\(\int \frac {(d+e x)^4}{(b x+c x^2)^{7/2}} \, dx\) [178]

Optimal result

Integrand size = 21, antiderivative size = 269 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {32 (c d-b e)^2 (2 c d-b e) x^3 (d+e x)^2}{5 b^4 d \left (b x+c x^2\right )^{5/2}}-\frac {16 (c d-b e) (2 c d-b e) x^2 (d+e x)^3}{3 b^3 d \left (b x+c x^2\right )^{5/2}}+\frac {2 (2 c d-b e) x (d+e x)^4}{3 b^2 d \left (b x+c x^2\right )^{5/2}}-\frac {2 (d+e x)^5}{5 b d \left (b x+c x^2\right )^{5/2}}-\frac {128 (c d-b e)^3 (2 c d-b e) x^2}{15 b^5 c \left (b x+c x^2\right )^{3/2}}-\frac {128 (c d-b e)^2 (2 c d-b e) (2 c d+b e) x}{15 b^6 c \sqrt {b x+c x^2}} \] Output:

-32/5*(-b*e+c*d)^2*(-b*e+2*c*d)*x^3*(e*x+d)^2/b^4/d/(c*x^2+b*x)^(5/2)-16/3 
*(-b*e+c*d)*(-b*e+2*c*d)*x^2*(e*x+d)^3/b^3/d/(c*x^2+b*x)^(5/2)+2/3*(-b*e+2 
*c*d)*x*(e*x+d)^4/b^2/d/(c*x^2+b*x)^(5/2)-2/5*(e*x+d)^5/b/d/(c*x^2+b*x)^(5 
/2)-128/15*(-b*e+c*d)^3*(-b*e+2*c*d)*x^2/b^5/c/(c*x^2+b*x)^(3/2)-128/15*(- 
b*e+c*d)^2*(-b*e+2*c*d)*(b*e+2*c*d)*x/b^6/c/(c*x^2+b*x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx=\frac {-512 c^5 d^4 x^5+256 b c^4 d^3 x^4 (-5 d+4 e x)-64 b^2 c^3 d^2 x^3 \left (15 d^2-40 d e x+9 e^2 x^2\right )+32 b^3 c^2 d x^2 \left (-5 d^3+60 d^2 e x-45 d e^2 x^2+2 e^3 x^3\right )-2 b^5 \left (3 d^4+20 d^3 e x+90 d^2 e^2 x^2-60 d e^3 x^3-5 e^4 x^4\right )+4 b^4 c x \left (5 d^4+80 d^3 e x-270 d^2 e^2 x^2+40 d e^3 x^3+e^4 x^4\right )}{15 b^6 (x (b+c x))^{5/2}} \] Input:

Integrate[(d + e*x)^4/(b*x + c*x^2)^(7/2),x]
 

Output:

(-512*c^5*d^4*x^5 + 256*b*c^4*d^3*x^4*(-5*d + 4*e*x) - 64*b^2*c^3*d^2*x^3* 
(15*d^2 - 40*d*e*x + 9*e^2*x^2) + 32*b^3*c^2*d*x^2*(-5*d^3 + 60*d^2*e*x - 
45*d*e^2*x^2 + 2*e^3*x^3) - 2*b^5*(3*d^4 + 20*d^3*e*x + 90*d^2*e^2*x^2 - 6 
0*d*e^3*x^3 - 5*e^4*x^4) + 4*b^4*c*x*(5*d^4 + 80*d^3*e*x - 270*d^2*e^2*x^2 
 + 40*d*e^3*x^3 + e^4*x^4))/(15*b^6*(x*(b + c*x))^(5/2))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.51, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1156, 1153, 1158}

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.

\(\displaystyle \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1156

\(\displaystyle -\frac {8 (2 c d-b e) \int \frac {(d+e x)^3}{\left (c x^2+b x\right )^{5/2}}dx}{5 b^2}-\frac {2 (b+2 c x) (d+e x)^4}{5 b^2 \left (b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1153

\(\displaystyle -\frac {8 (2 c d-b e) \left (-\frac {8 d (c d-b e) \int \frac {d+e x}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2}-\frac {2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\right )}{5 b^2}-\frac {2 (b+2 c x) (d+e x)^4}{5 b^2 \left (b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1158

