\(\int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx\) [506]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2 (d e-c f)^4}{3 d^4 (b c-a d) (c+d x)^{3/2}}+\frac {2 (d e-c f)^3 (b d e+3 b c f-4 a d f)}{d^4 (b c-a d)^2 \sqrt {c+d x}}+\frac {2 f^3 (4 b d e-3 b c f-a d f) \sqrt {c+d x}}{b^2 d^4}+\frac {2 f^4 (c+d x)^{3/2}}{3 b d^4}-\frac {2 (b e-a f)^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{5/2} (b c-a d)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.81 \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx=-\frac {2 \left (3 a^3 d^3 f^4 (c+d x)^2-a^2 b d^2 f^3 (c+d x)^2 (12 d e-2 c f+d f x)+b^3 \left (-4 c d^4 e^4+16 c^5 f^4-3 d^5 e^4 x+8 c^4 d f^3 (-4 e+3 f x)+6 c^3 d^2 f^2 \left (2 e^2-8 e f x+f^2 x^2\right )+c^2 d^3 f \left (4 e^3+18 e^2 f x-12 e f^2 x^2-f^3 x^3\right )\right )+a b^2 d \left (-24 c^4 f^4+4 c^3 d f^3 (14 e-9 f x)+d^4 e^3 (e+12 f x)-3 c^2 d^2 f^2 \left (10 e^2-28 e f x+3 f^2 x^2\right )+2 c d^3 f \left (4 e^3-18 e^2 f x+12 e f^2 x^2+f^3 x^3\right )\right )\right )}{3 b^2 d^4 (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 (b e-a f)^4 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{5/2} (-b c+a d)^{5/2}} \] Input:

Integrate[(e + f*x)^4/((a + b*x)*(c + d*x)^(5/2)),x]
 

Output:

(-2*(3*a^3*d^3*f^4*(c + d*x)^2 - a^2*b*d^2*f^3*(c + d*x)^2*(12*d*e - 2*c*f 
 + d*f*x) + b^3*(-4*c*d^4*e^4 + 16*c^5*f^4 - 3*d^5*e^4*x + 8*c^4*d*f^3*(-4 
*e + 3*f*x) + 6*c^3*d^2*f^2*(2*e^2 - 8*e*f*x + f^2*x^2) + c^2*d^3*f*(4*e^3 
 + 18*e^2*f*x - 12*e*f^2*x^2 - f^3*x^3)) + a*b^2*d*(-24*c^4*f^4 + 4*c^3*d* 
f^3*(14*e - 9*f*x) + d^4*e^3*(e + 12*f*x) - 3*c^2*d^2*f^2*(10*e^2 - 28*e*f 
*x + 3*f^2*x^2) + 2*c*d^3*f*(4*e^3 - 18*e^2*f*x + 12*e*f^2*x^2 + f^3*x^3)) 
))/(3*b^2*d^4*(b*c - a*d)^2*(c + d*x)^(3/2)) + (2*(b*e - a*f)^4*ArcTan[(Sq 
rt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(b^(5/2)*(-(b*c) + a*d)^(5/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {98, 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.

\(\displaystyle \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 98

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

\(\Big \downarrow \) 2009

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

Input:

Int[(e + f*x)^4/((a + b*x)*(c + d*x)^(5/2)),x]
 

Output:

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

Defintions of rubi rules used

rule 98

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x 
_)), x_] :> Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], (c + d*x)^n*(( 
e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] 
&& IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
 

