\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x) (f+g x)^4} \, dx\) [274]

Optimal result

Integrand size = 44, antiderivative size = 272 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^4} \, dx=-\frac {\left (c d^2-a e^2\right ) \left (c d^2 f+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (e f-d g)^2 (c d f-a e g) (f+g x)^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (e f-d g) (f+g x)^3}+\frac {\left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 (e f-d g)^{5/2} (c d f-a e g)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.79 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^4} \, dx=\frac {1}{24} \sqrt {(a e+c d x) (d+e x)} \left (\frac {2 a c d e \left (d^2 g (f-7 g x)-4 d e (f-g x)^2+e^2 f x (-7 f+g x)\right )+c^2 d^2 \left (-8 e^2 f^2 x^2-2 d e f x (f-7 g x)+d^2 \left (3 f^2+8 f g x-3 g^2 x^2\right )\right )+a^2 e^2 \left (-8 d^2 g^2-2 d e g (-7 f+g x)+e^2 \left (-3 f^2+8 f g x+3 g^2 x^2\right )\right )}{(e f-d g)^2 (-c d f+a e g) (f+g x)^3}-\frac {3 \left (c d^2-a e^2\right )^3 \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{(-e f+d g)^{5/2} (c d f-a e g)^{3/2} \sqrt {a e+c d x} \sqrt {d+e x}}\right ) \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)*(f + g* 
x)^4),x]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((2*a*c*d*e*(d^2*g*(f - 7*g*x) - 4*d*e*(f - 
 g*x)^2 + e^2*f*x*(-7*f + g*x)) + c^2*d^2*(-8*e^2*f^2*x^2 - 2*d*e*f*x*(f - 
 7*g*x) + d^2*(3*f^2 + 8*f*g*x - 3*g^2*x^2)) + a^2*e^2*(-8*d^2*g^2 - 2*d*e 
*g*(-7*f + g*x) + e^2*(-3*f^2 + 8*f*g*x + 3*g^2*x^2)))/((e*f - d*g)^2*(-(c 
*d*f) + a*e*g)*(f + g*x)^3) - (3*(c*d^2 - a*e^2)^3*ArcTan[(Sqrt[c*d*f - a* 
e*g]*Sqrt[d + e*x])/(Sqrt[-(e*f) + d*g]*Sqrt[a*e + c*d*x])])/((-(e*f) + d* 
g)^(5/2)*(c*d*f - a*e*g)^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/24
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1215, 1228, 1152, 1154, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)*(f + g*x)^4), 
x]
 

Output:

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

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(10983\) vs. \(2(252)=504\).

Time = 3.49 (sec) , antiderivative size = 10984, normalized size of antiderivative = 40.38

method	result	size
default	\(\text {Expression too large to display}\)	\(10984\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)/(g*x+f)^4,x,method=_RE 
TURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1684 vs. \(2 (252) = 504\).

Time = 90.33 (sec) , antiderivative size = 3425, normalized size of antiderivative = 12.59 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^4,x, alg 
orithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)/(g*x+f)**4,x 
)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^4,x, alg 
orithm="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*(g*x + 
f)^4), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5573 vs. \(2 (252) = 504\).

Time = 2.40 (sec) , antiderivative size = 5573, normalized size of antiderivative = 20.49 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^4,x, alg 
orithm="giac")
 

Output:

1/8*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*arctan(-((sqrt 
(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*g + sqrt(c*d*e)*f 
)/sqrt(-c*d*e*f^2 + c*d^2*f*g + a*e^2*f*g - a*d*e*g^2))/((c*d*e^2*f^3 - 2* 
c*d^2*e*f^2*g - a*e^3*f^2*g + c*d^3*f*g^2 + 2*a*d*e^2*f*g^2 - a*d^2*e*g^3) 
*sqrt(-c*d*e*f^2 + c*d^2*f*g + a*e^2*f*g - a*d*e*g^2)) + 1/24*(48*(sqrt(c* 
d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*c^5*d^7*e^3*f^5 + 96 
*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a*c^4*d^5*e 
^5*f^5 + 48*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))* 
a^2*c^3*d^3*e^7*f^5 - 84*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2 
*x + a*d*e))*c^5*d^8*e^2*f^4*g - 276*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d 
^2*x + a*e^2*x + a*d*e))*a*c^4*d^6*e^4*f^4*g - 204*(sqrt(c*d*e)*x - sqrt(c 
*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^2*c^3*d^4*e^6*f^4*g - 12*(sqrt(c* 
d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^3*c^2*d^2*e^8*f^4* 
g + 18*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*c^5*d 
^9*e*f^3*g^2 + 252*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a 
*d*e))*a*c^4*d^7*e^3*f^3*g^2 + 336*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2 
*x + a*e^2*x + a*d*e))*a^2*c^3*d^5*e^5*f^3*g^2 + 36*(sqrt(c*d*e)*x - sqrt( 
c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^3*c^2*d^3*e^7*f^3*g^2 - 18*(sqrt 
(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^4*c*d*e^9*f^3*g 
^2 + 3*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*c^...
 

Mupad [F(-1)]

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

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/((f + g*x)^4*(d + e*x)), 
x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/((f + g*x)^4*(d + e*x)), 
 x)
 

Reduce [F]

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

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^4,x)
 

Output:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^4,x)