Integrand size = 24, antiderivative size = 165 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=-\frac {8 (b c-a d) (2 b c+a d) \left (a x+b x^2\right )^{3/2}}{105 a^3 x^4}+\frac {8 (2 b c-5 a d) (b c-a d) (2 b c+a d) \left (a x+b x^2\right )^{3/2}}{315 a^4 c x^3}+\frac {2 (2 b c+a d) (c+d x)^2 \left (a x+b x^2\right )^{3/2}}{21 a^2 c x^5}-\frac {2 (c+d x)^3 \left (a x+b x^2\right )^{3/2}}{9 a c x^6} \] Output:
-8/105*(-a*d+b*c)*(a*d+2*b*c)*(b*x^2+a*x)^(3/2)/a^3/x^4+8/315*(-5*a*d+2*b* c)*(-a*d+b*c)*(a*d+2*b*c)*(b*x^2+a*x)^(3/2)/a^4/c/x^3+2/21*(a*d+2*b*c)*(d* x+c)^2*(b*x^2+a*x)^(3/2)/a^2/c/x^5-2/9*(d*x+c)^3*(b*x^2+a*x)^(3/2)/a/c/x^6
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.60 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=\frac {2 (x (a+b x))^{3/2} \left (16 b^3 c^2 x^3-24 a b^2 c x^2 (c+2 d x)+6 a^2 b x \left (5 c^2+12 c d x+7 d^2 x^2\right )-a^3 \left (35 c^2+90 c d x+63 d^2 x^2\right )\right )}{315 a^4 x^6} \] Input:
Integrate[((c + d*x)^2*Sqrt[a*x + b*x^2])/x^6,x]
Output:
(2*(x*(a + b*x))^(3/2)*(16*b^3*c^2*x^3 - 24*a*b^2*c*x^2*(c + 2*d*x) + 6*a^ 2*b*x*(5*c^2 + 12*c*d*x + 7*d^2*x^2) - a^3*(35*c^2 + 90*c*d*x + 63*d^2*x^2 )))/(315*a^4*x^6)
Time = 0.60 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1262, 27, 1220, 1129, 1129, 1123}
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.
| \(\displaystyle \int \frac {\sqrt {a x+b x^2} (c+d x)^2}{x^6} \, dx\) |
| \(\Big \downarrow \) 1262 |
| \(\displaystyle -\frac {\int -\frac {\left (4 b c^2+d (8 b c-7 a d) x\right ) \sqrt {b x^2+a x}}{2 x^6}dx}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{2 b x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (4 b c^2+d (8 b c-7 a d) x\right ) \sqrt {b x^2+a x}}{x^6}dx}{4 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{2 b x^5}\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {-\frac {\left (21 a^2 d^2-24 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {b x^2+a x}}{x^5}dx}{3 a}-\frac {8 b c^2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}}{4 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{2 b x^5}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {-\frac {\left (21 a^2 d^2-24 a b c d+8 b^2 c^2\right ) \left (-\frac {4 b \int \frac {\sqrt {b x^2+a x}}{x^4}dx}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {8 b c^2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}}{4 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{2 b x^5}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {-\frac {\left (21 a^2 d^2-24 a b c d+8 b^2 c^2\right ) \left (-\frac {4 b \left (-\frac {2 b \int \frac {\sqrt {b x^2+a x}}{x^3}dx}{5 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right )}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {8 b c^2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}}{4 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{2 b x^5}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {-\frac {\left (-\frac {4 b \left (\frac {4 b \left (a x+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right )}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right ) \left (21 a^2 d^2-24 a b c d+8 b^2 c^2\right )}{3 a}-\frac {8 b c^2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}}{4 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{2 b x^5}\) |
Input:
Int[((c + d*x)^2*Sqrt[a*x + b*x^2])/x^6,x]
Output:
-1/2*(d^2*(a*x + b*x^2)^(3/2))/(b*x^5) + ((-8*b*c^2*(a*x + b*x^2)^(3/2))/( 9*a*x^6) - ((8*b^2*c^2 - 24*a*b*c*d + 21*a^2*d^2)*((-2*(a*x + b*x^2)^(3/2) )/(7*a*x^5) - (4*b*((-2*(a*x + b*x^2)^(3/2))/(5*a*x^4) + (4*b*(a*x + b*x^2 )^(3/2))/(15*a^2*x^3)))/(7*a)))/(3*a))/(4*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n + e*g^n*( m + p + n)*(d + e*x)^(n - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[n, 0] && NeQ[m + n + 2*p + 1, 0]