rule 2009

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

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.64

method	result	size
risch	\(-\frac {2 f^{3} \left (-f b d x +3 a d f +8 b c f -12 b d e \right ) \sqrt {x d +c}}{3 d^{4} b^{2}}+\frac {\frac {2 d^{4} \left (a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {2 b^{2} \left (c^{4} f^{4}-4 e \,f^{3} d \,c^{3}+6 d^{2} f^{2} e^{2} c^{2}-4 e^{3} f \,d^{3} c +d^{4} e^{4}\right )}{3 \left (a d -b c \right ) \left (x d +c \right )^{\frac {3}{2}}}+\frac {2 b^{2} \left (4 a \,c^{3} d \,f^{4}-12 a \,c^{2} d^{2} e \,f^{3}+12 a c \,d^{3} e^{2} f^{2}-4 a \,d^{4} e^{3} f -3 b \,c^{4} f^{4}+8 b \,c^{3} d e \,f^{3}-6 b \,c^{2} d^{2} e^{2} f^{2}+b \,d^{4} e^{4}\right )}{\left (a d -b c \right )^{2} \sqrt {x d +c}}}{b^{2} d^{4}}\)	\(330\)
derivativedivides	\(\frac {-\frac {2 f^{3} \left (-\frac {f \left (x d +c \right )^{\frac {3}{2}} b}{3}+a d f \sqrt {x d +c}+3 b c f \sqrt {x d +c}-4 b d e \sqrt {x d +c}\right )}{b^{2}}+\frac {2 d^{4} \left (a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{2} \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {2 \left (c^{4} f^{4}-4 e \,f^{3} d \,c^{3}+6 d^{2} f^{2} e^{2} c^{2}-4 e^{3} f \,d^{3} c +d^{4} e^{4}\right )}{3 \left (a d -b c \right ) \left (x d +c \right )^{\frac {3}{2}}}-\frac {2 \left (-4 a \,c^{3} d \,f^{4}+12 a \,c^{2} d^{2} e \,f^{3}-12 a c \,d^{3} e^{2} f^{2}+4 a \,d^{4} e^{3} f +3 b \,c^{4} f^{4}-8 b \,c^{3} d e \,f^{3}+6 b \,c^{2} d^{2} e^{2} f^{2}-b \,d^{4} e^{4}\right )}{\left (a d -b c \right )^{2} \sqrt {x d +c}}}{d^{4}}\)	\(340\)
default	\(\frac {-\frac {2 f^{3} \left (-\frac {f \left (x d +c \right )^{\frac {3}{2}} b}{3}+a d f \sqrt {x d +c}+3 b c f \sqrt {x d +c}-4 b d e \sqrt {x d +c}\right )}{b^{2}}+\frac {2 d^{4} \left (a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{2} \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {2 \left (c^{4} f^{4}-4 e \,f^{3} d \,c^{3}+6 d^{2} f^{2} e^{2} c^{2}-4 e^{3} f \,d^{3} c +d^{4} e^{4}\right )}{3 \left (a d -b c \right ) \left (x d +c \right )^{\frac {3}{2}}}-\frac {2 \left (-4 a \,c^{3} d \,f^{4}+12 a \,c^{2} d^{2} e \,f^{3}-12 a c \,d^{3} e^{2} f^{2}+4 a \,d^{4} e^{3} f +3 b \,c^{4} f^{4}-8 b \,c^{3} d e \,f^{3}+6 b \,c^{2} d^{2} e^{2} f^{2}-b \,d^{4} e^{4}\right )}{\left (a d -b c \right )^{2} \sqrt {x d +c}}}{d^{4}}\)	\(340\)
pseudoelliptic	\(\frac {2 d^{4} \left (x d +c \right )^{\frac {3}{2}} \left (a f -b e \right )^{4} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-2 \sqrt {\left (a d -b c \right ) b}\, \left (\left (-b^{3} e^{4} x +\frac {a \,e^{3} \left (12 f x +e \right ) b^{2}}{3}-4 a^{2} \left (\frac {f x}{12}+e \right ) x^{2} f^{3} b +a^{3} f^{4} x^{2}\right ) d^{5}+2 \left (-\frac {2 e^{4} b^{3}}{3}+\frac {4 a \left (\frac {1}{4} f^{3} x^{3}+3 x^{2} f^{2} e -\frac {9}{2} e^{2} f x +e^{3}\right ) f \,b^{2}}{3}-4 a^{2} b e \,f^{3} x +a^{3} f^{4} x \right ) c \,d^{4}+c^{2} \left (\left (-\frac {1}{3} f^{3} x^{3}+\frac {4}{3} e^{3}-4 x^{2} f^{2} e +6 e^{2} f x \right ) b^{3}-10 a \left (\frac {3}{10} f^{2} x^{2}-\frac {14}{5} e f x +e^{2}\right ) f \,b^{2}-4 a^{2} f^{2} \left (-\frac {f x}{4}+e \right ) b +a^{3} f^{3}\right ) f \,d^{3}+\frac {2 c^{3} b \,f^{2} \left (3 \left (f^{2} x^{2}-8 e f x +2 e^{2}\right ) b^{2}+28 a \left (-\frac {9 f x}{14}+e \right ) f b +a^{2} f^{2}\right ) d^{2}}{3}-8 \left (\left (-f x +\frac {4 e}{3}\right ) b +a f \right ) c^{4} b^{2} f^{3} d +\frac {16 b^{3} c^{5} f^{4}}{3}\right )}{d^{4} b^{2} \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}\, \left (x d +c \right )^{\frac {3}{2}}}\)	\(403\)

