\(\int \frac {(d+e x)^2 (f+g x)^2}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [290]

Optimal result

Integrand size = 44, antiderivative size = 231 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (c d f-a e g)^2 (d+e x)^2}{3 c^2 d^2 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 (c d f-a e g) \left (2 a e^2 g+c d (e f-3 d g)\right ) (d+e x)}{3 c^2 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 g^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{5/2} d^{5/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \left (-\frac {\sqrt {c} \sqrt {d} (c d f-a e g) (d+e x) \left (-3 a^2 e^3 g+a c d e (-3 e f+5 d g-4 e g x)+c^2 d^2 (-2 e f x+d (f+6 g x))\right )}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {3 g^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {e}}\right )}{3 c^{5/2} d^{5/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[((d + e*x)^2*(f + g*x)^2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
)^(5/2),x]
 

Output:

(2*(-((Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)*(d + e*x)*(-3*a^2*e^3*g + a*c*d*e*( 
-3*e*f + 5*d*g - 4*e*g*x) + c^2*d^2*(-2*e*f*x + d*(f + 6*g*x))))/((c*d^2 - 
 a*e^2)^2*(a*e + c*d*x))) + (3*g^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh 
[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/Sqrt[e]))/( 
3*c^(5/2)*d^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1242, 27, 1211, 27, 1092, 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.

\(\displaystyle \int \frac {(d+e x)^2 (f+g x)^2}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1242

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1092

\(\displaystyle \frac {6 g^2 \left (c d^2-a e^2\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\frac {4 (d+e x) (c d f-a e g) \left (2 a e^2 g-3 c d^2 g+c d e f\right )}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 c^2 d^2 \left (c d^2-a e^2\right )}-\frac {2 (d+e x)^2 (c d f-a e g)^2}{3 c^2 d^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

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

Input:

Int[((d + e*x)^2*(f + g*x)^2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2 
),x]
 

Output:

(-2*(c*d*f - a*e*g)^2*(d + e*x)^2)/(3*c^2*d^2*(c*d^2 - a*e^2)*(a*d*e + (c* 
d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + ((4*(c*d*f - a*e*g)*(c*d*e*f - 3*c*d^ 
2*g + 2*a*e^2*g)*(d + e*x))/((c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)* 
x + c*d*e*x^2]) + (3*(c*d^2 - a*e^2)*g^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e* 
x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) 
])/(Sqrt[c]*Sqrt[d]*Sqrt[e]))/(3*c^2*d^2*(c*d^2 - a*e^2))
 

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1211

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( 
e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* 
x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2))   Int[ExpandToSum[((2*c*d - b 
*e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) 
*(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free 
Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] 
&& IGtQ[n, 0]
 

rule 1242

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x) 
^n, a*e + c*d*x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, 
 Simp[R*(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^ 
2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*( 
a + b*x + c*x^2)^(p + 1)*ExpandToSum[d*e*(p + 1)*(b^2 - 4*a*c)*Q - R*(2*c*d 
 - b*e)*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IG 
tQ[n, 1] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2628\) vs. \(2(209)=418\).

Time = 2.55 (sec) , antiderivative size = 2629, normalized size of antiderivative = 11.38

method	result	size
default	\(\text {Expression too large to display}\)	\(2629\)

Input:

int((e*x+d)^2*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_ 
RETURNVERBOSE)
 

Output:

d^2*f^2*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d* 
e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c*d^2) 
^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))+2*e 
*g*(d*g+e*f)*(-x^2/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+1/2*(a*e^ 
2+c*d^2)/d/e/c*(-1/2*x/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-1/4*( 
a*e^2+c*d^2)/d/e/c*(-1/3/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-1/2 
*(a*e^2+c*d^2)/d/e/c*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c* 
d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2- 
(a*e^2+c*d^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)))+1/2*a/c*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^ 
2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a 
*e^2+c*d^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e) 
^(1/2)))+2*a/c*(-1/3/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-1/2*(a* 
e^2+c*d^2)/d/e/c*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2) 
^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e 
^2+c*d^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^( 
1/2))))+2*d*f*(d*g+e*f)*(-1/3/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2 
)-1/2*(a*e^2+c*d^2)/d/e/c*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e 
^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2 
*e^2-(a*e^2+c*d^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (209) = 418\).