\(\displaystyle -\frac {2 (b+2 c x) (d+e x)^4}{5 b^2 \left (b x+c x^2\right )^{5/2}}-\frac {8 (2 c d-b e) \left (\frac {16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\right )}{5 b^2}\)

Input:

Int[(d + e*x)^4/(b*x + c*x^2)^(7/2),x]
 

Output:

(-2*(b + 2*c*x)*(d + e*x)^4)/(5*b^2*(b*x + c*x^2)^(5/2)) - (8*(2*c*d - b*e 
)*((-2*(d + e*x)^2*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + 
(16*d*(c*d - b*e)*(b*d + (2*c*d - b*e)*x))/(3*b^4*Sqrt[b*x + c*x^2])))/(5* 
b^2)
 

Defintions of rubi rules used

rule 1153

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[2*(2*p + 3)*((c*d^2 - 
b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c)))   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] 
&& LtQ[p, -1]
 

rule 1156

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^m*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* 
(b^2 - 4*a*c))), x] + Simp[m*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a*c)))   Int[ 
(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
 

rule 1158

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x 
+ c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {2 d^{2} \left (c x +b \right ) \left (90 b^{2} e^{2} x^{2}-220 b c d e \,x^{2}+128 d^{2} c^{2} x^{2}+20 b^{2} d e x -19 x b c \,d^{2}+3 b^{2} d^{2}\right )}{15 b^{6} x^{2} \sqrt {x \left (c x +b \right )}}+\frac {2 x \left (2 b^{2} c \,e^{2} x^{2}+36 b \,c^{2} d e \,x^{2}-128 c^{3} d^{2} x^{2}+5 x \,b^{3} e^{2}+90 x \,b^{2} c d e -275 x b \,c^{2} d^{2}+60 b^{3} d e -150 b^{2} c \,d^{2}\right ) \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}{15 \sqrt {x \left (c x +b \right )}\, \left (c^{2} x^{2}+2 c b x +b^{2}\right ) b^{6}}\)	\(216\)
pseudoelliptic	\(\frac {\left (10 e^{4} x^{4}+120 d \,e^{3} x^{3}-180 d^{2} e^{2} x^{2}-40 d^{3} e x -6 d^{4}\right ) b^{5}+20 x \left (\frac {1}{5} e^{4} x^{4}+8 d \,e^{3} x^{3}-54 d^{2} e^{2} x^{2}+16 d^{3} e x +d^{4}\right ) c \,b^{4}-160 \left (-\frac {2}{5} e^{3} x^{3}+9 d \,e^{2} x^{2}-12 d^{2} e x +d^{3}\right ) x^{2} c^{2} d \,b^{3}-960 x^{3} c^{3} \left (\frac {3}{5} e^{2} x^{2}-\frac {8}{3} d e x +d^{2}\right ) d^{2} b^{2}-1280 x^{4} \left (-\frac {4 e x}{5}+d \right ) c^{4} d^{3} b -512 c^{5} d^{4} x^{5}}{15 \sqrt {x \left (c x +b \right )}\, x^{2} \left (c x +b \right )^{2} b^{6}}\)	\(217\)
gosper	\(-\frac {2 x \left (c x +b \right ) \left (-2 b^{4} c \,e^{4} x^{5}-32 b^{3} c^{2} d \,e^{3} x^{5}+288 b^{2} c^{3} d^{2} e^{2} x^{5}-512 b \,c^{4} d^{3} e \,x^{5}+256 c^{5} d^{4} x^{5}-5 b^{5} e^{4} x^{4}-80 b^{4} c d \,e^{3} x^{4}+720 b^{3} c^{2} d^{2} e^{2} x^{4}-1280 b^{2} c^{3} d^{3} e \,x^{4}+640 b \,c^{4} d^{4} x^{4}-60 b^{5} d \,e^{3} x^{3}+540 b^{4} c \,d^{2} e^{2} x^{3}-960 b^{3} c^{2} d^{3} e \,x^{3}+480 b^{2} c^{3} d^{4} x^{3}+90 b^{5} d^{2} e^{2} x^{2}-160 b^{4} c \,d^{3} e \,x^{2}+80 b^{3} c^{2} d^{4} x^{2}+20 b^{5} d^{3} e x -10 b^{4} c \,d^{4} x +3 d^{4} b^{5}\right )}{15 b^{6} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}\)	\(285\)
orering	\(-\frac {2 x \left (c x +b \right ) \left (-2 b^{4} c \,e^{4} x^{5}-32 b^{3} c^{2} d \,e^{3} x^{5}+288 b^{2} c^{3} d^{2} e^{2} x^{5}-512 b \,c^{4} d^{3} e \,x^{5}+256 c^{5} d^{4} x^{5}-5 b^{5} e^{4} x^{4}-80 b^{4} c d \,e^{3} x^{4}+720 b^{3} c^{2} d^{2} e^{2} x^{4}-1280 b^{2} c^{3} d^{3} e \,x^{4}+640 b \,c^{4} d^{4} x^{4}-60 b^{5} d \,e^{3} x^{3}+540 b^{4} c \,d^{2} e^{2} x^{3}-960 b^{3} c^{2} d^{3} e \,x^{3}+480 b^{2} c^{3} d^{4} x^{3}+90 b^{5} d^{2} e^{2} x^{2}-160 b^{4} c \,d^{3} e \,x^{2}+80 b^{3} c^{2} d^{4} x^{2}+20 b^{5} d^{3} e x -10 b^{4} c \,d^{4} x +3 d^{4} b^{5}\right )}{15 b^{6} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}\)	\(285\)
trager	\(-\frac {2 \left (-2 b^{4} c \,e^{4} x^{5}-32 b^{3} c^{2} d \,e^{3} x^{5}+288 b^{2} c^{3} d^{2} e^{2} x^{5}-512 b \,c^{4} d^{3} e \,x^{5}+256 c^{5} d^{4} x^{5}-5 b^{5} e^{4} x^{4}-80 b^{4} c d \,e^{3} x^{4}+720 b^{3} c^{2} d^{2} e^{2} x^{4}-1280 b^{2} c^{3} d^{3} e \,x^{4}+640 b \,c^{4} d^{4} x^{4}-60 b^{5} d \,e^{3} x^{3}+540 b^{4} c \,d^{2} e^{2} x^{3}-960 b^{3} c^{2} d^{3} e \,x^{3}+480 b^{2} c^{3} d^{4} x^{3}+90 b^{5} d^{2} e^{2} x^{2}-160 b^{4} c \,d^{3} e \,x^{2}+80 b^{3} c^{2} d^{4} x^{2}+20 b^{5} d^{3} e x -10 b^{4} c \,d^{4} x +3 d^{4} b^{5}\right ) \sqrt {c \,x^{2}+b x}}{15 b^{6} \left (c x +b \right )^{3} x^{3}}\)	\(289\)
default	\(d^{4} \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )+e^{4} \left (-\frac {x^{3}}{2 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}+\frac {b \left (-\frac {x^{2}}{3 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {x}{4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {1}{5 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{2 c}\right )}{8 c}\right )}{6 c}\right )}{4 c}\right )+4 d \,e^{3} \left (-\frac {x^{2}}{3 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {x}{4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {1}{5 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{2 c}\right )}{8 c}\right )}{6 c}\right )+6 d^{2} e^{2} \left (-\frac {x}{4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {1}{5 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{2 c}\right )}{8 c}\right )+4 d^{3} e \left (-\frac {1}{5 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{2 c}\right )\)	\(647\)