Input:

int((f*x+e)^4/(b*x+a)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3*f^3*(-b*d*f*x+3*a*d*f+8*b*c*f-12*b*d*e)/d^4*(d*x+c)^(1/2)/b^2+2/b^2/d 
^4*(d^4*(a^4*f^4-4*a^3*b*e*f^3+6*a^2*b^2*e^2*f^2-4*a*b^3*e^3*f+b^4*e^4)/(a 
*d-b*c)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))- 
1/3*b^2*(c^4*f^4-4*c^3*d*e*f^3+6*c^2*d^2*e^2*f^2-4*c*d^3*e^3*f+d^4*e^4)/(a 
*d-b*c)/(d*x+c)^(3/2)+b^2*(4*a*c^3*d*f^4-12*a*c^2*d^2*e*f^3+12*a*c*d^3*e^2 
*f^2-4*a*d^4*e^3*f-3*b*c^4*f^4+8*b*c^3*d*e*f^3-6*b*c^2*d^2*e^2*f^2+b*d^4*e 
^4)/(a*d-b*c)^2/(d*x+c)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1064 vs. \(2 (179) = 358\).

Time = 0.13 (sec) , antiderivative size = 2142, normalized size of antiderivative = 10.66 \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^4/(b*x+a)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/3*(3*(b^4*c^2*d^4*e^4 - 4*a*b^3*c^2*d^4*e^3*f + 6*a^2*b^2*c^2*d^4*e^2*f 
^2 - 4*a^3*b*c^2*d^4*e*f^3 + a^4*c^2*d^4*f^4 + (b^4*d^6*e^4 - 4*a*b^3*d^6* 
e^3*f + 6*a^2*b^2*d^6*e^2*f^2 - 4*a^3*b*d^6*e*f^3 + a^4*d^6*f^4)*x^2 + 2*( 
b^4*c*d^5*e^4 - 4*a*b^3*c*d^5*e^3*f + 6*a^2*b^2*c*d^5*e^2*f^2 - 4*a^3*b*c* 
d^5*e*f^3 + a^4*c*d^5*f^4)*x)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d 
 - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + 2*((b^5*c^3*d^3 - 3*a 
*b^4*c^2*d^4 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)*f^4*x^3 + (4*b^5*c^2*d^4 - 5 
*a*b^4*c*d^5 + a^2*b^3*d^6)*e^4 - 4*(b^5*c^3*d^3 + a*b^4*c^2*d^4 - 2*a^2*b 
^3*c*d^5)*e^3*f - 6*(2*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 + 5*a^2*b^3*c^2*d^4)* 
e^2*f^2 + 4*(8*b^5*c^5*d - 22*a*b^4*c^4*d^2 + 17*a^2*b^3*c^3*d^3 - 3*a^3*b 
^2*c^2*d^4)*e*f^3 - (16*b^5*c^6 - 40*a*b^4*c^5*d + 26*a^2*b^3*c^4*d^2 + a^ 
3*b^2*c^3*d^3 - 3*a^4*b*c^2*d^4)*f^4 + 3*(4*(b^5*c^3*d^3 - 3*a*b^4*c^2*d^4 
 + 3*a^2*b^3*c*d^5 - a^3*b^2*d^6)*e*f^3 - (2*b^5*c^4*d^2 - 5*a*b^4*c^3*d^3 
 + 3*a^2*b^3*c^2*d^4 + a^3*b^2*c*d^5 - a^4*b*d^6)*f^4)*x^2 + 3*((b^5*c*d^5 
 - a*b^4*d^6)*e^4 - 4*(a*b^4*c*d^5 - a^2*b^3*d^6)*e^3*f - 6*(b^5*c^3*d^3 - 
 3*a*b^4*c^2*d^4 + 2*a^2*b^3*c*d^5)*e^2*f^2 + 4*(4*b^5*c^4*d^2 - 11*a*b^4* 
c^3*d^3 + 9*a^2*b^3*c^2*d^4 - 2*a^3*b^2*c*d^5)*e*f^3 - (8*b^5*c^5*d - 20*a 
*b^4*c^4*d^2 + 13*a^2*b^3*c^3*d^3 + a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5)*f^4)* 
x)*sqrt(d*x + c))/(b^6*c^5*d^4 - 3*a*b^5*c^4*d^5 + 3*a^2*b^4*c^3*d^6 - a^3 
*b^3*c^2*d^7 + (b^6*c^3*d^6 - 3*a*b^5*c^2*d^7 + 3*a^2*b^4*c*d^8 - a^3*b...
 

