\(\int \frac {(e+f x)^{7/2}}{(a c+(b c+a d) x+b d x^2)^2} \, dx\) [395]

Optimal result

Integrand size = 31, antiderivative size = 343 \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {f \left (3 a^2 d^2 f^2-2 a b d f (d e+2 c f)+b^2 \left (2 d^2 e^2-2 c d e f+3 c^2 f^2\right )\right ) \sqrt {e+f x}}{b^2 d^2 (b c-a d)^2}+\frac {f (2 b d e-b c f-a d f) (e+f x)^{3/2}}{b d (b c-a d)^2}-\frac {(e+f x)^{5/2} (b c e+a d e-2 a c f+(2 b d e-(b c+a d) f) x)}{(b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )}+\frac {(b e-a f)^{5/2} (4 b d e-7 b c f+3 a d f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{5/2} (b c-a d)^3}-\frac {(d e-c f)^{5/2} (4 b d e+3 b c f-7 a d f) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} (b c-a d)^3} \] Output:

f*(3*a^2*d^2*f^2-2*a*b*d*f*(2*c*f+d*e)+b^2*(3*c^2*f^2-2*c*d*e*f+2*d^2*e^2) 
)*(f*x+e)^(1/2)/b^2/d^2/(-a*d+b*c)^2+f*(-a*d*f-b*c*f+2*b*d*e)*(f*x+e)^(3/2 
)/b/d/(-a*d+b*c)^2-(f*x+e)^(5/2)*(b*c*e+a*d*e-2*a*c*f+(2*b*d*e-(a*d+b*c)*f 
)*x)/(-a*d+b*c)^2/(a*c+(a*d+b*c)*x+b*d*x^2)+(-a*f+b*e)^(5/2)*(3*a*d*f-7*b* 
c*f+4*b*d*e)*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(5/2)/(-a*d 
+b*c)^3-(-c*f+d*e)^(5/2)*(-7*a*d*f+3*b*c*f+4*b*d*e)*arctanh(d^(1/2)*(f*x+e 
)^(1/2)/(-c*f+d*e)^(1/2))/d^(5/2)/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 2.74 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.01 \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {\sqrt {e+f x} \left (-3 a^3 d^2 f^3 (c+d x)+a^2 b d f^2 (c+d x) (3 d e+4 c f-2 d f x)+b^3 \left (2 d^3 e^3 x-3 c^3 f^3 x+c d^2 e^2 (e-3 f x)+c^2 d f^2 x (3 e-2 f x)\right )+a b^2 \left (-3 c^3 f^3+d^3 e^2 (e-3 f x)+c^2 d f^2 (3 e+2 f x)+c d^2 \left (-6 e^2 f+4 f^3 x^2\right )\right )\right )}{b^2 d^2 (b c-a d)^2 (a+b x) (c+d x)}+\frac {(-b e+a f)^{5/2} (4 b d e-7 b c f+3 a d f) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{5/2} (b c-a d)^3}+\frac {(-d e+c f)^{5/2} (4 b d e+3 b c f-7 a d f) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{5/2} (-b c+a d)^3} \] Input:

Integrate[(e + f*x)^(7/2)/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
 

Output:

-((Sqrt[e + f*x]*(-3*a^3*d^2*f^3*(c + d*x) + a^2*b*d*f^2*(c + d*x)*(3*d*e 
+ 4*c*f - 2*d*f*x) + b^3*(2*d^3*e^3*x - 3*c^3*f^3*x + c*d^2*e^2*(e - 3*f*x 
) + c^2*d*f^2*x*(3*e - 2*f*x)) + a*b^2*(-3*c^3*f^3 + d^3*e^2*(e - 3*f*x) + 
 c^2*d*f^2*(3*e + 2*f*x) + c*d^2*(-6*e^2*f + 4*f^3*x^2))))/(b^2*d^2*(b*c - 
 a*d)^2*(a + b*x)*(c + d*x))) + ((-(b*e) + a*f)^(5/2)*(4*b*d*e - 7*b*c*f + 
 3*a*d*f)*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(b^(5/2)*(b* 
c - a*d)^3) + ((-(d*e) + c*f)^(5/2)*(4*b*d*e + 3*b*c*f - 7*a*d*f)*ArcTan[( 
Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(5/2)*(-(b*c) + a*d)^3)
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {1164, 27, 25, 1196, 25, 1196, 1197, 27, 1480, 221}

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.