Input:

int((e*x+d)^4/(c*x^2+b*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-2/15*d^2*(c*x+b)*(90*b^2*e^2*x^2-220*b*c*d*e*x^2+128*c^2*d^2*x^2+20*b^2*d 
*e*x-19*b*c*d^2*x+3*b^2*d^2)/b^6/x^2/(x*(c*x+b))^(1/2)+2/15*x*(2*b^2*c*e^2 
*x^2+36*b*c^2*d*e*x^2-128*c^3*d^2*x^2+5*b^3*e^2*x+90*b^2*c*d*e*x-275*b*c^2 
*d^2*x+60*b^3*d*e-150*b^2*c*d^2)*(b^2*e^2-2*b*c*d*e+c^2*d^2)/(x*(c*x+b))^( 
1/2)/(c^2*x^2+2*b*c*x+b^2)/b^6
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (3 \, b^{5} d^{4} + 2 \, {\left (128 \, c^{5} d^{4} - 256 \, b c^{4} d^{3} e + 144 \, b^{2} c^{3} d^{2} e^{2} - 16 \, b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x^{5} + 5 \, {\left (128 \, b c^{4} d^{4} - 256 \, b^{2} c^{3} d^{3} e + 144 \, b^{3} c^{2} d^{2} e^{2} - 16 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x^{4} + 60 \, {\left (8 \, b^{2} c^{3} d^{4} - 16 \, b^{3} c^{2} d^{3} e + 9 \, b^{4} c d^{2} e^{2} - b^{5} d e^{3}\right )} x^{3} + 10 \, {\left (8 \, b^{3} c^{2} d^{4} - 16 \, b^{4} c d^{3} e + 9 \, b^{5} d^{2} e^{2}\right )} x^{2} - 10 \, {\left (b^{4} c d^{4} - 2 \, b^{5} d^{3} e\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, {\left (b^{6} c^{3} x^{6} + 3 \, b^{7} c^{2} x^{5} + 3 \, b^{8} c x^{4} + b^{9} x^{3}\right )}} \] Input:

integrate((e*x+d)^4/(c*x^2+b*x)^(7/2),x, algorithm="fricas")
 

Output:

-2/15*(3*b^5*d^4 + 2*(128*c^5*d^4 - 256*b*c^4*d^3*e + 144*b^2*c^3*d^2*e^2 
- 16*b^3*c^2*d*e^3 - b^4*c*e^4)*x^5 + 5*(128*b*c^4*d^4 - 256*b^2*c^3*d^3*e 
 + 144*b^3*c^2*d^2*e^2 - 16*b^4*c*d*e^3 - b^5*e^4)*x^4 + 60*(8*b^2*c^3*d^4 
 - 16*b^3*c^2*d^3*e + 9*b^4*c*d^2*e^2 - b^5*d*e^3)*x^3 + 10*(8*b^3*c^2*d^4 
 - 16*b^4*c*d^3*e + 9*b^5*d^2*e^2)*x^2 - 10*(b^4*c*d^4 - 2*b^5*d^3*e)*x)*s 
qrt(c*x^2 + b*x)/(b^6*c^3*x^6 + 3*b^7*c^2*x^5 + 3*b^8*c*x^4 + b^9*x^3)
 

Sympy [F]

\[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((e*x+d)**4/(c*x**2+b*x)**(7/2),x)
 

Output:

Integral((d + e*x)**4/(x*(b + c*x))**(7/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (245) = 490\).

Time = 0.04 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^4/(c*x^2+b*x)^(7/2),x, algorithm="maxima")
 

Output:

-1/2*e^4*x^3/((c*x^2 + b*x)^(5/2)*c) - 4/3*d*e^3*x^2/((c*x^2 + b*x)^(5/2)* 
c) - 1/12*b*e^4*x^2/((c*x^2 + b*x)^(5/2)*c^2) - 4/5*c*d^4*x/((c*x^2 + b*x) 
^(5/2)*b^2) + 64/15*c^2*d^4*x/((c*x^2 + b*x)^(3/2)*b^4) - 512/15*c^3*d^4*x 
/(sqrt(c*x^2 + b*x)*b^6) + 8/5*d^3*e*x/((c*x^2 + b*x)^(5/2)*b) - 128/15*c* 
d^3*e*x/((c*x^2 + b*x)^(3/2)*b^3) + 1024/15*c^2*d^3*e*x/(sqrt(c*x^2 + b*x) 
*b^5) + 24/5*d^2*e^2*x/((c*x^2 + b*x)^(3/2)*b^2) - 12/5*d^2*e^2*x/((c*x^2 
+ b*x)^(5/2)*c) - 192/5*c*d^2*e^2*x/(sqrt(c*x^2 + b*x)*b^4) + 64/15*d*e^3* 
x/(sqrt(c*x^2 + b*x)*b^3) + 4/15*b*d*e^3*x/((c*x^2 + b*x)^(5/2)*c^2) - 8/1 
5*d*e^3*x/((c*x^2 + b*x)^(3/2)*b*c) + 1/60*b^2*e^4*x/((c*x^2 + b*x)^(5/2)* 
c^3) - 1/30*e^4*x/((c*x^2 + b*x)^(3/2)*c^2) + 4/15*e^4*x/(sqrt(c*x^2 + b*x 
)*b^2*c) - 2/5*d^4/((c*x^2 + b*x)^(5/2)*b) + 32/15*c*d^4/((c*x^2 + b*x)^(3 
/2)*b^3) - 256/15*c^2*d^4/(sqrt(c*x^2 + b*x)*b^5) - 64/15*d^3*e/((c*x^2 + 
b*x)^(3/2)*b^2) + 512/15*c*d^3*e/(sqrt(c*x^2 + b*x)*b^4) - 96/5*d^2*e^2/(s 
qrt(c*x^2 + b*x)*b^3) + 12/5*d^2*e^2/((c*x^2 + b*x)^(3/2)*b*c) - 4/15*d*e^ 
3/((c*x^2 + b*x)^(3/2)*c^2) + 32/15*d*e^3/(sqrt(c*x^2 + b*x)*b^2*c) - 1/60 
*b*e^4/((c*x^2 + b*x)^(3/2)*c^3) + 2/15*e^4/(sqrt(c*x^2 + b*x)*b*c^2)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (\frac {3 \, d^{4}}{b} + {\left ({\left ({\left (x {\left (\frac {2 \, {\left (128 \, c^{5} d^{4} - 256 \, b c^{4} d^{3} e + 144 \, b^{2} c^{3} d^{2} e^{2} - 16 \, b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x}{b^{6}} + \frac {5 \, {\left (128 \, b c^{4} d^{4} - 256 \, b^{2} c^{3} d^{3} e + 144 \, b^{3} c^{2} d^{2} e^{2} - 16 \, b^{4} c d e^{3} - b^{5} e^{4}\right )}}{b^{6}}\right )} + \frac {60 \, {\left (8 \, b^{2} c^{3} d^{4} - 16 \, b^{3} c^{2} d^{3} e + 9 \, b^{4} c d^{2} e^{2} - b^{5} d e^{3}\right )}}{b^{6}}\right )} x + \frac {10 \, {\left (8 \, b^{3} c^{2} d^{4} - 16 \, b^{4} c d^{3} e + 9 \, b^{5} d^{2} e^{2}\right )}}{b^{6}}\right )} x - \frac {10 \, {\left (b^{4} c d^{4} - 2 \, b^{5} d^{3} e\right )}}{b^{6}}\right )} x\right )}}{15 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \] Input:

integrate((e*x+d)^4/(c*x^2+b*x)^(7/2),x, algorithm="giac")
 

Output:

-2/15*(3*d^4/b + (((x*(2*(128*c^5*d^4 - 256*b*c^4*d^3*e + 144*b^2*c^3*d^2* 
e^2 - 16*b^3*c^2*d*e^3 - b^4*c*e^4)*x/b^6 + 5*(128*b*c^4*d^4 - 256*b^2*c^3 
*d^3*e + 144*b^3*c^2*d^2*e^2 - 16*b^4*c*d*e^3 - b^5*e^4)/b^6) + 60*(8*b^2* 
c^3*d^4 - 16*b^3*c^2*d^3*e + 9*b^4*c*d^2*e^2 - b^5*d*e^3)/b^6)*x + 10*(8*b 
^3*c^2*d^4 - 16*b^4*c*d^3*e + 9*b^5*d^2*e^2)/b^6)*x - 10*(b^4*c*d^4 - 2*b^ 
5*d^3*e)/b^6)*x)/(c*x^2 + b*x)^(5/2)
 

Mupad [B] (verification not implemented)

Time = 5.33 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2\,\left (3\,b^5\,d^4+20\,b^5\,d^3\,e\,x+90\,b^5\,d^2\,e^2\,x^2-60\,b^5\,d\,e^3\,x^3-5\,b^5\,e^4\,x^4-10\,b^4\,c\,d^4\,x-160\,b^4\,c\,d^3\,e\,x^2+540\,b^4\,c\,d^2\,e^2\,x^3-80\,b^4\,c\,d\,e^3\,x^4-2\,b^4\,c\,e^4\,x^5+80\,b^3\,c^2\,d^4\,x^2-960\,b^3\,c^2\,d^3\,e\,x^3+720\,b^3\,c^2\,d^2\,e^2\,x^4-32\,b^3\,c^2\,d\,e^3\,x^5+480\,b^2\,c^3\,d^4\,x^3-1280\,b^2\,c^3\,d^3\,e\,x^4+288\,b^2\,c^3\,d^2\,e^2\,x^5+640\,b\,c^4\,d^4\,x^4-512\,b\,c^4\,d^3\,e\,x^5+256\,c^5\,d^4\,x^5\right )}{15\,b^6\,{\left (c\,x^2+b\,x\right )}^{5/2}} \] Input:

int((d + e*x)^4/(b*x + c*x^2)^(7/2),x)
 