Sympy [A] (verification not implemented)

Time = 21.20 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.73 \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {\left (c f - d e\right )^{3} \cdot \left (4 a d f - 3 b c f - b d e\right )}{d^{3} \sqrt {c + d x} \left (a d - b c\right )^{2}} - \frac {\left (c f - d e\right )^{4}}{3 d^{3} \left (c + d x\right )^{\frac {3}{2}} \left (a d - b c\right )} + \frac {f^{4} \left (c + d x\right )^{\frac {3}{2}}}{3 b d^{3}} + \frac {\sqrt {c + d x} \left (- a d f^{4} - 3 b c f^{4} + 4 b d e f^{3}\right )}{b^{2} d^{3}} + \frac {d \left (a f - b e\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{3} \sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )^{2}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\frac {f^{4} x^{4}}{4 b} + \frac {x^{3} \left (- a f^{4} + 4 b e f^{3}\right )}{3 b^{2}} + \frac {x^{2} \left (a^{2} f^{4} - 4 a b e f^{3} + 6 b^{2} e^{2} f^{2}\right )}{2 b^{3}} + \frac {x \left (- a^{3} f^{4} + 4 a^{2} b e f^{3} - 6 a b^{2} e^{2} f^{2} + 4 b^{3} e^{3} f\right )}{b^{4}} + \frac {\left (a f - b e\right )^{4} \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{4}}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:

integrate((f*x+e)**4/(b*x+a)/(d*x+c)**(5/2),x)
 

Output:

Piecewise((2*((c*f - d*e)**3*(4*a*d*f - 3*b*c*f - b*d*e)/(d**3*sqrt(c + d* 
x)*(a*d - b*c)**2) - (c*f - d*e)**4/(3*d**3*(c + d*x)**(3/2)*(a*d - b*c)) 
+ f**4*(c + d*x)**(3/2)/(3*b*d**3) + sqrt(c + d*x)*(-a*d*f**4 - 3*b*c*f**4 
 + 4*b*d*e*f**3)/(b**2*d**3) + d*(a*f - b*e)**4*atan(sqrt(c + d*x)/sqrt((a 
*d - b*c)/b))/(b**3*sqrt((a*d - b*c)/b)*(a*d - b*c)**2))/d, Ne(d, 0)), ((f 
**4*x**4/(4*b) + x**3*(-a*f**4 + 4*b*e*f**3)/(3*b**2) + x**2*(a**2*f**4 - 
4*a*b*e*f**3 + 6*b**2*e**2*f**2)/(2*b**3) + x*(-a**3*f**4 + 4*a**2*b*e*f** 
3 - 6*a*b**2*e**2*f**2 + 4*b**3*e**3*f)/b**4 + (a*f - b*e)**4*Piecewise((x 
/a, Eq(b, 0)), (log(a + b*x)/b, True))/b**4)/c**(5/2), True))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((f*x+e)^4/(b*x+a)/(d*x+c)^(5/2),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(a*d-b*c>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. 479 vs. \(2 (179) = 358\).

