\(\int \frac {d+e x}{(b x+c x^2)^{7/2}} \, dx\) [181]

Optimal result

Integrand size = 19, antiderivative size = 178 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 d}{5 b \left (b x+c x^2\right )^{5/2}}-\frac {2 (2 c d-b e) x}{5 b^2 \left (b x+c x^2\right )^{5/2}}-\frac {16 (2 c d-b e)}{15 b^3 \left (b x+c x^2\right )^{3/2}}-\frac {32 (2 c d-b e)}{5 b^4 x \sqrt {b x+c x^2}}+\frac {128 (2 c d-b e) \sqrt {b x+c x^2}}{15 b^5 x^2}-\frac {256 c (2 c d-b e) \sqrt {b x+c x^2}}{15 b^6 x} \] Output:

-2/5*d/b/(c*x^2+b*x)^(5/2)-2/5*(-b*e+2*c*d)*x/b^2/(c*x^2+b*x)^(5/2)-16/15* 
(-b*e+2*c*d)/b^3/(c*x^2+b*x)^(3/2)-32/5*(-b*e+2*c*d)/b^4/x/(c*x^2+b*x)^(1/ 
2)+128/15*(-b*e+2*c*d)*(c*x^2+b*x)^(1/2)/b^5/x^2-256/15*c*(-b*e+2*c*d)*(c* 
x^2+b*x)^(1/2)/b^6/x
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.59 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \left (256 c^5 d x^5+80 b^3 c^2 x^2 (d-3 e x)+160 b^2 c^3 x^3 (3 d-2 e x)-128 b c^4 x^4 (-5 d+e x)-10 b^4 c x (d+4 e x)+b^5 (3 d+5 e x)\right )}{15 b^6 (x (b+c x))^{5/2}} \] Input:

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

Output:

(-2*(256*c^5*d*x^5 + 80*b^3*c^2*x^2*(d - 3*e*x) + 160*b^2*c^3*x^3*(3*d - 2 
*e*x) - 128*b*c^4*x^4*(-5*d + e*x) - 10*b^4*c*x*(d + 4*e*x) + b^5*(3*d + 5 
*e*x)))/(15*b^6*(x*(b + c*x))^(5/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.60, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1159, 1089, 1088}

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.

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

\(\Big \downarrow \) 1159

\(\displaystyle -\frac {8 (2 c d-b e) \int \frac {1}{\left (c x^2+b x\right )^{5/2}}dx}{5 b^2}-\frac {2 (x (2 c d-b e)+b d)}{5 b^2 \left (b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1089

\(\displaystyle -\frac {8 (2 c d-b e) \left (-\frac {8 c \int \frac {1}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2}-\frac {2 (b+2 c x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\right )}{5 b^2}-\frac {2 (x (2 c d-b e)+b d)}{5 b^2 \left (b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1088

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1088

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 
2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0]
 

rule 1089

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1159

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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(\frac {\left (-10 e x -6 d \right ) b^{5}+20 c x \left (4 e x +d \right ) b^{4}-160 c^{2} x^{2} \left (-3 e x +d \right ) b^{3}-960 x^{3} c^{3} \left (-\frac {2 e x}{3}+d \right ) b^{2}-1280 \left (-\frac {e x}{5}+d \right ) x^{4} c^{4} b -512 c^{5} d \,x^{5}}{15 \sqrt {x \left (c x +b \right )}\, x^{2} \left (c x +b \right )^{2} b^{6}}\)	\(109\)
gosper	\(-\frac {2 x \left (c x +b \right ) \left (-128 b \,c^{4} e \,x^{5}+256 c^{5} d \,x^{5}-320 b^{2} c^{3} e \,x^{4}+640 b \,c^{4} d \,x^{4}-240 b^{3} c^{2} e \,x^{3}+480 c^{3} b^{2} d \,x^{3}-40 b^{4} c e \,x^{2}+80 b^{3} c^{2} d \,x^{2}+5 b^{5} e x -10 b^{4} c d x +3 d \,b^{5}\right )}{15 b^{6} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}\)	\(132\)
orering	\(-\frac {2 x \left (c x +b \right ) \left (-128 b \,c^{4} e \,x^{5}+256 c^{5} d \,x^{5}-320 b^{2} c^{3} e \,x^{4}+640 b \,c^{4} d \,x^{4}-240 b^{3} c^{2} e \,x^{3}+480 c^{3} b^{2} d \,x^{3}-40 b^{4} c e \,x^{2}+80 b^{3} c^{2} d \,x^{2}+5 b^{5} e x -10 b^{4} c d x +3 d \,b^{5}\right )}{15 b^{6} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}\)	\(132\)
trager	\(-\frac {2 \left (-128 b \,c^{4} e \,x^{5}+256 c^{5} d \,x^{5}-320 b^{2} c^{3} e \,x^{4}+640 b \,c^{4} d \,x^{4}-240 b^{3} c^{2} e \,x^{3}+480 c^{3} b^{2} d \,x^{3}-40 b^{4} c e \,x^{2}+80 b^{3} c^{2} d \,x^{2}+5 b^{5} e x -10 b^{4} c d x +3 d \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{15 b^{6} \left (c x +b \right )^{3} x^{3}}\)	\(136\)
risch	\(-\frac {2 \left (c x +b \right ) \left (-55 b c e \,x^{2}+128 c^{2} d \,x^{2}+5 b^{2} e x -19 b c d x +3 b^{2} d \right )}{15 b^{6} x^{2} \sqrt {x \left (c x +b \right )}}+\frac {2 c^{2} \left (73 b e \,x^{2} c^{2}-128 c^{3} d \,x^{2}+160 x \,b^{2} c e -275 b \,c^{2} d x +90 e \,b^{3}-150 c d \,b^{2}\right ) x}{15 \sqrt {x \left (c x +b \right )}\, \left (c^{2} x^{2}+2 c b x +b^{2}\right ) b^{6}}\)	\(146\)
default	\(d \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )+e \left (-\frac {1}{5 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{2 c}\right )\)	\(179\)