Output:

-(2*(3*b^5*d^4 - 5*b^5*e^4*x^4 + 256*c^5*d^4*x^5 + 640*b*c^4*d^4*x^4 - 2*b 
^4*c*e^4*x^5 - 60*b^5*d*e^3*x^3 + 80*b^3*c^2*d^4*x^2 + 480*b^2*c^3*d^4*x^3 
 + 90*b^5*d^2*e^2*x^2 - 10*b^4*c*d^4*x + 20*b^5*d^3*e*x + 720*b^3*c^2*d^2* 
e^2*x^4 + 288*b^2*c^3*d^2*e^2*x^5 - 160*b^4*c*d^3*e*x^2 - 80*b^4*c*d*e^3*x 
^4 - 512*b*c^4*d^3*e*x^5 - 960*b^3*c^2*d^3*e*x^3 + 540*b^4*c*d^2*e^2*x^3 - 
 1280*b^2*c^3*d^3*e*x^4 - 32*b^3*c^2*d*e^3*x^5))/(15*b^6*(b*x + c*x^2)^(5/ 
2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.58 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^4/(c*x^2+b*x)^(7/2),x)
 

Output:

(2*( - 14*sqrt(c)*sqrt(b + c*x)*b**6*e**4*x**3 - 32*sqrt(c)*sqrt(b + c*x)* 
b**5*c*d*e**3*x**3 - 28*sqrt(c)*sqrt(b + c*x)*b**5*c*e**4*x**4 + 288*sqrt( 
c)*sqrt(b + c*x)*b**4*c**2*d**2*e**2*x**3 - 64*sqrt(c)*sqrt(b + c*x)*b**4* 
c**2*d*e**3*x**4 - 14*sqrt(c)*sqrt(b + c*x)*b**4*c**2*e**4*x**5 - 512*sqrt 
(c)*sqrt(b + c*x)*b**3*c**3*d**3*e*x**3 + 576*sqrt(c)*sqrt(b + c*x)*b**3*c 
**3*d**2*e**2*x**4 - 32*sqrt(c)*sqrt(b + c*x)*b**3*c**3*d*e**3*x**5 + 256* 
sqrt(c)*sqrt(b + c*x)*b**2*c**4*d**4*x**3 - 1024*sqrt(c)*sqrt(b + c*x)*b** 
2*c**4*d**3*e*x**4 + 288*sqrt(c)*sqrt(b + c*x)*b**2*c**4*d**2*e**2*x**5 + 
512*sqrt(c)*sqrt(b + c*x)*b*c**5*d**4*x**4 - 512*sqrt(c)*sqrt(b + c*x)*b*c 
**5*d**3*e*x**5 + 256*sqrt(c)*sqrt(b + c*x)*c**6*d**4*x**5 - 3*sqrt(x)*b** 
5*c**2*d**4 - 20*sqrt(x)*b**5*c**2*d**3*e*x - 90*sqrt(x)*b**5*c**2*d**2*e* 
*2*x**2 + 60*sqrt(x)*b**5*c**2*d*e**3*x**3 + 5*sqrt(x)*b**5*c**2*e**4*x**4 
 + 10*sqrt(x)*b**4*c**3*d**4*x + 160*sqrt(x)*b**4*c**3*d**3*e*x**2 - 540*s 
qrt(x)*b**4*c**3*d**2*e**2*x**3 + 80*sqrt(x)*b**4*c**3*d*e**3*x**4 + 2*sqr 
t(x)*b**4*c**3*e**4*x**5 - 80*sqrt(x)*b**3*c**4*d**4*x**2 + 960*sqrt(x)*b* 
*3*c**4*d**3*e*x**3 - 720*sqrt(x)*b**3*c**4*d**2*e**2*x**4 + 32*sqrt(x)*b* 
*3*c**4*d*e**3*x**5 - 480*sqrt(x)*b**2*c**5*d**4*x**3 + 1280*sqrt(x)*b**2* 
c**5*d**3*e*x**4 - 288*sqrt(x)*b**2*c**5*d**2*e**2*x**5 - 640*sqrt(x)*b*c* 
*6*d**4*x**4 + 512*sqrt(x)*b*c**6*d**3*e*x**5 - 256*sqrt(x)*c**7*d**4*x**5 
))/(15*sqrt(b + c*x)*b**6*c**2*x**3*(b**2 + 2*b*c*x + c**2*x**2))