\(\int \frac {(d+e x)^3}{(b x+c x^2)^{5/2}} \, dx\) [172]

Optimal result

Integrand size = 21, antiderivative size = 140 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {16 (c d-b e)^3 x^2}{3 b^3 c \left (b x+c x^2\right )^{3/2}}+\frac {4 (c d-b e) x (d+e x)^2}{b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x)^3}{3 b \left (b x+c x^2\right )^{3/2}}+\frac {16 (c d-b e)^2 (2 c d+b e) x}{3 b^4 c \sqrt {b x+c x^2}} \] Output:

16/3*(-b*e+c*d)^3*x^2/b^3/c/(c*x^2+b*x)^(3/2)+4*(-b*e+c*d)*x*(e*x+d)^2/b^2 
/(c*x^2+b*x)^(3/2)-2/3*(e*x+d)^3/b/(c*x^2+b*x)^(3/2)+16/3*(-b*e+c*d)^2*(b* 
e+2*c*d)*x/b^4/c/(c*x^2+b*x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (16 c^3 d^3 x^3+24 b c^2 d^2 x^2 (d-e x)+6 b^2 c d x \left (d^2-6 d e x+e^2 x^2\right )+b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \] Input:

Integrate[(d + e*x)^3/(b*x + c*x^2)^(5/2),x]
 

Output:

(2*(16*c^3*d^3*x^3 + 24*b*c^2*d^2*x^2*(d - e*x) + 6*b^2*c*d*x*(d^2 - 6*d*e 
*x + e^2*x^2) + b^3*(-d^3 - 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3)))/(3*b^4*(x 
*(b + c*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {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.

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

\(\Big \downarrow \) 1153

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

\(\Big \downarrow \) 1158

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

Input:

Int[(d + e*x)^3/(b*x + c*x^2)^(5/2),x]
 

Output:

(-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])
 

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 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 = 0.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.59

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (e^{2} x^{2}+10 d e x +d^{2}\right ) b^{2}-8 d c x \left (-e x +d \right ) b -8 d^{2} c^{2} x^{2}\right ) \left (2 c d x +b \left (-e x +d \right )\right )}{3 \sqrt {x \left (c x +b \right )}\, x \left (c x +b \right ) b^{4}}\)	\(83\)
risch	\(-\frac {2 d^{2} \left (c x +b \right ) \left (9 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {2 x \left (b e x +8 c d x +9 b d \right ) \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\)	\(98\)
gosper	\(-\frac {2 x \left (c x +b \right ) \left (-b^{3} e^{3} x^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 d^{3} c^{3} x^{3}-9 b^{3} d \,e^{2} x^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +b^{3} d^{3}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\)	\(136\)
orering	\(-\frac {2 x \left (c x +b \right ) \left (-b^{3} e^{3} x^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 d^{3} c^{3} x^{3}-9 b^{3} d \,e^{2} x^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +b^{3} d^{3}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\)	\(136\)
trager	\(-\frac {2 \left (-b^{3} e^{3} x^{3}-6 b^{2} c d \,e^{2} x^{3}+24 b \,c^{2} d^{2} e \,x^{3}-16 d^{3} c^{3} x^{3}-9 b^{3} d \,e^{2} x^{2}+36 b^{2} c \,d^{2} e \,x^{2}-24 b \,c^{2} d^{3} x^{2}+9 b^{3} d^{2} e x -6 b^{2} c \,d^{3} x +b^{3} d^{3}\right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\)	\(140\)
default	\(d^{3} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )+e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )}{2 c}\right )+3 d \,e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+3 d^{2} e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )\)	\(349\)

Input:

int((e*x+d)^3/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3/(x*(c*x+b))^(1/2)*((e^2*x^2+10*d*e*x+d^2)*b^2-8*d*c*x*(-e*x+d)*b-8*d^ 
2*c^2*x^2)*(2*c*d*x+b*(-e*x+d))/x/(c*x+b)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (b^{3} d^{3} - {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{3} - 3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )} x^{2} - 3 \, {\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \] Input:

integrate((e*x+d)^3/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
 

Output:

-2/3*(b^3*d^3 - (16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)*x^ 
3 - 3*(8*b*c^2*d^3 - 12*b^2*c*d^2*e + 3*b^3*d*e^2)*x^2 - 3*(2*b^2*c*d^3 - 
3*b^3*d^2*e)*x)*sqrt(c*x^2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)
 

Sympy [F]

\[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((e*x+d)**3/(c*x**2+b*x)**(5/2),x)
 

Output:

Integral((d + e*x)**3/(x*(b + c*x))**(5/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (126) = 252\).

Time = 0.03 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.12 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {e^{3} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {4 \, c d^{3} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d^{3} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {2 \, d^{2} e x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, c d^{2} e x}{\sqrt {c x^{2} + b x} b^{3}} + \frac {4 \, d e^{2} x}{\sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, d e^{2} x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {b e^{3} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, e^{3} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {2 \, d^{3}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d^{3}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, d^{2} e}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, d e^{2}}{\sqrt {c x^{2} + b x} b c} + \frac {e^{3}}{3 \, \sqrt {c x^{2} + b x} c^{2}} \] Input:

integrate((e*x+d)^3/(c*x^2+b*x)^(5/2),x, algorithm="maxima")
 

Output:

-e^3*x^2/((c*x^2 + b*x)^(3/2)*c) - 4/3*c*d^3*x/((c*x^2 + b*x)^(3/2)*b^2) + 
 32/3*c^2*d^3*x/(sqrt(c*x^2 + b*x)*b^4) + 2*d^2*e*x/((c*x^2 + b*x)^(3/2)*b 
) - 16*c*d^2*e*x/(sqrt(c*x^2 + b*x)*b^3) + 4*d*e^2*x/(sqrt(c*x^2 + b*x)*b^ 
2) - 2*d*e^2*x/((c*x^2 + b*x)^(3/2)*c) - 1/3*b*e^3*x/((c*x^2 + b*x)^(3/2)* 
c^2) + 2/3*e^3*x/(sqrt(c*x^2 + b*x)*b*c) - 2/3*d^3/((c*x^2 + b*x)^(3/2)*b) 
 + 16/3*c*d^3/(sqrt(c*x^2 + b*x)*b^3) - 8*d^2*e/(sqrt(c*x^2 + b*x)*b^2) + 
2*d*e^2/(sqrt(c*x^2 + b*x)*b*c) + 1/3*e^3/(sqrt(c*x^2 + b*x)*c^2)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (\frac {d^{3}}{b} - {\left (x {\left (\frac {{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x}{b^{4}} + \frac {3 \, {\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )}}{b^{4}}\right )} + \frac {3 \, {\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} e\right )}}{b^{4}}\right )} x\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \] Input:

integrate((e*x+d)^3/(c*x^2+b*x)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-2/3*(d^3/b - (x*((16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3)* 
x/b^4 + 3*(8*b*c^2*d^3 - 12*b^2*c*d^2*e + 3*b^3*d*e^2)/b^4) + 3*(2*b^2*c*d 
^3 - 3*b^3*d^2*e)/b^4)*x)/(c*x^2 + b*x)^(3/2)
 

Mupad [B] (verification not implemented)

Time = 5.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2\,\left (-b^3\,d^3-9\,b^3\,d^2\,e\,x+9\,b^3\,d\,e^2\,x^2+b^3\,e^3\,x^3+6\,b^2\,c\,d^3\,x-36\,b^2\,c\,d^2\,e\,x^2+6\,b^2\,c\,d\,e^2\,x^3+24\,b\,c^2\,d^3\,x^2-24\,b\,c^2\,d^2\,e\,x^3+16\,c^3\,d^3\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \] Input:

int((d + e*x)^3/(b*x + c*x^2)^(5/2),x)
 

Output:

(2*(b^3*e^3*x^3 - b^3*d^3 + 16*c^3*d^3*x^3 + 24*b*c^2*d^3*x^2 + 9*b^3*d*e^ 
2*x^2 + 6*b^2*c*d^3*x - 9*b^3*d^2*e*x - 36*b^2*c*d^2*e*x^2 - 24*b*c^2*d^2* 
e*x^3 + 6*b^2*c*d*e^2*x^3))/(3*b^4*(b*x + c*x^2)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.45 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {\frac {10 \sqrt {c}\, \sqrt {c x +b}\, b^{4} e^{3} x^{2}}{3}-12 \sqrt {c}\, \sqrt {c x +b}\, b^{3} c d \,e^{2} x^{2}+\frac {10 \sqrt {c}\, \sqrt {c x +b}\, b^{3} c \,e^{3} x^{3}}{3}+16 \sqrt {c}\, \sqrt {c x +b}\, b^{2} c^{2} d^{2} e \,x^{2}-12 \sqrt {c}\, \sqrt {c x +b}\, b^{2} c^{2} d \,e^{2} x^{3}-\frac {32 \sqrt {c}\, \sqrt {c x +b}\, b \,c^{3} d^{3} x^{2}}{3}+16 \sqrt {c}\, \sqrt {c x +b}\, b \,c^{3} d^{2} e \,x^{3}-\frac {32 \sqrt {c}\, \sqrt {c x +b}\, c^{4} d^{3} x^{3}}{3}-\frac {2 \sqrt {x}\, b^{3} c^{2} d^{3}}{3}-6 \sqrt {x}\, b^{3} c^{2} d^{2} e x +6 \sqrt {x}\, b^{3} c^{2} d \,e^{2} x^{2}+\frac {2 \sqrt {x}\, b^{3} c^{2} e^{3} x^{3}}{3}+4 \sqrt {x}\, b^{2} c^{3} d^{3} x -24 \sqrt {x}\, b^{2} c^{3} d^{2} e \,x^{2}+4 \sqrt {x}\, b^{2} c^{3} d \,e^{2} x^{3}+16 \sqrt {x}\, b \,c^{4} d^{3} x^{2}-16 \sqrt {x}\, b \,c^{4} d^{2} e \,x^{3}+\frac {32 \sqrt {x}\, c^{5} d^{3} x^{3}}{3}}{\sqrt {c x +b}\, b^{4} c^{2} x^{2} \left (c x +b \right )} \] Input:

int((e*x+d)^3/(c*x^2+b*x)^(5/2),x)
 

Output:

(2*(5*sqrt(c)*sqrt(b + c*x)*b**4*e**3*x**2 - 18*sqrt(c)*sqrt(b + c*x)*b**3 
*c*d*e**2*x**2 + 5*sqrt(c)*sqrt(b + c*x)*b**3*c*e**3*x**3 + 24*sqrt(c)*sqr 
t(b + c*x)*b**2*c**2*d**2*e*x**2 - 18*sqrt(c)*sqrt(b + c*x)*b**2*c**2*d*e* 
*2*x**3 - 16*sqrt(c)*sqrt(b + c*x)*b*c**3*d**3*x**2 + 24*sqrt(c)*sqrt(b + 
c*x)*b*c**3*d**2*e*x**3 - 16*sqrt(c)*sqrt(b + c*x)*c**4*d**3*x**3 - sqrt(x 
)*b**3*c**2*d**3 - 9*sqrt(x)*b**3*c**2*d**2*e*x + 9*sqrt(x)*b**3*c**2*d*e* 
*2*x**2 + sqrt(x)*b**3*c**2*e**3*x**3 + 6*sqrt(x)*b**2*c**3*d**3*x - 36*sq 
rt(x)*b**2*c**3*d**2*e*x**2 + 6*sqrt(x)*b**2*c**3*d*e**2*x**3 + 24*sqrt(x) 
*b*c**4*d**3*x**2 - 24*sqrt(x)*b*c**4*d**2*e*x**3 + 16*sqrt(x)*c**5*d**3*x 
**3))/(3*sqrt(b + c*x)*b**4*c**2*x**2*(b + c*x))