Time = 0.14 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.38 \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (b^{4} e^{4} - 4 \, a b^{3} e^{3} f + 6 \, a^{2} b^{2} e^{2} f^{2} - 4 \, a^{3} b e f^{3} + a^{4} f^{4}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )} b d^{4} e^{4} + b c d^{4} e^{4} - a d^{5} e^{4} - 4 \, b c^{2} d^{3} e^{3} f - 12 \, {\left (d x + c\right )} a d^{4} e^{3} f + 4 \, a c d^{4} e^{3} f - 18 \, {\left (d x + c\right )} b c^{2} d^{2} e^{2} f^{2} + 6 \, b c^{3} d^{2} e^{2} f^{2} + 36 \, {\left (d x + c\right )} a c d^{3} e^{2} f^{2} - 6 \, a c^{2} d^{3} e^{2} f^{2} + 24 \, {\left (d x + c\right )} b c^{3} d e f^{3} - 4 \, b c^{4} d e f^{3} - 36 \, {\left (d x + c\right )} a c^{2} d^{2} e f^{3} + 4 \, a c^{3} d^{2} e f^{3} - 9 \, {\left (d x + c\right )} b c^{4} f^{4} + b c^{5} f^{4} + 12 \, {\left (d x + c\right )} a c^{3} d f^{4} - a c^{4} d f^{4}\right )}}{3 \, {\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (12 \, \sqrt {d x + c} b^{2} d^{9} e f^{3} + {\left (d x + c\right )}^{\frac {3}{2}} b^{2} d^{8} f^{4} - 9 \, \sqrt {d x + c} b^{2} c d^{8} f^{4} - 3 \, \sqrt {d x + c} a b d^{9} f^{4}\right )}}{3 \, b^{3} d^{12}} \] Input:

integrate((f*x+e)^4/(b*x+a)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

2*(b^4*e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)* 
arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^4*c^2 - 2*a*b^3*c*d + a^2 
*b^2*d^2)*sqrt(-b^2*c + a*b*d)) + 2/3*(3*(d*x + c)*b*d^4*e^4 + b*c*d^4*e^4 
 - a*d^5*e^4 - 4*b*c^2*d^3*e^3*f - 12*(d*x + c)*a*d^4*e^3*f + 4*a*c*d^4*e^ 
3*f - 18*(d*x + c)*b*c^2*d^2*e^2*f^2 + 6*b*c^3*d^2*e^2*f^2 + 36*(d*x + c)* 
a*c*d^3*e^2*f^2 - 6*a*c^2*d^3*e^2*f^2 + 24*(d*x + c)*b*c^3*d*e*f^3 - 4*b*c 
^4*d*e*f^3 - 36*(d*x + c)*a*c^2*d^2*e*f^3 + 4*a*c^3*d^2*e*f^3 - 9*(d*x + c 
)*b*c^4*f^4 + b*c^5*f^4 + 12*(d*x + c)*a*c^3*d*f^4 - a*c^4*d*f^4)/((b^2*c^ 
2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*(d*x + c)^(3/2)) + 2/3*(12*sqrt(d*x + c)*b^ 
2*d^9*e*f^3 + (d*x + c)^(3/2)*b^2*d^8*f^4 - 9*sqrt(d*x + c)*b^2*c*d^8*f^4 
- 3*sqrt(d*x + c)*a*b*d^9*f^4)/(b^3*d^12)
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.15 \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2\,f^4\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^4}-\left (\frac {8\,c\,f^4-8\,d\,e\,f^3}{b\,d^4}+\frac {2\,f^4\,\left (a\,d^5-b\,c\,d^4\right )}{b^2\,d^8}\right )\,\sqrt {c+d\,x}+\frac {2\,\mathrm {atan}\left (\frac {2\,{\left (a\,f-b\,e\right )}^4\,\sqrt {c+d\,x}\,\left (a^2\,b^2\,d^2-2\,a\,b^3\,c\,d+b^4\,c^2\right )}{b^{3/2}\,{\left (a\,d-b\,c\right )}^{5/2}\,\left (2\,a^4\,f^4-8\,a^3\,b\,e\,f^3+12\,a^2\,b^2\,e^2\,f^2-8\,a\,b^3\,e^3\,f+2\,b^4\,e^4\right )}\right )\,{\left (a\,f-b\,e\right )}^4}{b^{5/2}\,{\left (a\,d-b\,c\right )}^{5/2}}-\frac {\frac {2\,\left (b^2\,c^4\,f^4-4\,b^2\,c^3\,d\,e\,f^3+6\,b^2\,c^2\,d^2\,e^2\,f^2-4\,b^2\,c\,d^3\,e^3\,f+b^2\,d^4\,e^4\right )}{3\,\left (a\,d-b\,c\right )}+\frac {2\,\left (c+d\,x\right )\,\left (3\,b^3\,c^4\,f^4-8\,b^3\,c^3\,d\,e\,f^3+6\,b^3\,c^2\,d^2\,e^2\,f^2-b^3\,d^4\,e^4-4\,a\,b^2\,c^3\,d\,f^4+12\,a\,b^2\,c^2\,d^2\,e\,f^3-12\,a\,b^2\,c\,d^3\,e^2\,f^2+4\,a\,b^2\,d^4\,e^3\,f\right )}{{\left (a\,d-b\,c\right )}^2}}{b^2\,d^4\,{\left (c+d\,x\right )}^{3/2}} \] Input:

int((e + f*x)^4/((a + b*x)*(c + d*x)^(5/2)),x)
 

Output:

(2*f^4*(c + d*x)^(3/2))/(3*b*d^4) - ((8*c*f^4 - 8*d*e*f^3)/(b*d^4) + (2*f^ 
4*(a*d^5 - b*c*d^4))/(b^2*d^8))*(c + d*x)^(1/2) + (2*atan((2*(a*f - b*e)^4 
*(c + d*x)^(1/2)*(b^4*c^2 + a^2*b^2*d^2 - 2*a*b^3*c*d))/(b^(3/2)*(a*d - b* 
c)^(5/2)*(2*a^4*f^4 + 2*b^4*e^4 + 12*a^2*b^2*e^2*f^2 - 8*a*b^3*e^3*f - 8*a 
^3*b*e*f^3)))*(a*f - b*e)^4)/(b^(5/2)*(a*d - b*c)^(5/2)) - ((2*(b^2*c^4*f^ 
4 + b^2*d^4*e^4 + 6*b^2*c^2*d^2*e^2*f^2 - 4*b^2*c*d^3*e^3*f - 4*b^2*c^3*d* 
e*f^3))/(3*(a*d - b*c)) + (2*(c + d*x)*(3*b^3*c^4*f^4 - b^3*d^4*e^4 + 6*b^ 
3*c^2*d^2*e^2*f^2 - 4*a*b^2*c^3*d*f^4 + 4*a*b^2*d^4*e^3*f - 8*b^3*c^3*d*e* 
f^3 - 12*a*b^2*c*d^3*e^2*f^2 + 12*a*b^2*c^2*d^2*e*f^3))/(a*d - b*c)^2)/(b^ 
2*d^4*(c + d*x)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1422, normalized size of antiderivative = 7.07 \[ \int \frac {(e+f x)^4}{(a+b x) (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int((f*x+e)^4/(b*x+a)/(d*x+c)^(5/2),x)
 

Output:

(2*(3*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b 
)*sqrt(a*d - b*c)))*a**4*c*d**4*f**4 + 3*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - 
b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**4*d**5*f**4*x - 
12*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(a*d - b*c)))*a**3*b*c*d**4*e*f**3 - 12*sqrt(b)*sqrt(c + d*x)*sqrt(a*d 
- b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b*d**5*e*f** 
3*x + 18*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqr 
t(b)*sqrt(a*d - b*c)))*a**2*b**2*c*d**4*e**2*f**2 + 18*sqrt(b)*sqrt(c + d* 
x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2* 
b**2*d**5*e**2*f**2*x - 12*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqr 
t(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**3*c*d**4*e**3*f - 12*sqrt(b) 
*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - 
b*c)))*a*b**3*d**5*e**3*f*x + 3*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan 
((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**4*c*d**4*e**4 + 3*sqrt(b) 
*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - 
b*c)))*b**4*d**5*e**4*x - 3*a**4*b*c**2*d**4*f**4 - 6*a**4*b*c*d**5*f**4*x 
 - 3*a**4*b*d**6*f**4*x**2 + a**3*b**2*c**3*d**3*f**4 + 12*a**3*b**2*c**2* 
d**4*e*f**3 + 3*a**3*b**2*c**2*d**4*f**4*x + 24*a**3*b**2*c*d**5*e*f**3*x 
+ 3*a**3*b**2*c*d**5*f**4*x**2 + 12*a**3*b**2*d**6*e*f**3*x**2 + a**3*b**2 
*d**6*f**4*x**3 + 26*a**2*b**3*c**4*d**2*f**4 - 68*a**2*b**3*c**3*d**3*...