Time = 4.09 (sec) , antiderivative size = 1058, normalized size of antiderivative = 4.58 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^2*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, a 
lgorithm="fricas")
 

Output:

[1/6*(3*((c^4*d^6 - 2*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*g^2*x^2 + 2*(a*c^3* 
d^5*e - 2*a^2*c^2*d^3*e^3 + a^3*c*d*e^5)*g^2*x + (a^2*c^2*d^4*e^2 - 2*a^3* 
c*d^2*e^4 + a^4*e^6)*g^2)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6* 
a*c*d^2*e^2 + a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c 
*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 4*(4* 
a*c^3*d^4*e^2*f*g + (c^4*d^5*e - 3*a*c^3*d^3*e^3)*f^2 - (5*a^2*c^2*d^3*e^3 
 - 3*a^3*c*d*e^5)*g^2 - 2*(c^4*d^4*e^2*f^2 - (3*c^4*d^5*e - a*c^3*d^3*e^3) 
*f*g + (3*a*c^3*d^4*e^2 - 2*a^2*c^2*d^2*e^4)*g^2)*x)*sqrt(c*d*e*x^2 + a*d* 
e + (c*d^2 + a*e^2)*x))/(a^2*c^5*d^7*e^3 - 2*a^3*c^4*d^5*e^5 + a^4*c^3*d^3 
*e^7 + (c^7*d^9*e - 2*a*c^6*d^7*e^3 + a^2*c^5*d^5*e^5)*x^2 + 2*(a*c^6*d^8* 
e^2 - 2*a^2*c^5*d^6*e^4 + a^3*c^4*d^4*e^6)*x), -1/3*(3*((c^4*d^6 - 2*a*c^3 
*d^4*e^2 + a^2*c^2*d^2*e^4)*g^2*x^2 + 2*(a*c^3*d^5*e - 2*a^2*c^2*d^3*e^3 + 
 a^3*c*d*e^5)*g^2*x + (a^2*c^2*d^4*e^2 - 2*a^3*c*d^2*e^4 + a^4*e^6)*g^2)*s 
qrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d* 
e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^ 
3*e + a*c*d*e^3)*x)) + 2*(4*a*c^3*d^4*e^2*f*g + (c^4*d^5*e - 3*a*c^3*d^3*e 
^3)*f^2 - (5*a^2*c^2*d^3*e^3 - 3*a^3*c*d*e^5)*g^2 - 2*(c^4*d^4*e^2*f^2 - ( 
3*c^4*d^5*e - a*c^3*d^3*e^3)*f*g + (3*a*c^3*d^4*e^2 - 2*a^2*c^2*d^2*e^4)*g 
^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*c^5*d^7*e^3 - 2*a 
^3*c^4*d^5*e^5 + a^4*c^3*d^3*e^7 + (c^7*d^9*e - 2*a*c^6*d^7*e^3 + a^2*c...
 

Sympy [F]

\[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2} \left (f + g x\right )^{2}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((e*x+d)**2*(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2 
),x)
 

Output:

Integral((d + e*x)**2*(f + g*x)**2/((d + e*x)*(a*e + c*d*x))**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^2*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, a 
lgorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x+d)^2*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, a 
lgorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{1,[2,2,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[4,4] 
%%%}+%%%{
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,{\left (d+e\,x\right )}^2}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:

int(((f + g*x)^2*(d + e*x)^2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2 
),x)
 

Output:

int(((f + g*x)^2*(d + e*x)^2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 943, normalized size of antiderivative = 4.08 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2*(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

(2*(3*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c* 
d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**3*e**5*g** 
2 - 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c* 
d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d**2*e 
**3*g**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a 
*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c 
*d*e**4*g**2*x + 3*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)* 
sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))* 
a*c**2*d**4*e*g**2 - 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt 
(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d** 
2))*a*c**2*d**3*e**2*g**2*x + 3*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)* 
log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e** 
2 - c*d**2))*c**3*d**5*g**2*x - 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x 
)*a**2*c*d**2*e**3*g**2 + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2 
*c*d*e**4*f*g + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**3*e* 
*2*f*g - 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**2*g**2 
*x - 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**2*e**3*f**2 + 2 
*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**2*e**3*f*g*x + 2*sqrt 
(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**4*e*f*g*x - 2*sqrt(e)*sqrt(d 
)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**3*e**2*f**2*x - 3*sqrt(d + e*x)*a**...