\(\displaystyle \int \frac {(e+f x)^{7/2}}{\left (x (a d+b c)+a c+b d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {(e+f x)^{3/2} \left (4 b d e^2-7 b c f e-7 a d f e+10 a c f^2-3 f (2 b d e-b c f-a d f) x\right )}{2 \left (b d x^2+(b c+a d) x+a c\right )}dx}{(b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -\frac {(e+f x)^{3/2} (a f (7 d e-10 c f)-b e (4 d e-7 c f)+3 f (2 b d e-b c f-a d f) x)}{b d x^2+(b c+a d) x+a c}dx}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(e+f x)^{3/2} (a f (7 d e-10 c f)-b e (4 d e-7 c f)+3 f (2 b d e-b c f-a d f) x)}{b d x^2+(b c+a d) x+a c}dx}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {\frac {\int -\frac {\sqrt {e+f x} \left (3 a c (2 b d e-b c f-a d f) f^2-\left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right ) x f-b d e (a f (7 d e-10 c f)-b e (4 d e-7 c f))\right )}{b d x^2+(b c+a d) x+a c}dx}{b d}+\frac {2 f (e+f x)^{3/2} (-a d f-b c f+2 b d e)}{b d}}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 f (e+f x)^{3/2} (-a d f-b c f+2 b d e)}{b d}-\frac {\int \frac {\sqrt {e+f x} \left (3 a c (2 b d e-b c f-a d f) f^2-\left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right ) x f-b d e (a f (7 d e-10 c f)-b e (4 d e-7 c f))\right )}{b d x^2+(b c+a d) x+a c}dx}{b d}}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {\frac {2 f (e+f x)^{3/2} (-a d f-b c f+2 b d e)}{b d}-\frac {\frac {\int \frac {3 a^3 c d^2 f^4-a^2 b c d (5 d e+4 c f) f^3-a b^2 \left (7 d^3 e^3-18 c d^2 f e^2+5 c^2 d f^2 e-3 c^3 f^3\right ) f-(2 b d e-b c f-a d f) \left (-\left (\left (d^2 e^2-c d f e-3 c^2 f^2\right ) b^2\right )+a d f (d e-7 c f) b+3 a^2 d^2 f^2\right ) x f+b^3 d^2 e^3 (4 d e-7 c f)}{\sqrt {e+f x} \left (b d x^2+(b c+a d) x+a c\right )}dx}{b d}-\frac {2 f \sqrt {e+f x} \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{b d}}{b d}}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {\frac {2 f (e+f x)^{3/2} (-a d f-b c f+2 b d e)}{b d}-\frac {\frac {2 \int \frac {f \left ((b e-a f) (d e-c f) \left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right )-(2 b d e-b c f-a d f) \left (-\left (\left (d^2 e^2-c d f e-3 c^2 f^2\right ) b^2\right )+a d f (d e-7 c f) b+3 a^2 d^2 f^2\right ) (e+f x)\right )}{b d (e+f x)^2-(2 b d e-b c f-a d f) (e+f x)+(b e-a f) (d e-c f)}d\sqrt {e+f x}}{b d}-\frac {2 f \sqrt {e+f x} \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{b d}}{b d}}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 f (e+f x)^{3/2} (-a d f-b c f+2 b d e)}{b d}-\frac {\frac {2 f \int \frac {(b e-a f) (d e-c f) \left (\left (2 d^2 e^2-2 c d f e+3 c^2 f^2\right ) b^2-2 a d f (d e+2 c f) b+3 a^2 d^2 f^2\right )-(2 b d e-b c f-a d f) \left (-\left (\left (d^2 e^2-c d f e-3 c^2 f^2\right ) b^2\right )+a d f (d e-7 c f) b+3 a^2 d^2 f^2\right ) (e+f x)}{b d (e+f x)^2-(2 b d e-b c f-a d f) (e+f x)+(b e-a f) (d e-c f)}d\sqrt {e+f x}}{b d}-\frac {2 f \sqrt {e+f x} \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{b d}}{b d}}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {2 f (e+f x)^{3/2} (-a d f-b c f+2 b d e)}{b d}-\frac {\frac {2 f \left (\frac {d^3 (b e-a f)^3 (3 a d f-7 b c f+4 b d e) \int \frac {1}{b d (e+f x)-d (b e-a f)}d\sqrt {e+f x}}{f (b c-a d)}-\frac {b^3 (d e-c f)^3 (-7 a d f+3 b c f+4 b d e) \int \frac {1}{b d (e+f x)-b (d e-c f)}d\sqrt {e+f x}}{f (b c-a d)}\right )}{b d}-\frac {2 f \sqrt {e+f x} \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{b d}}{b d}}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 f (e+f x)^{3/2} (-a d f-b c f+2 b d e)}{b d}-\frac {\frac {2 f \left (\frac {b^2 (d e-c f)^{5/2} (-7 a d f+3 b c f+4 b d e) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} f (b c-a d)}-\frac {d^2 (b e-a f)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right ) (3 a d f-7 b c f+4 b d e)}{\sqrt {b} f (b c-a d)}\right )}{b d}-\frac {2 f \sqrt {e+f x} \left (3 a^2 d^2 f^2-2 a b d f (2 c f+d e)+b^2 \left (3 c^2 f^2-2 c d e f+2 d^2 e^2\right )\right )}{b d}}{b d}}{2 (b c-a d)^2}-\frac {(e+f x)^{5/2} (x (2 b d e-f (a d+b c))-2 a c f+a d e+b c e)}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}\)

