Integrand size = 21, antiderivative size = 169 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {8 (c d-b e)^2 (2 c d-b e) x^2}{3 b^3 c d \left (b x+c x^2\right )^{3/2}}+\frac {2 (2 c d-b e) x (d+e x)^2}{b^2 d \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x)^3}{3 b d \left (b x+c x^2\right )^{3/2}}+\frac {8 (c d-b e) (2 c d-b e) (2 c d+b e) x}{3 b^4 c d \sqrt {b x+c x^2}} \] Output:
8/3*(-b*e+c*d)^2*(-b*e+2*c*d)*x^2/b^3/c/d/(c*x^2+b*x)^(3/2)+2*(-b*e+2*c*d) *x*(e*x+d)^2/b^2/d/(c*x^2+b*x)^(3/2)-2/3*(e*x+d)^3/b/d/(c*x^2+b*x)^(3/2)+8 /3*(-b*e+c*d)*(-b*e+2*c*d)*(b*e+2*c*d)*x/b^4/c/d/(c*x^2+b*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.56 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {32 c^3 d^2 x^3+16 b c^2 d x^2 (3 d-2 e x)-2 b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+4 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )}{3 b^4 (x (b+c x))^{3/2}} \] Input:
Integrate[(d + e*x)^2/(b*x + c*x^2)^(5/2),x]
Output:
(32*c^3*d^2*x^3 + 16*b*c^2*d*x^2*(3*d - 2*e*x) - 2*b^3*(d^2 + 6*d*e*x - 3* e^2*x^2) + 4*b^2*c*x*(3*d^2 - 12*d*e*x + e^2*x^2))/(3*b^4*(x*(b + c*x))^(3 /2))
Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1156, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1156 |
| \(\displaystyle -\frac {4 (2 c d-b e) \int \frac {d+e x}{\left (c x^2+b x\right )^{3/2}}dx}{3 b^2}-\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1158 |
| \(\displaystyle \frac {8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}}\) |
Input:
Int[(d + e*x)^2/(b*x + c*x^2)^(5/2),x]
Output:
(-2*(b + 2*c*x)*(d + e*x)^2)/(3*b^2*(b*x + c*x^2)^(3/2)) + (8*(2*c*d - b*e )*(b*d + (2*c*d - b*e)*x))/(3*b^4*Sqrt[b*x + c*x^2])
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]
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]
Time = 0.75 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {2 d \left (c x +b \right ) \left (6 b e x -8 c d x +b d \right )}{3 b^{4} x \sqrt {x \left (c x +b \right )}}+\frac {2 x \left (2 b c e x -8 c^{2} d x +3 e \,b^{2}-9 d b c \right ) \left (b e -c d \right )}{3 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right ) b^{4}}\) | \(94\) |
| pseudoelliptic | \(-\frac {2 \left (\left (-3 e^{2} x^{2}+6 d e x +d^{2}\right ) b^{3}-6 \left (\frac {1}{3} e^{2} x^{2}-4 d e x +d^{2}\right ) x c \,b^{2}-24 x^{2} b d \left (-\frac {2 e x}{3}+d \right ) c^{2}-16 c^{3} d^{2} x^{3}\right )}{3 \sqrt {x \left (c x +b \right )}\, x \left (c x +b \right ) b^{4}}\) | \(98\) |
| gosper | \(-\frac {2 x \left (c x +b \right ) \left (-2 b^{2} c \,e^{2} x^{3}+16 b \,c^{2} d e \,x^{3}-16 c^{3} d^{2} x^{3}-3 b^{3} e^{2} x^{2}+24 b^{2} c d e \,x^{2}-24 b \,c^{2} d^{2} x^{2}+6 b^{3} d e x -6 b^{2} c \,d^{2} x +b^{3} d^{2}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(117\) |
