Integrand size = 19, antiderivative size = 113 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d}{3 b \left (b x+c x^2\right )^{3/2}}-\frac {2 (2 c d-b e) x}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {8 (2 c d-b e)}{3 b^3 \sqrt {b x+c x^2}}+\frac {16 (2 c d-b e) \sqrt {b x+c x^2}}{3 b^4 x} \] Output:
-2/3*d/b/(c*x^2+b*x)^(3/2)-2/3*(-b*e+2*c*d)*x/b^2/(c*x^2+b*x)^(3/2)-8/3*(- b*e+2*c*d)/b^3/(c*x^2+b*x)^(1/2)+16/3*(-b*e+2*c*d)*(c*x^2+b*x)^(1/2)/b^4/x
Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.59 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (-16 c^3 d x^3-6 b^2 c x (d-2 e x)+8 b c^2 x^2 (-3 d+e x)+b^3 (d+3 e x)\right )}{3 b^4 (x (b+c x))^{3/2}} \] Input:
Integrate[(d + e*x)/(b*x + c*x^2)^(5/2),x]
Output:
(-2*(-16*c^3*d*x^3 - 6*b^2*c*x*(d - 2*e*x) + 8*b*c^2*x^2*(-3*d + e*x) + b^ 3*(d + 3*e*x)))/(3*b^4*(x*(b + c*x))^(3/2))
Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.63, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1159, 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.
\(\displaystyle \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle -\frac {4 (2 c d-b e) \int \frac {1}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2}-\frac {2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle \frac {8 (b+2 c x) (2 c d-b e)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\) |
Input:
Int[(d + e*x)/(b*x + c*x^2)^(5/2),x]
Output:
(-2*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (8*(2*c*d - b*e )*(b + 2*c*x))/(3*b^4*Sqrt[b*x + c*x^2])
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]
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]
Time = 0.67 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (3 e x +d \right ) b^{3}-6 c x \left (-2 e x +d \right ) b^{2}-24 x^{2} \left (-\frac {e x}{3}+d \right ) b \,c^{2}-16 c^{3} d \,x^{3}\right )}{3 \sqrt {x \left (c x +b \right )}\, x \left (c x +b \right ) b^{4}}\) | \(73\) |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+3 b^{3} e x -6 b^{2} c x d +b^{3} d \right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(83\) |
orering | \(-\frac {2 x \left (c x +b \right ) \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+3 b^{3} e x -6 b^{2} c x d +b^{3} d \right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(83\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (3 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}-\frac {2 c \left (5 b c e x -8 c^{2} d x +6 e \,b^{2}-9 d b c \right ) x}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\) | \(86\) |
trager | \(-\frac {2 \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+3 b^{3} e x -6 b^{2} c x d +b^{3} d \right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\) | \(87\) |
default | \(d \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )+e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )\) | \(121\) |
Input:
int((e*x+d)/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3/(x*(c*x+b))^(1/2)*((3*e*x+d)*b^3-6*c*x*(-2*e*x+d)*b^2-24*x^2*(-1/3*e* x+d)*b*c^2-16*c^3*d*x^3)/x/(c*x+b)/b^4
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (b^{3} d - 8 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} - 12 \, {\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - 3 \, {\left (2 \, b^{2} c d - b^{3} e\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \] Input:
integrate((e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
Output:
-2/3*(b^3*d - 8*(2*c^3*d - b*c^2*e)*x^3 - 12*(2*b*c^2*d - b^2*c*e)*x^2 - 3 *(2*b^2*c*d - b^3*e)*x)*sqrt(c*x^2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6 *x^2)
\[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {d + e x}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)/(c*x**2+b*x)**(5/2),x)
Output:
Integral((d + e*x)/(x*(b + c*x))**(5/2), x)
Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {4 \, c d x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {2 \, e x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, c e x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, d}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, e}{3 \, \sqrt {c x^{2} + b x} b^{2}} \] Input:
integrate((e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")
Output:
-4/3*c*d*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*c^2*d*x/(sqrt(c*x^2 + b*x)*b^4 ) + 2/3*e*x/((c*x^2 + b*x)^(3/2)*b) - 16/3*c*e*x/(sqrt(c*x^2 + b*x)*b^3) - 2/3*d/((c*x^2 + b*x)^(3/2)*b) + 16/3*c*d/(sqrt(c*x^2 + b*x)*b^3) - 8/3*e/ (sqrt(c*x^2 + b*x)*b^2)
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.76 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, x {\left (\frac {2 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x}{b^{4}} + \frac {3 \, {\left (2 \, b c^{2} d - b^{2} c e\right )}}{b^{4}}\right )} + \frac {3 \, {\left (2 \, b^{2} c d - b^{3} e\right )}}{b^{4}}\right )} x - \frac {d}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \] Input:
integrate((e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="giac")
Output:
2/3*((4*x*(2*(2*c^3*d - b*c^2*e)*x/b^4 + 3*(2*b*c^2*d - b^2*c*e)/b^4) + 3* (2*b^2*c*d - b^3*e)/b^4)*x - d/b)/(c*x^2 + b*x)^(3/2)
Time = 5.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\left (3\,e\,b^3\,x+d\,b^3+12\,e\,b^2\,c\,x^2-6\,d\,b^2\,c\,x+8\,e\,b\,c^2\,x^3-24\,d\,b\,c^2\,x^2-16\,d\,c^3\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \] Input:
int((d + e*x)/(b*x + c*x^2)^(5/2),x)
Output:
-(2*(b^3*d - 16*c^3*d*x^3 + 3*b^3*e*x - 6*b^2*c*d*x - 24*b*c^2*d*x^2 + 12* b^2*c*e*x^2 + 8*b*c^2*e*x^3))/(3*b^4*(b*x + c*x^2)^(3/2))
Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.45 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {\frac {16 \sqrt {c}\, \sqrt {c x +b}\, b^{2} e \,x^{2}}{3}-\frac {32 \sqrt {c}\, \sqrt {c x +b}\, b c d \,x^{2}}{3}+\frac {16 \sqrt {c}\, \sqrt {c x +b}\, b c e \,x^{3}}{3}-\frac {32 \sqrt {c}\, \sqrt {c x +b}\, c^{2} d \,x^{3}}{3}-\frac {2 \sqrt {x}\, b^{3} d}{3}-2 \sqrt {x}\, b^{3} e x +4 \sqrt {x}\, b^{2} c d x -8 \sqrt {x}\, b^{2} c e \,x^{2}+16 \sqrt {x}\, b \,c^{2} d \,x^{2}-\frac {16 \sqrt {x}\, b \,c^{2} e \,x^{3}}{3}+\frac {32 \sqrt {x}\, c^{3} d \,x^{3}}{3}}{\sqrt {c x +b}\, b^{4} x^{2} \left (c x +b \right )} \] Input:
int((e*x+d)/(c*x^2+b*x)^(5/2),x)
Output:
(2*(8*sqrt(c)*sqrt(b + c*x)*b**2*e*x**2 - 16*sqrt(c)*sqrt(b + c*x)*b*c*d*x **2 + 8*sqrt(c)*sqrt(b + c*x)*b*c*e*x**3 - 16*sqrt(c)*sqrt(b + c*x)*c**2*d *x**3 - sqrt(x)*b**3*d - 3*sqrt(x)*b**3*e*x + 6*sqrt(x)*b**2*c*d*x - 12*sq rt(x)*b**2*c*e*x**2 + 24*sqrt(x)*b*c**2*d*x**2 - 8*sqrt(x)*b*c**2*e*x**3 + 16*sqrt(x)*c**3*d*x**3))/(3*sqrt(b + c*x)*b**4*x**2*(b + c*x))