Input:

Int[(e + f*x)^(7/2)/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
 

Output:

-(((e + f*x)^(5/2)*(b*c*e + a*d*e - 2*a*c*f + (2*b*d*e - (b*c + a*d)*f)*x) 
)/((b*c - a*d)^2*(a*c + (b*c + a*d)*x + b*d*x^2))) + ((2*f*(2*b*d*e - b*c* 
f - a*d*f)*(e + f*x)^(3/2))/(b*d) - ((-2*f*(3*a^2*d^2*f^2 - 2*a*b*d*f*(d*e 
 + 2*c*f) + b^2*(2*d^2*e^2 - 2*c*d*e*f + 3*c^2*f^2))*Sqrt[e + f*x])/(b*d) 
+ (2*f*(-((d^2*(b*e - a*f)^(5/2)*(4*b*d*e - 7*b*c*f + 3*a*d*f)*ArcTanh[(Sq 
rt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c - a*d)*f)) + (b^2*(d* 
e - c*f)^(5/2)*(4*b*d*e + 3*b*c*f - 7*a*d*f)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x 
])/Sqrt[d*e - c*f]])/(Sqrt[d]*(b*c - a*d)*f)))/(b*d))/(b*d))/(2*(b*c - a*d 
)^2)
 

Defintions of rubi rules used

rule 25

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

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1164

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 + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1196

