-4*a*d*e^2*f^2 - (b^2*d - 4*a*b*e)*f^4 + 
4*(b*e^3*f^2 - 2*d*e^4)*x^2 + (3*b^2*e*f^4 - 4*(2*b*d*e^2 - a*e^3)*f^2)*x 
- (b^2*f^5 - 4*(b*d*e - a*e^2)*f^3 + 4*(b*e^2*f^3 - 2*d*e^3*f)*x)*sqrt((b* 
f^2*x + e^2*x^2 + a*f^2)/f^2)) - 2*(a*b^2*f^6 + 4*a*d^2*e^2*f^2 - (b^2*d^2 
 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^2)*f^4 
)*x)*log(a*f^2 - d^2 + (b*f^2 - 2*d*e)*x) + 2*(a*b^2*f^6 + 4*a*d^2*e^2*f^2 
 - (b^2*d^2 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a 
*b*e^2)*f^4)*x)*log(-e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) - d) - 
4*((b^2*d - 2*a*b*e)*f^5 - 2*(b*d^2*e - 2*a*d*e^2)*f^3 - (b^2*e*f^5 - 4*b* 
d*e^2*f^3 + 4*d^2*e^3*f)*x)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))/(a*b^3* 
f^8 + 8*d^5*e^3 - (b^3*d^2 + 6*a*b^2*d*e)*f^6 + 6*(b^2*d^3*e + 2*a*b*d^2*e 
^2)*f^4 - 4*(3*b*d^4*e^2 + 2*a*d^3*e^3)*f^2 + (b^4*f^8 - 8*b^3*d*e*f^6 + 2 
4*b^2*d^2*e^2*f^4 - 32*b*d^3*e^3*f^2 + 16*d^4*e^4)*x)
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx=\int \frac {1}{\left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{2}}\, dx \] Input:

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

Output:

Integral((d + e*x + f*sqrt(a + b*x + e**2*x**2/f**2))**(-2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx=\int { \frac {1}{{\left (e x + \sqrt {b x + \frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{2}} \,d x } \] Input:

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

Output:

integrate((e*x + sqrt(b*x + e^2*x^2/f^2 + a)*f + d)^(-2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (251) = 502\).

Time = 4.93 (sec) , antiderivative size = 1618, normalized size of antiderivative = 6.08 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*e^2*x/(b^2*f^4 - 4*b*d*e*f^2 + 4*d^2*e^2) + 1/5*(b^2*e*f^3*abs(f) - 4*a* 
e^3*f*abs(f))*log(abs((x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^4*b^3*f 
^6 - 2*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^2*b^3*d^2*f^6 + b^3*d^ 
4*f^6 + 4*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^2*a*b^2*d*e*f^6 - 4 
*a*b^2*d^3*e*f^6 + 4*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^2*a^2*b* 
e^2*f^6 + 4*a^2*b*d^2*e^2*f^6 + 4*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f 
^2))^3*a*b^2*f^6*abs(e) - 4*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))*a 
*b^2*d^2*f^6*abs(e) + 8*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))*a^2*b 
*d*e*f^6*abs(e) - 4*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^4*b^2*d*e 
*f^4 + 4*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^2*b^2*d^3*e*f^4 + 8* 
(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^4*a*b*e^2*f^4 - 24*(x*abs(e) 
- sqrt(b*f^2*x + e^2*x^2 + a*f^2))^2*a*b*d^2*e^2*f^4 + 16*(x*abs(e) - sqrt 
(b*f^2*x + e^2*x^2 + a*f^2))^2*a^2*d*e^3*f^4 + 2*(x*abs(e) - sqrt(b*f^2*x 
+ e^2*x^2 + a*f^2))^5*b^2*f^4*abs(e) - 8*(x*abs(e) - sqrt(b*f^2*x + e^2*x^ 
2 + a*f^2))^3*b^2*d^2*f^4*abs(e) + 6*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + 
a*f^2))*b^2*d^4*f^4*abs(e) - 16*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2 
))*a*b*d^3*e*f^4*abs(e) + 8*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^3 
*a^2*e^2*f^4*abs(e) + 8*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))*a^2*d 
^2*e^2*f^4*abs(e) - 4*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^4*b*d^2 
*e^2*f^2 + 12*(x*abs(e) - sqrt(b*f^2*x + e^2*x^2 + a*f^2))^2*b*d^4*e^2*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx=\int \frac {1}{{\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 2122, normalized size of antiderivative = 7.98 \[ \int \frac {1}{\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*( - 2*sqrt(a*f**2 + b*f**2*x + e**2*x**2)*a**2*b*e*f**6 + 4*sqrt(a*f**2 
 + b*f**2*x + e**2*x**2)*a**2*d*e**2*f**4 + sqrt(a*f**2 + b*f**2*x + e**2* 
x**2)*a*b**2*d*f**6 - sqrt(a*f**2 + b*f**2*x + e**2*x**2)*a*b**2*e*f**6*x 
+ 4*sqrt(a*f**2 + b*f**2*x + e**2*x**2)*a*b*d*e**2*f**4*x - 4*sqrt(a*f**2 
+ b*f**2*x + e**2*x**2)*a*d**3*e**2*f**2 - 4*sqrt(a*f**2 + b*f**2*x + e**2 
*x**2)*a*d**2*e**3*f**2*x - sqrt(a*f**2 + b*f**2*x + e**2*x**2)*b**2*d**3* 
f**4 + sqrt(a*f**2 + b*f**2*x + e**2*x**2)*b**2*d**2*e*f**4*x + 2*sqrt(a*f 
**2 + b*f**2*x + e**2*x**2)*b*d**4*e*f**2 - 4*sqrt(a*f**2 + b*f**2*x + e** 
2*x**2)*b*d**3*e**2*f**2*x + 4*sqrt(a*f**2 + b*f**2*x + e**2*x**2)*d**4*e* 
*3*x - 4*log( - sqrt(a*f**2 + b*f**2*x + e**2*x**2)*b*f**2 + 2*sqrt(a*f**2 
 + b*f**2*x + e**2*x**2)*d*e + 2*a*e*f**2 - b*d*f**2 + b*e*f**2*x - 2*d*e* 
*2*x)*a**3*e**2*f**6 + log( - sqrt(a*f**2 + b*f**2*x + e**2*x**2)*b*f**2 + 
 2*sqrt(a*f**2 + b*f**2*x + e**2*x**2)*d*e + 2*a*e*f**2 - b*d*f**2 + b*e*f 
**2*x - 2*d*e**2*x)*a**2*b**2*f**8 - 4*log( - sqrt(a*f**2 + b*f**2*x + e** 
2*x**2)*b*f**2 + 2*sqrt(a*f**2 + b*f**2*x + e**2*x**2)*d*e + 2*a*e*f**2 - 
b*d*f**2 + b*e*f**2*x - 2*d*e**2*x)*a**2*b*e**2*f**6*x + 8*log( - sqrt(a*f 
**2 + b*f**2*x + e**2*x**2)*b*f**2 + 2*sqrt(a*f**2 + b*f**2*x + e**2*x**2) 
*d*e + 2*a*e*f**2 - b*d*f**2 + b*e*f**2*x - 2*d*e**2*x)*a**2*d**2*e**2*f** 
4 + 8*log( - sqrt(a*f**2 + b*f**2*x + e**2*x**2)*b*f**2 + 2*sqrt(a*f**2 + 
b*f**2*x + e**2*x**2)*d*e + 2*a*e*f**2 - b*d*f**2 + b*e*f**2*x - 2*d*e*...