| orering | \(-\frac {2 x \left (c x +b \right ) \left (-2 b^{2} c \,e^{2} x^{3}+16 b \,c^{2} d e \,x^{3}-16 c^{3} d^{2} x^{3}-3 b^{3} e^{2} x^{2}+24 b^{2} c d e \,x^{2}-24 b \,c^{2} d^{2} x^{2}+6 b^{3} d e x -6 b^{2} c \,d^{2} x +b^{3} d^{2}\right )}{3 b^{4} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}\) | \(117\) |
| trager | \(-\frac {2 \left (-2 b^{2} c \,e^{2} x^{3}+16 b \,c^{2} d e \,x^{3}-16 c^{3} d^{2} x^{3}-3 b^{3} e^{2} x^{2}+24 b^{2} c d e \,x^{2}-24 b \,c^{2} d^{2} x^{2}+6 b^{3} d e x -6 b^{2} c \,d^{2} x +b^{3} d^{2}\right ) \sqrt {c \,x^{2}+b x}}{3 b^{4} x^{2} \left (c x +b \right )^{2}}\) | \(121\) |
| default | \(d^{2} \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )+e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )}{4 c}\right )+2 d e \left (-\frac {1}{3 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{2 c}\right )\) | \(222\) |
Input:
int((e*x+d)^2/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*d*(c*x+b)*(6*b*e*x-8*c*d*x+b*d)/b^4/x/(x*(c*x+b))^(1/2)+2/3*x*(2*b*c* e*x-8*c^2*d*x+3*b^2*e-9*b*c*d)*(b*e-c*d)/(x*(c*x+b))^(1/2)/(c*x+b)/b^4
Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (b^{3} d^{2} - 2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x^{3} - 3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )} x^{2} - 6 \, {\left (b^{2} c d^{2} - b^{3} d e\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \] Input:
integrate((e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
Output:
-2/3*(b^3*d^2 - 2*(8*c^3*d^2 - 8*b*c^2*d*e + b^2*c*e^2)*x^3 - 3*(8*b*c^2*d ^2 - 8*b^2*c*d*e + b^3*e^2)*x^2 - 6*(b^2*c*d^2 - b^3*d*e)*x)*sqrt(c*x^2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)
\[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)**2/(c*x**2+b*x)**(5/2),x)
Output:
Integral((d + e*x)**2/(x*(b + c*x))**(5/2), x)
Time = 0.03 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {4 \, c d^{2} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} d^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {4 \, d e x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {32 \, c d e x}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {4 \, e^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, e^{2} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {2 \, d^{2}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c d^{2}}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {16 \, d e}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, e^{2}}{3 \, \sqrt {c x^{2} + b x} b c} \] Input:
integrate((e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="maxima")
Output:
-4/3*c*d^2*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*c^2*d^2*x/(sqrt(c*x^2 + b*x) *b^4) + 4/3*d*e*x/((c*x^2 + b*x)^(3/2)*b) - 32/3*c*d*e*x/(sqrt(c*x^2 + b*x )*b^3) + 4/3*e^2*x/(sqrt(c*x^2 + b*x)*b^2) - 2/3*e^2*x/((c*x^2 + b*x)^(3/2 )*c) - 2/3*d^2/((c*x^2 + b*x)^(3/2)*b) + 16/3*c*d^2/(sqrt(c*x^2 + b*x)*b^3 ) - 16/3*d*e/(sqrt(c*x^2 + b*x)*b^2) + 2/3*e^2/(sqrt(c*x^2 + b*x)*b*c)
Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (x {\left (\frac {2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x}{b^{4}} + \frac {3 \, {\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )}}{b^{4}}\right )} + \frac {6 \, {\left (b^{2} c d^{2} - b^{3} d e\right )}}{b^{4}}\right )} x - \frac {d^{2}}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \] Input:
integrate((e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="giac")
Output:
2/3*((x*(2*(8*c^3*d^2 - 8*b*c^2*d*e + b^2*c*e^2)*x/b^4 + 3*(8*b*c^2*d^2 - 8*b^2*c*d*e + b^3*e^2)/b^4) + 6*(b^2*c*d^2 - b^3*d*e)/b^4)*x - d^2/b)/(c*x ^2 + b*x)^(3/2)
Time = 5.35 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.66 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2\,\left (-b^3\,d^2-6\,b^3\,d\,e\,x+3\,b^3\,e^2\,x^2+6\,b^2\,c\,d^2\,x-24\,b^2\,c\,d\,e\,x^2+2\,b^2\,c\,e^2\,x^3+24\,b\,c^2\,d^2\,x^2-16\,b\,c^2\,d\,e\,x^3+16\,c^3\,d^2\,x^3\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \] Input:
int((d + e*x)^2/(b*x + c*x^2)^(5/2),x)
Output:
(2*(3*b^3*e^2*x^2 - b^3*d^2 + 16*c^3*d^2*x^3 + 24*b*c^2*d^2*x^2 + 2*b^2*c* e^2*x^3 - 6*b^3*d*e*x + 6*b^2*c*d^2*x - 24*b^2*c*d*e*x^2 - 16*b*c^2*d*e*x^ 3))/(3*b^4*(b*x + c*x^2)^(3/2))
Time = 0.22 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.56 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {-4 \sqrt {c}\, \sqrt {c x +b}\, b^{3} e^{2} x^{2}+\frac {32 \sqrt {c}\, \sqrt {c x +b}\, b^{2} c d e \,x^{2}}{3}-4 \sqrt {c}\, \sqrt {c x +b}\, b^{2} c \,e^{2} x^{3}-\frac {32 \sqrt {c}\, \sqrt {c x +b}\, b \,c^{2} d^{2} x^{2}}{3}+\frac {32 \sqrt {c}\, \sqrt {c x +b}\, b \,c^{2} d e \,x^{3}}{3}-\frac {32 \sqrt {c}\, \sqrt {c x +b}\, c^{3} d^{2} x^{3}}{3}-\frac {2 \sqrt {x}\, b^{3} c \,d^{2}}{3}-4 \sqrt {x}\, b^{3} c d e x +2 \sqrt {x}\, b^{3} c \,e^{2} x^{2}+4 \sqrt {x}\, b^{2} c^{2} d^{2} x -16 \sqrt {x}\, b^{2} c^{2} d e \,x^{2}+\frac {4 \sqrt {x}\, b^{2} c^{2} e^{2} x^{3}}{3}+16 \sqrt {x}\, b \,c^{3} d^{2} x^{2}-\frac {32 \sqrt {x}\, b \,c^{3} d e \,x^{3}}{3}+\frac {32 \sqrt {x}\, c^{4} d^{2} x^{3}}{3}}{\sqrt {c x +b}\, b^{4} c \,x^{2} \left (c x +b \right )} \] Input:
int((e*x+d)^2/(c*x^2+b*x)^(5/2),x)
Output:
(2*( - 6*sqrt(c)*sqrt(b + c*x)*b**3*e**2*x**2 + 16*sqrt(c)*sqrt(b + c*x)*b **2*c*d*e*x**2 - 6*sqrt(c)*sqrt(b + c*x)*b**2*c*e**2*x**3 - 16*sqrt(c)*sqr t(b + c*x)*b*c**2*d**2*x**2 + 16*sqrt(c)*sqrt(b + c*x)*b*c**2*d*e*x**3 - 1 6*sqrt(c)*sqrt(b + c*x)*c**3*d**2*x**3 - sqrt(x)*b**3*c*d**2 - 6*sqrt(x)*b **3*c*d*e*x + 3*sqrt(x)*b**3*c*e**2*x**2 + 6*sqrt(x)*b**2*c**2*d**2*x - 24 *sqrt(x)*b**2*c**2*d*e*x**2 + 2*sqrt(x)*b**2*c**2*e**2*x**3 + 24*sqrt(x)*b *c**3*d**2*x**2 - 16*sqrt(x)*b*c**3*d*e*x**3 + 16*sqrt(x)*c**4*d**2*x**3)) /(3*sqrt(b + c*x)*b**4*c*x**2*(b + c*x))