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

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 2.19 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {-3 d^{2} \sqrt {\left (c f -d e \right ) d}\, \left (\frac {\left (-7 c f +4 d e \right ) b}{3}+a d f \right ) \left (a f -b e \right )^{3} \left (d x +c \right ) \left (b x +a \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+3 \sqrt {\left (a f -b e \right ) b}\, \left (-\frac {7 b^{2} \left (c f -d e \right )^{3} \left (d x +c \right ) \left (\frac {\left (-3 c f -4 d e \right ) b}{7}+a d f \right ) \left (b x +a \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{3}+\sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\, \left (a d -b c \right ) \left (\left (-\frac {2 d^{3} e^{3} x}{3}-\frac {c \,e^{2} \left (-3 f x +e \right ) d^{2}}{3}-\left (-\frac {2 f x}{3}+e \right ) f^{2} x \,c^{2} d +c^{3} f^{3} x \right ) b^{3}+\left (\left (e^{2} f x -\frac {1}{3} e^{3}\right ) d^{3}+2 f \left (-\frac {2 f^{2} x^{2}}{3}+e^{2}\right ) c \,d^{2}-f^{2} \left (\frac {2 f x}{3}+e \right ) c^{2} d +f^{3} c^{3}\right ) a \,b^{2}-\frac {4 d \,f^{2} \left (\frac {\left (-f x +\frac {3 e}{2}\right ) d}{2}+c f \right ) \left (d x +c \right ) a^{2} b}{3}+a^{3} d^{2} f^{3} \left (d x +c \right )\right )\right )}{\sqrt {\left (c f -d e \right ) d}\, \left (a d -b c \right )^{3} \left (d x +c \right ) \left (b x +a \right ) \sqrt {\left (a f -b e \right ) b}\, d^{2} b^{2}}\)	\(402\)
derivativedivides	\(2 f^{3} \left (\frac {\sqrt {f x +e}}{b^{2} d^{2}}-\frac {\frac {\left (-\frac {1}{2} a \,c^{3} d \,f^{4}+\frac {3}{2} a \,c^{2} d^{2} e \,f^{3}-\frac {3}{2} a c \,d^{3} e^{2} f^{2}+\frac {1}{2} a \,d^{4} e^{3} f +\frac {1}{2} b \,c^{4} f^{4}-\frac {3}{2} b \,c^{3} d e \,f^{3}+\frac {3}{2} b \,c^{2} d^{2} e^{2} f^{2}-\frac {1}{2} b c \,d^{3} e^{3} f \right ) \sqrt {f x +e}}{d \left (f x +e \right )+c f -d e}+\frac {\left (7 a \,c^{3} d \,f^{4}-21 a \,c^{2} d^{2} e \,f^{3}+21 a c \,d^{3} e^{2} f^{2}-7 a \,d^{4} e^{3} f -3 b \,c^{4} f^{4}+5 b \,c^{3} d e \,f^{3}+3 b \,c^{2} d^{2} e^{2} f^{2}-9 b c \,d^{3} e^{3} f +4 b \,d^{4} e^{4}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}}{d^{2} f^{3} \left (a d -b c \right )^{3}}-\frac {\frac {\left (-\frac {1}{2} a^{4} d \,f^{4}+\frac {1}{2} a^{3} b c \,f^{4}+\frac {3}{2} a^{3} b d e \,f^{3}-\frac {3}{2} a^{2} b^{2} c e \,f^{3}-\frac {3}{2} a^{2} b^{2} d \,e^{2} f^{2}+\frac {3}{2} a \,b^{3} c \,e^{2} f^{2}+\frac {1}{2} a \,b^{3} d \,e^{3} f -\frac {1}{2} b^{4} c \,e^{3} f \right ) \sqrt {f x +e}}{b \left (f x +e \right )+a f -b e}+\frac {\left (3 a^{4} d \,f^{4}-7 a^{3} b c \,f^{4}-5 a^{3} b d e \,f^{3}+21 a^{2} b^{2} c e \,f^{3}-3 a^{2} b^{2} d \,e^{2} f^{2}-21 a \,b^{3} c \,e^{2} f^{2}+9 a \,b^{3} d \,e^{3} f +7 b^{4} c \,e^{3} f -4 b^{4} d \,e^{4}\right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}}{b^{2} f^{3} \left (a d -b c \right )^{3}}\right )\)	\(571\)
default	\(2 f^{3} \left (\frac {\sqrt {f x +e}}{b^{2} d^{2}}-\frac {\frac {\left (-\frac {1}{2} a \,c^{3} d \,f^{4}+\frac {3}{2} a \,c^{2} d^{2} e \,f^{3}-\frac {3}{2} a c \,d^{3} e^{2} f^{2}+\frac {1}{2} a \,d^{4} e^{3} f +\frac {1}{2} b \,c^{4} f^{4}-\frac {3}{2} b \,c^{3} d e \,f^{3}+\frac {3}{2} b \,c^{2} d^{2} e^{2} f^{2}-\frac {1}{2} b c \,d^{3} e^{3} f \right ) \sqrt {f x +e}}{d \left (f x +e \right )+c f -d e}+\frac {\left (7 a \,c^{3} d \,f^{4}-21 a \,c^{2} d^{2} e \,f^{3}+21 a c \,d^{3} e^{2} f^{2}-7 a \,d^{4} e^{3} f -3 b \,c^{4} f^{4}+5 b \,c^{3} d e \,f^{3}+3 b \,c^{2} d^{2} e^{2} f^{2}-9 b c \,d^{3} e^{3} f +4 b \,d^{4} e^{4}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}}{d^{2} f^{3} \left (a d -b c \right )^{3}}-\frac {\frac {\left (-\frac {1}{2} a^{4} d \,f^{4}+\frac {1}{2} a^{3} b c \,f^{4}+\frac {3}{2} a^{3} b d e \,f^{3}-\frac {3}{2} a^{2} b^{2} c e \,f^{3}-\frac {3}{2} a^{2} b^{2} d \,e^{2} f^{2}+\frac {3}{2} a \,b^{3} c \,e^{2} f^{2}+\frac {1}{2} a \,b^{3} d \,e^{3} f -\frac {1}{2} b^{4} c \,e^{3} f \right ) \sqrt {f x +e}}{b \left (f x +e \right )+a f -b e}+\frac {\left (3 a^{4} d \,f^{4}-7 a^{3} b c \,f^{4}-5 a^{3} b d e \,f^{3}+21 a^{2} b^{2} c e \,f^{3}-3 a^{2} b^{2} d \,e^{2} f^{2}-21 a \,b^{3} c \,e^{2} f^{2}+9 a \,b^{3} d \,e^{3} f +7 b^{4} c \,e^{3} f -4 b^{4} d \,e^{4}\right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}}{b^{2} f^{3} \left (a d -b c \right )^{3}}\right )\)	\(571\)
risch	\(\frac {2 f^{3} \sqrt {f x +e}}{b^{2} d^{2}}-\frac {2 f^{3} \left (\frac {d^{2} \left (\frac {\left (-\frac {1}{2} a^{4} d \,f^{4}+\frac {1}{2} a^{3} b c \,f^{4}+\frac {3}{2} a^{3} b d e \,f^{3}-\frac {3}{2} a^{2} b^{2} c e \,f^{3}-\frac {3}{2} a^{2} b^{2} d \,e^{2} f^{2}+\frac {3}{2} a \,b^{3} c \,e^{2} f^{2}+\frac {1}{2} a \,b^{3} d \,e^{3} f -\frac {1}{2} b^{4} c \,e^{3} f \right ) \sqrt {f x +e}}{b \left (f x +e \right )+a f -b e}+\frac {\left (3 a^{4} d \,f^{4}-7 a^{3} b c \,f^{4}-5 a^{3} b d e \,f^{3}+21 a^{2} b^{2} c e \,f^{3}-3 a^{2} b^{2} d \,e^{2} f^{2}-21 a \,b^{3} c \,e^{2} f^{2}+9 a \,b^{3} d \,e^{3} f +7 b^{4} c \,e^{3} f -4 b^{4} d \,e^{4}\right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{f^{3} \left (a d -b c \right )^{3}}+\frac {b^{2} \left (\frac {\left (-\frac {1}{2} a \,c^{3} d \,f^{4}+\frac {3}{2} a \,c^{2} d^{2} e \,f^{3}-\frac {3}{2} a c \,d^{3} e^{2} f^{2}+\frac {1}{2} a \,d^{4} e^{3} f +\frac {1}{2} b \,c^{4} f^{4}-\frac {3}{2} b \,c^{3} d e \,f^{3}+\frac {3}{2} b \,c^{2} d^{2} e^{2} f^{2}-\frac {1}{2} b c \,d^{3} e^{3} f \right ) \sqrt {f x +e}}{d \left (f x +e \right )+c f -d e}+\frac {\left (7 a \,c^{3} d \,f^{4}-21 a \,c^{2} d^{2} e \,f^{3}+21 a c \,d^{3} e^{2} f^{2}-7 a \,d^{4} e^{3} f -3 b \,c^{4} f^{4}+5 b \,c^{3} d e \,f^{3}+3 b \,c^{2} d^{2} e^{2} f^{2}-9 b c \,d^{3} e^{3} f +4 b \,d^{4} e^{4}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} \left (a d -b c \right )^{3}}\right )}{b^{2} d^{2}}\)	\(580\)

