Integrand size = 21, antiderivative size = 169 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {8 (b c-a d)^2 (2 b c-a d) x^2}{3 a^3 b c \left (a x+b x^2\right )^{3/2}}+\frac {2 (2 b c-a d) x (c+d x)^2}{a^2 c \left (a x+b x^2\right )^{3/2}}-\frac {2 (c+d x)^3}{3 a c \left (a x+b x^2\right )^{3/2}}+\frac {8 (b c-a d) (2 b c-a d) (2 b c+a d) x}{3 a^4 b c \sqrt {a x+b x^2}} \] Output:
8/3*(-a*d+b*c)^2*(-a*d+2*b*c)*x^2/a^3/b/c/(b*x^2+a*x)^(3/2)+2*(-a*d+2*b*c) *x*(d*x+c)^2/a^2/c/(b*x^2+a*x)^(3/2)-2/3*(d*x+c)^3/a/c/(b*x^2+a*x)^(3/2)+8 /3*(-a*d+b*c)*(-a*d+2*b*c)*(a*d+2*b*c)*x/a^4/b/c/(b*x^2+a*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.56 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {32 b^3 c^2 x^3+16 a b^2 c x^2 (3 c-2 d x)-2 a^3 \left (c^2+6 c d x-3 d^2 x^2\right )+4 a^2 b x \left (3 c^2-12 c d x+d^2 x^2\right )}{3 a^4 (x (a+b x))^{3/2}} \] Input:
Integrate[(c + d*x)^2/(a*x + b*x^2)^(5/2),x]
Output:
(32*b^3*c^2*x^3 + 16*a*b^2*c*x^2*(3*c - 2*d*x) - 2*a^3*(c^2 + 6*c*d*x - 3* d^2*x^2) + 4*a^2*b*x*(3*c^2 - 12*c*d*x + d^2*x^2))/(3*a^4*(x*(a + b*x))^(3 /2))
Time = 0.33 (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.
\(\displaystyle \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1156 |
\(\displaystyle -\frac {4 (2 b c-a d) \int \frac {c+d x}{\left (b x^2+a x\right )^{3/2}}dx}{3 a^2}-\frac {2 (a+2 b x) (c+d x)^2}{3 a^2 \left (a x+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle \frac {8 (2 b c-a d) (x (2 b c-a d)+a c)}{3 a^4 \sqrt {a x+b x^2}}-\frac {2 (a+2 b x) (c+d x)^2}{3 a^2 \left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[(c + d*x)^2/(a*x + b*x^2)^(5/2),x]
Output:
(-2*(a + 2*b*x)*(c + d*x)^2)/(3*a^2*(a*x + b*x^2)^(3/2)) + (8*(2*b*c - a*d )*(a*c + (2*b*c - a*d)*x))/(3*a^4*Sqrt[a*x + b*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 = 0.55 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.56
method | result | size |
risch | \(-\frac {2 c \left (b x +a \right ) \left (6 a d x -8 c b x +a c \right )}{3 a^{4} x \sqrt {x \left (b x +a \right )}}+\frac {2 x \left (2 a b d x -8 b^{2} c x +3 a^{2} d -9 a b c \right ) \left (a d -b c \right )}{3 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right ) a^{4}}\) | \(94\) |
pseudoelliptic | \(-\frac {2 \left (\left (-3 d^{2} x^{2}+6 c d x +c^{2}\right ) a^{3}-6 x \left (\frac {1}{3} d^{2} x^{2}-4 c d x +c^{2}\right ) b \,a^{2}-24 x^{2} \left (-\frac {2 d x}{3}+c \right ) b^{2} c a -16 b^{3} c^{2} x^{3}\right )}{3 \sqrt {x \left (b x +a \right )}\, x \left (b x +a \right ) a^{4}}\) | \(98\) |
gosper | \(-\frac {2 x \left (b x +a \right ) \left (-2 d^{2} x^{3} a^{2} b +16 a \,b^{2} c d \,x^{3}-16 b^{3} c^{2} x^{3}-3 a^{3} d^{2} x^{2}+24 x^{2} a^{2} b c d -24 a \,b^{2} c^{2} x^{2}+6 a^{3} c d x -6 a^{2} b \,c^{2} x +c^{2} a^{3}\right )}{3 a^{4} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}\) | \(117\) |