Input:

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

Output:

1/15*((-10*e*x-6*d)*b^5+20*c*x*(4*e*x+d)*b^4-160*c^2*x^2*(-3*e*x+d)*b^3-96 
0*x^3*c^3*(-2/3*e*x+d)*b^2-1280*(-1/5*e*x+d)*x^4*c^4*b-512*c^5*d*x^5)/(x*( 
c*x+b))^(1/2)/x^2/(c*x+b)^2/b^6
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (3 \, b^{5} d + 128 \, {\left (2 \, c^{5} d - b c^{4} e\right )} x^{5} + 320 \, {\left (2 \, b c^{4} d - b^{2} c^{3} e\right )} x^{4} + 240 \, {\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )} x^{3} + 40 \, {\left (2 \, b^{3} c^{2} d - b^{4} c e\right )} x^{2} - 5 \, {\left (2 \, b^{4} c d - b^{5} e\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, {\left (b^{6} c^{3} x^{6} + 3 \, b^{7} c^{2} x^{5} + 3 \, b^{8} c x^{4} + b^{9} x^{3}\right )}} \] Input:

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

Output:

-2/15*(3*b^5*d + 128*(2*c^5*d - b*c^4*e)*x^5 + 320*(2*b*c^4*d - b^2*c^3*e) 
*x^4 + 240*(2*b^2*c^3*d - b^3*c^2*e)*x^3 + 40*(2*b^3*c^2*d - b^4*c*e)*x^2 
- 5*(2*b^4*c*d - b^5*e)*x)*sqrt(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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.18 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {4 \, c d x}{5 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2}} + \frac {64 \, c^{2} d x}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4}} - \frac {512 \, c^{3} d x}{15 \, \sqrt {c x^{2} + b x} b^{6}} + \frac {2 \, e x}{5 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b} - \frac {32 \, c e x}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}} + \frac {256 \, c^{2} e x}{15 \, \sqrt {c x^{2} + b x} b^{5}} - \frac {2 \, d}{5 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b} + \frac {32 \, c d}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}} - \frac {256 \, c^{2} d}{15 \, \sqrt {c x^{2} + b x} b^{5}} - \frac {16 \, e}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {128 \, c e}{15 \, \sqrt {c x^{2} + b x} b^{4}} \] Input:

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

Output:

-4/5*c*d*x/((c*x^2 + b*x)^(5/2)*b^2) + 64/15*c^2*d*x/((c*x^2 + b*x)^(3/2)* 
b^4) - 512/15*c^3*d*x/(sqrt(c*x^2 + b*x)*b^6) + 2/5*e*x/((c*x^2 + b*x)^(5/ 
2)*b) - 32/15*c*e*x/((c*x^2 + b*x)^(3/2)*b^3) + 256/15*c^2*e*x/(sqrt(c*x^2 
 + b*x)*b^5) - 2/5*d/((c*x^2 + b*x)^(5/2)*b) + 32/15*c*d/((c*x^2 + b*x)^(3 
/2)*b^3) - 256/15*c^2*d/(sqrt(c*x^2 + b*x)*b^5) - 16/15*e/((c*x^2 + b*x)^( 
3/2)*b^2) + 128/15*c*e/(sqrt(c*x^2 + b*x)*b^4)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.80 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (4 \, x {\left (\frac {2 \, {\left (2 \, c^{5} d - b c^{4} e\right )} x}{b^{6}} + \frac {5 \, {\left (2 \, b c^{4} d - b^{2} c^{3} e\right )}}{b^{6}}\right )} + \frac {15 \, {\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )}}{b^{6}}\right )} x + \frac {5 \, {\left (2 \, b^{3} c^{2} d - b^{4} c e\right )}}{b^{6}}\right )} x - \frac {5 \, {\left (2 \, b^{4} c d - b^{5} e\right )}}{b^{6}}\right )} x + \frac {3 \, d}{b}\right )}}{15 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \] Input:

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

Output:

-2/15*((8*(2*(4*x*(2*(2*c^5*d - b*c^4*e)*x/b^6 + 5*(2*b*c^4*d - b^2*c^3*e) 
/b^6) + 15*(2*b^2*c^3*d - b^3*c^2*e)/b^6)*x + 5*(2*b^3*c^2*d - b^4*c*e)/b^ 
6)*x - 5*(2*b^4*c*d - b^5*e)/b^6)*x + 3*d/b)/(c*x^2 + b*x)^(5/2)
 

Mupad [B] (verification not implemented)

Time = 5.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2\,\left (5\,e\,b^5\,x+3\,d\,b^5-40\,e\,b^4\,c\,x^2-10\,d\,b^4\,c\,x-240\,e\,b^3\,c^2\,x^3+80\,d\,b^3\,c^2\,x^2-320\,e\,b^2\,c^3\,x^4+480\,d\,b^2\,c^3\,x^3-128\,e\,b\,c^4\,x^5+640\,d\,b\,c^4\,x^4+256\,d\,c^5\,x^5\right )}{15\,b^6\,{\left (c\,x^2+b\,x\right )}^{5/2}} \] Input:

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

Output:

-(2*(3*b^5*d + 256*c^5*d*x^5 + 5*b^5*e*x + 80*b^3*c^2*d*x^2 + 480*b^2*c^3* 
d*x^3 - 240*b^3*c^2*e*x^3 - 320*b^2*c^3*e*x^4 - 10*b^4*c*d*x + 640*b*c^4*d 
*x^4 - 40*b^4*c*e*x^2 - 128*b*c^4*e*x^5))/(15*b^6*(b*x + c*x^2)^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.55 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=\frac {-\frac {256 \sqrt {c}\, \sqrt {c x +b}\, b^{3} c e \,x^{3}}{15}+\frac {512 \sqrt {c}\, \sqrt {c x +b}\, b^{2} c^{2} d \,x^{3}}{15}-\frac {512 \sqrt {c}\, \sqrt {c x +b}\, b^{2} c^{2} e \,x^{4}}{15}+\frac {1024 \sqrt {c}\, \sqrt {c x +b}\, b \,c^{3} d \,x^{4}}{15}-\frac {256 \sqrt {c}\, \sqrt {c x +b}\, b \,c^{3} e \,x^{5}}{15}+\frac {512 \sqrt {c}\, \sqrt {c x +b}\, c^{4} d \,x^{5}}{15}-\frac {2 \sqrt {x}\, b^{5} d}{5}-\frac {2 \sqrt {x}\, b^{5} e x}{3}+\frac {4 \sqrt {x}\, b^{4} c d x}{3}+\frac {16 \sqrt {x}\, b^{4} c e \,x^{2}}{3}-\frac {32 \sqrt {x}\, b^{3} c^{2} d \,x^{2}}{3}+32 \sqrt {x}\, b^{3} c^{2} e \,x^{3}-64 \sqrt {x}\, b^{2} c^{3} d \,x^{3}+\frac {128 \sqrt {x}\, b^{2} c^{3} e \,x^{4}}{3}-\frac {256 \sqrt {x}\, b \,c^{4} d \,x^{4}}{3}+\frac {256 \sqrt {x}\, b \,c^{4} e \,x^{5}}{15}-\frac {512 \sqrt {x}\, c^{5} d \,x^{5}}{15}}{\sqrt {c x +b}\, b^{6} x^{3} \left (c^{2} x^{2}+2 b c x +b^{2}\right )} \] Input:

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

Output:

(2*( - 128*sqrt(c)*sqrt(b + c*x)*b**3*c*e*x**3 + 256*sqrt(c)*sqrt(b + c*x) 
*b**2*c**2*d*x**3 - 256*sqrt(c)*sqrt(b + c*x)*b**2*c**2*e*x**4 + 512*sqrt( 
c)*sqrt(b + c*x)*b*c**3*d*x**4 - 128*sqrt(c)*sqrt(b + c*x)*b*c**3*e*x**5 + 
 256*sqrt(c)*sqrt(b + c*x)*c**4*d*x**5 - 3*sqrt(x)*b**5*d - 5*sqrt(x)*b**5 
*e*x + 10*sqrt(x)*b**4*c*d*x + 40*sqrt(x)*b**4*c*e*x**2 - 80*sqrt(x)*b**3* 
c**2*d*x**2 + 240*sqrt(x)*b**3*c**2*e*x**3 - 480*sqrt(x)*b**2*c**3*d*x**3 
+ 320*sqrt(x)*b**2*c**3*e*x**4 - 640*sqrt(x)*b*c**4*d*x**4 + 128*sqrt(x)*b 
*c**4*e*x**5 - 256*sqrt(x)*c**5*d*x**5))/(15*sqrt(b + c*x)*b**6*x**3*(b**2 
 + 2*b*c*x + c**2*x**2))