Input:

int((f*x+e)^(7/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

3*(-d^2*((c*f-d*e)*d)^(1/2)*(1/3*(-7*c*f+4*d*e)*b+a*d*f)*(a*f-b*e)^3*(d*x+ 
c)*(b*x+a)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+((a*f-b*e)*b)^(1/2) 
*(-7/3*b^2*(c*f-d*e)^3*(d*x+c)*(1/7*(-3*c*f-4*d*e)*b+a*d*f)*(b*x+a)*arctan 
(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+((c*f-d*e)*d)^(1/2)*(f*x+e)^(1/2)*(a 
*d-b*c)*((-2/3*d^3*e^3*x-1/3*c*e^2*(-3*f*x+e)*d^2-(-2/3*f*x+e)*f^2*x*c^2*d 
+c^3*f^3*x)*b^3+((e^2*f*x-1/3*e^3)*d^3+2*f*(-2/3*f^2*x^2+e^2)*c*d^2-f^2*(2 
/3*f*x+e)*c^2*d+f^3*c^3)*a*b^2-4/3*d*f^2*(1/2*(-f*x+3/2*e)*d+c*f)*(d*x+c)* 
a^2*b+a^3*d^2*f^3*(d*x+c))))/((c*f-d*e)*d)^(1/2)/((a*f-b*e)*b)^(1/2)/(a*d- 
b*c)^3/(d*x+c)/(b*x+a)/d^2/b^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (317) = 634\).