orering | \(-\frac {2 x \left (b x +a \right ) \left (-2 d^{2} x^{3} a^{2} b +16 a \,b^{2} c d \,x^{3}-16 b^{3} c^{2} x^{3}-3 a^{3} d^{2} x^{2}+24 x^{2} a^{2} b c d -24 a \,b^{2} c^{2} x^{2}+6 a^{3} c d x -6 a^{2} b \,c^{2} x +c^{2} a^{3}\right )}{3 a^{4} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}\) | \(117\) |
trager | \(-\frac {2 \left (-2 d^{2} x^{3} a^{2} b +16 a \,b^{2} c d \,x^{3}-16 b^{3} c^{2} x^{3}-3 a^{3} d^{2} x^{2}+24 x^{2} a^{2} b c d -24 a \,b^{2} c^{2} x^{2}+6 a^{3} c d x -6 a^{2} b \,c^{2} x +c^{2} a^{3}\right ) \sqrt {b \,x^{2}+a x}}{3 a^{4} x^{2} \left (b x +a \right )^{2}}\) | \(121\) |
default | \(c^{2} \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )+d^{2} \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )+2 c d \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )\) | \(222\) |
Input:
int((d*x+c)^2/(b*x^2+a*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*c*(b*x+a)*(6*a*d*x-8*b*c*x+a*c)/a^4/x/(x*(b*x+a))^(1/2)+2/3*x*(2*a*b* d*x-8*b^2*c*x+3*a^2*d-9*a*b*c)*(a*d-b*c)/(x*(b*x+a))^(1/2)/(b*x+a)/a^4
Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (a^{3} c^{2} - 2 \, {\left (8 \, b^{3} c^{2} - 8 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} - 3 \, {\left (8 \, a b^{2} c^{2} - 8 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 6 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(5/2),x, algorithm="fricas")
Output:
-2/3*(a^3*c^2 - 2*(8*b^3*c^2 - 8*a*b^2*c*d + a^2*b*d^2)*x^3 - 3*(8*a*b^2*c ^2 - 8*a^2*b*c*d + a^3*d^2)*x^2 - 6*(a^2*b*c^2 - a^3*c*d)*x)*sqrt(b*x^2 + a*x)/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)
\[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x+c)**2/(b*x**2+a*x)**(5/2),x)
Output:
Integral((c + d*x)**2/(x*(a + b*x))**(5/2), x)
Time = 0.03 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=-\frac {4 \, b c^{2} x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} + \frac {32 \, b^{2} c^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {4 \, c d x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} - \frac {32 \, b c d x}{3 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {4 \, d^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, d^{2} x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {2 \, c^{2}}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} + \frac {16 \, b c^{2}}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {16 \, c d}{3 \, \sqrt {b x^{2} + a x} a^{2}} + \frac {2 \, d^{2}}{3 \, \sqrt {b x^{2} + a x} a b} \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(5/2),x, algorithm="maxima")
Output:
-4/3*b*c^2*x/((b*x^2 + a*x)^(3/2)*a^2) + 32/3*b^2*c^2*x/(sqrt(b*x^2 + a*x) *a^4) + 4/3*c*d*x/((b*x^2 + a*x)^(3/2)*a) - 32/3*b*c*d*x/(sqrt(b*x^2 + a*x )*a^3) + 4/3*d^2*x/(sqrt(b*x^2 + a*x)*a^2) - 2/3*d^2*x/((b*x^2 + a*x)^(3/2 )*b) - 2/3*c^2/((b*x^2 + a*x)^(3/2)*a) + 16/3*b*c^2/(sqrt(b*x^2 + a*x)*a^3 ) - 16/3*c*d/(sqrt(b*x^2 + a*x)*a^2) + 2/3*d^2/(sqrt(b*x^2 + a*x)*a*b)
Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (x {\left (\frac {2 \, {\left (8 \, b^{3} c^{2} - 8 \, a b^{2} c d + a^{2} b d^{2}\right )} x}{a^{4}} + \frac {3 \, {\left (8 \, a b^{2} c^{2} - 8 \, a^{2} b c d + a^{3} d^{2}\right )}}{a^{4}}\right )} + \frac {6 \, {\left (a^{2} b c^{2} - a^{3} c d\right )}}{a^{4}}\right )} x - \frac {c^{2}}{a}\right )}}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}}} \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(5/2),x, algorithm="giac")
Output:
2/3*((x*(2*(8*b^3*c^2 - 8*a*b^2*c*d + a^2*b*d^2)*x/a^4 + 3*(8*a*b^2*c^2 - 8*a^2*b*c*d + a^3*d^2)/a^4) + 6*(a^2*b*c^2 - a^3*c*d)/a^4)*x - c^2/a)/(b*x ^2 + a*x)^(3/2)
Time = 9.60 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {2\,\left (-a^3\,c^2-6\,a^3\,c\,d\,x+3\,a^3\,d^2\,x^2+6\,a^2\,b\,c^2\,x-24\,a^2\,b\,c\,d\,x^2+2\,a^2\,b\,d^2\,x^3+24\,a\,b^2\,c^2\,x^2-16\,a\,b^2\,c\,d\,x^3+16\,b^3\,c^2\,x^3\right )}{3\,a^4\,{\left (b\,x^2+a\,x\right )}^{3/2}} \] Input:
int((c + d*x)^2/(a*x + b*x^2)^(5/2),x)
Output:
(2*(3*a^3*d^2*x^2 - a^3*c^2 + 16*b^3*c^2*x^3 + 24*a*b^2*c^2*x^2 + 2*a^2*b* d^2*x^3 - 6*a^3*c*d*x + 6*a^2*b*c^2*x - 24*a^2*b*c*d*x^2 - 16*a*b^2*c*d*x^ 3))/(3*a^4*(a*x + b*x^2)^(3/2))
Time = 0.25 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.56 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {-4 \sqrt {b}\, \sqrt {b x +a}\, a^{3} d^{2} x^{2}+\frac {32 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b c d \,x^{2}}{3}-4 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b \,d^{2} x^{3}-\frac {32 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c^{2} x^{2}}{3}+\frac {32 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c d \,x^{3}}{3}-\frac {32 \sqrt {b}\, \sqrt {b x +a}\, b^{3} c^{2} x^{3}}{3}-\frac {2 \sqrt {x}\, a^{3} b \,c^{2}}{3}-4 \sqrt {x}\, a^{3} b c d x +2 \sqrt {x}\, a^{3} b \,d^{2} x^{2}+4 \sqrt {x}\, a^{2} b^{2} c^{2} x -16 \sqrt {x}\, a^{2} b^{2} c d \,x^{2}+\frac {4 \sqrt {x}\, a^{2} b^{2} d^{2} x^{3}}{3}+16 \sqrt {x}\, a \,b^{3} c^{2} x^{2}-\frac {32 \sqrt {x}\, a \,b^{3} c d \,x^{3}}{3}+\frac {32 \sqrt {x}\, b^{4} c^{2} x^{3}}{3}}{\sqrt {b x +a}\, a^{4} b \,x^{2} \left (b x +a \right )} \] Input:
int((d*x+c)^2/(b*x^2+a*x)^(5/2),x)
Output:
(2*( - 6*sqrt(b)*sqrt(a + b*x)*a**3*d**2*x**2 + 16*sqrt(b)*sqrt(a + b*x)*a **2*b*c*d*x**2 - 6*sqrt(b)*sqrt(a + b*x)*a**2*b*d**2*x**3 - 16*sqrt(b)*sqr t(a + b*x)*a*b**2*c**2*x**2 + 16*sqrt(b)*sqrt(a + b*x)*a*b**2*c*d*x**3 - 1 6*sqrt(b)*sqrt(a + b*x)*b**3*c**2*x**3 - sqrt(x)*a**3*b*c**2 - 6*sqrt(x)*a **3*b*c*d*x + 3*sqrt(x)*a**3*b*d**2*x**2 + 6*sqrt(x)*a**2*b**2*c**2*x - 24 *sqrt(x)*a**2*b**2*c*d*x**2 + 2*sqrt(x)*a**2*b**2*d**2*x**3 + 24*sqrt(x)*a *b**3*c**2*x**2 - 16*sqrt(x)*a*b**3*c*d*x**3 + 16*sqrt(x)*b**4*c**2*x**3)) /(3*sqrt(a + b*x)*a**4*b*x**2*(a + b*x))