Time = 56.61 (sec) , antiderivative size = 5115, normalized size of antiderivative = 14.91 \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^(7/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)**(7/2)/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((f*x+e)^(7/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="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(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 870 vs. \(2 (317) = 634\).

Time = 0.52 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.54 \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((f*x+e)^(7/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")
 

Output:

-(4*b^4*d*e^4 - 7*b^4*c*e^3*f - 9*a*b^3*d*e^3*f + 21*a*b^3*c*e^2*f^2 + 3*a 
^2*b^2*d*e^2*f^2 - 21*a^2*b^2*c*e*f^3 + 5*a^3*b*d*e*f^3 + 7*a^3*b*c*f^4 - 
3*a^4*d*f^4)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^5*c^3 - 3*a* 
b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*sqrt(-b^2*e + a*b*f)) + (4*b*d^ 
4*e^4 - 9*b*c*d^3*e^3*f - 7*a*d^4*e^3*f + 3*b*c^2*d^2*e^2*f^2 + 21*a*c*d^3 
*e^2*f^2 + 5*b*c^3*d*e*f^3 - 21*a*c^2*d^2*e*f^3 - 3*b*c^4*f^4 + 7*a*c^3*d* 
f^4)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^3*c^3*d^2 - 3*a*b^2* 
c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*sqrt(-d^2*e + c*d*f)) + 2*sqrt(f*x + e) 
*f^3/(b^2*d^2) - (2*(f*x + e)^(3/2)*b^3*d^3*e^3*f - 2*sqrt(f*x + e)*b^3*d^ 
3*e^4*f - 3*(f*x + e)^(3/2)*b^3*c*d^2*e^2*f^2 - 3*(f*x + e)^(3/2)*a*b^2*d^ 
3*e^2*f^2 + 4*sqrt(f*x + e)*b^3*c*d^2*e^3*f^2 + 4*sqrt(f*x + e)*a*b^2*d^3* 
e^3*f^2 + 3*(f*x + e)^(3/2)*b^3*c^2*d*e*f^3 + 3*(f*x + e)^(3/2)*a^2*b*d^3* 
e*f^3 - 3*sqrt(f*x + e)*b^3*c^2*d*e^2*f^3 - 6*sqrt(f*x + e)*a*b^2*c*d^2*e^ 
2*f^3 - 3*sqrt(f*x + e)*a^2*b*d^3*e^2*f^3 - (f*x + e)^(3/2)*b^3*c^3*f^4 - 
(f*x + e)^(3/2)*a^3*d^3*f^4 + sqrt(f*x + e)*b^3*c^3*e*f^4 + 3*sqrt(f*x + e 
)*a*b^2*c^2*d*e*f^4 + 3*sqrt(f*x + e)*a^2*b*c*d^2*e*f^4 + sqrt(f*x + e)*a^ 
3*d^3*e*f^4 - sqrt(f*x + e)*a*b^2*c^3*f^5 - sqrt(f*x + e)*a^3*c*d^2*f^5)/( 
(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*((f*x + e)^2*b*d - 2*(f*x + e) 
*b*d*e + b*d*e^2 + (f*x + e)*b*c*f + (f*x + e)*a*d*f - b*c*e*f - a*d*e*f + 
 a*c*f^2))
 

Mupad [B] (verification not implemented)

Time = 9.23 (sec) , antiderivative size = 28140, normalized size of antiderivative = 82.04 \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int((e + f*x)^(7/2)/(a*c + x*(a*d + b*c) + b*d*x^2)^2,x)
 

Output:

(((e + f*x)^(3/2)*(a^3*d^3*f^4 + b^3*c^3*f^4 - 2*b^3*d^3*e^3*f - 3*a^2*b*d 
^3*e*f^3 - 3*b^3*c^2*d*e*f^3 + 3*a*b^2*d^3*e^2*f^2 + 3*b^3*c*d^2*e^2*f^2)) 
/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + ((e + f*x)^(1/2)*(a*b^2*c^3*f^5 + a^3*c 
*d^2*f^5 - a^3*d^3*e*f^4 - b^3*c^3*e*f^4 + 2*b^3*d^3*e^4*f - 4*a*b^2*d^3*e 
^3*f^2 + 3*a^2*b*d^3*e^2*f^3 - 4*b^3*c*d^2*e^3*f^2 + 3*b^3*c^2*d*e^2*f^3 + 
 6*a*b^2*c*d^2*e^2*f^3 - 3*a*b^2*c^2*d*e*f^4 - 3*a^2*b*c*d^2*e*f^4))/(a^2* 
d^2 + b^2*c^2 - 2*a*b*c*d))/((e + f*x)*(a*b^2*d^3*f - 2*b^3*d^3*e + b^3*c* 
d^2*f) + b^3*d^3*(e + f*x)^2 + b^3*d^3*e^2 - a*b^2*d^3*e*f - b^3*c*d^2*e*f 
 + a*b^2*c*d^2*f^2) - atan(((((288*a^3*b^9*c^7*d^5*f^7 - 88*a^2*b^10*c^8*d 
^4*f^7 - 552*a^4*b^8*c^6*d^6*f^7 + 680*a^5*b^7*c^5*d^7*f^7 - 552*a^6*b^6*c 
^4*d^8*f^7 + 288*a^7*b^5*c^3*d^9*f^7 - 88*a^8*b^4*c^2*d^10*f^7 + 8*a^6*b^6 
*d^12*e^4*f^3 - 16*a^7*b^5*d^12*e^3*f^4 + 20*a^8*b^4*d^12*e^2*f^5 + 8*b^12 
*c^6*d^6*e^4*f^3 - 16*b^12*c^7*d^5*e^3*f^4 + 20*b^12*c^8*d^4*e^2*f^5 + 12* 
a*b^11*c^9*d^3*f^7 + 12*a^9*b^3*c*d^11*f^7 - 12*a^9*b^3*d^12*e*f^6 - 12*b^ 
12*c^9*d^3*e*f^6 + 68*a*b^11*c^8*d^4*e*f^6 + 68*a^8*b^4*c*d^11*e*f^6 - 48* 
a*b^11*c^5*d^7*e^4*f^3 + 80*a*b^11*c^6*d^6*e^3*f^4 - 112*a*b^11*c^7*d^5*e^ 
2*f^5 - 160*a^2*b^10*c^7*d^5*e*f^6 + 192*a^3*b^9*c^6*d^6*e*f^6 - 88*a^4*b^ 
8*c^5*d^7*e*f^6 - 48*a^5*b^7*c*d^11*e^4*f^3 - 88*a^5*b^7*c^4*d^8*e*f^6 + 8 
0*a^6*b^6*c*d^11*e^3*f^4 + 192*a^6*b^6*c^3*d^9*e*f^6 - 112*a^7*b^5*c*d^11* 
e^2*f^5 - 160*a^7*b^5*c^2*d^10*e*f^6 + 120*a^2*b^10*c^4*d^8*e^4*f^3 - 1...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 3219, normalized size of antiderivative = 9.38 \[ \int \frac {(e+f x)^{7/2}}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((f*x+e)^(7/2)/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)
 

Output:

( - 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b 
*e)))*a**4*c*d**4*f**3 - 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/ 
(sqrt(b)*sqrt(a*f - b*e)))*a**4*d**5*f**3*x + 7*sqrt(b)*sqrt(a*f - b*e)*at 
an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c**2*d**3*f**3 + 2* 
sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
a**3*b*c*d**4*e*f**2 + 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(s 
qrt(b)*sqrt(a*f - b*e)))*a**3*b*c*d**4*f**3*x + 2*sqrt(b)*sqrt(a*f - b*e)* 
atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*d**5*e*f**2*x - 3 
*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) 
*a**3*b*d**5*f**3*x**2 - 14*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) 
/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c**2*d**3*e*f**2 + 7*sqrt(b)*sqrt(a* 
f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c**2* 
d**3*f**3*x + 5*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sq 
rt(a*f - b*e)))*a**2*b**2*c*d**4*e**2*f - 12*sqrt(b)*sqrt(a*f - b*e)*atan( 
(sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c*d**4*e*f**2*x + 7 
*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) 
*a**2*b**2*c*d**4*f**3*x**2 + 5*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d**5*e**2*f*x + 2*sqrt(b)*sqrt(a 
*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d**5 
*e*f**2*x**2 + 7*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b...