Time = 0.55 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.55
| method | result | size |
| pseudoelliptic | \(-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right ) \left (\left (\frac {9}{5} d^{2} x^{2}+\frac {18}{7} c d x +c^{2}\right ) a^{3}-\frac {6 \left (d x +c \right ) \left (\frac {7 d x}{5}+c \right ) x b \,a^{2}}{7}+\frac {24 b^{2} c \,x^{2} \left (2 d x +c \right ) a}{35}-\frac {16 b^{3} c^{2} x^{3}}{35}\right )}{9 x^{5} a^{4}}\) | \(90\) |
| gosper | \(-\frac {2 \left (b x +a \right ) \left (-42 d^{2} x^{3} a^{2} b +48 a \,b^{2} c d \,x^{3}-16 b^{3} c^{2} x^{3}+63 a^{3} d^{2} x^{2}-72 x^{2} a^{2} b c d +24 a \,b^{2} c^{2} x^{2}+90 a^{3} c d x -30 a^{2} b \,c^{2} x +35 c^{2} a^{3}\right ) \sqrt {b \,x^{2}+a x}}{315 x^{5} a^{4}}\) | \(120\) |
| orering | \(-\frac {2 \left (b x +a \right ) \left (-42 d^{2} x^{3} a^{2} b +48 a \,b^{2} c d \,x^{3}-16 b^{3} c^{2} x^{3}+63 a^{3} d^{2} x^{2}-72 x^{2} a^{2} b c d +24 a \,b^{2} c^{2} x^{2}+90 a^{3} c d x -30 a^{2} b \,c^{2} x +35 c^{2} a^{3}\right ) \sqrt {b \,x^{2}+a x}}{315 x^{5} a^{4}}\) | \(120\) |
| trager | \(-\frac {2 \left (-42 a^{2} b^{2} d^{2} x^{4}+48 a \,b^{3} c d \,x^{4}-16 b^{4} c^{2} x^{4}+21 a^{3} b \,d^{2} x^{3}-24 a^{2} b^{2} c d \,x^{3}+8 a \,b^{3} c^{2} x^{3}+63 a^{4} d^{2} x^{2}+18 a^{3} d c b \,x^{2}-6 a^{2} b^{2} c^{2} x^{2}+90 a^{4} c d x +5 a^{3} b \,c^{2} x +35 a^{4} c^{2}\right ) \sqrt {b \,x^{2}+a x}}{315 x^{5} a^{4}}\) | \(156\) |
| risch | \(-\frac {2 \left (b x +a \right ) \left (-42 a^{2} b^{2} d^{2} x^{4}+48 a \,b^{3} c d \,x^{4}-16 b^{4} c^{2} x^{4}+21 a^{3} b \,d^{2} x^{3}-24 a^{2} b^{2} c d \,x^{3}+8 a \,b^{3} c^{2} x^{3}+63 a^{4} d^{2} x^{2}+18 a^{3} d c b \,x^{2}-6 a^{2} b^{2} c^{2} x^{2}+90 a^{4} c d x +5 a^{3} b \,c^{2} x +35 a^{4} c^{2}\right )}{315 x^{4} \sqrt {x \left (b x +a \right )}\, a^{4}}\) | \(159\) |
| default | \(c^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{9 a \,x^{6}}-\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )+d^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )+2 c d \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )\) | \(212\) |
Input:
int((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-2/9*(x*(b*x+a))^(1/2)*(b*x+a)*((9/5*d^2*x^2+18/7*c*d*x+c^2)*a^3-6/7*(d*x+ c)*(7/5*d*x+c)*x*b*a^2+24/35*b^2*c*x^2*(2*d*x+c)*a-16/35*b^3*c^2*x^3)/x^5/ a^4
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=-\frac {2 \, {\left (35 \, a^{4} c^{2} - 2 \, {\left (8 \, b^{4} c^{2} - 24 \, a b^{3} c d + 21 \, a^{2} b^{2} d^{2}\right )} x^{4} + {\left (8 \, a b^{3} c^{2} - 24 \, a^{2} b^{2} c d + 21 \, a^{3} b d^{2}\right )} x^{3} - 3 \, {\left (2 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d - 21 \, a^{4} d^{2}\right )} x^{2} + 5 \, {\left (a^{3} b c^{2} + 18 \, a^{4} c d\right )} x\right )} \sqrt {b x^{2} + a x}}{315 \, a^{4} x^{5}} \] Input:
integrate((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^6,x, algorithm="fricas")
Output:
-2/315*(35*a^4*c^2 - 2*(8*b^4*c^2 - 24*a*b^3*c*d + 21*a^2*b^2*d^2)*x^4 + ( 8*a*b^3*c^2 - 24*a^2*b^2*c*d + 21*a^3*b*d^2)*x^3 - 3*(2*a^2*b^2*c^2 - 6*a^ 3*b*c*d - 21*a^4*d^2)*x^2 + 5*(a^3*b*c^2 + 18*a^4*c*d)*x)*sqrt(b*x^2 + a*x )/(a^4*x^5)
\[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=\int \frac {\sqrt {x \left (a + b x\right )} \left (c + d x\right )^{2}}{x^{6}}\, dx \] Input:
integrate((d*x+c)**2*(b*x**2+a*x)**(1/2)/x**6,x)
Output:
Integral(sqrt(x*(a + b*x))*(c + d*x)**2/x**6, x)
Time = 0.04 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} b^{4} c^{2}}{315 \, a^{4} x} - \frac {32 \, \sqrt {b x^{2} + a x} b^{3} c d}{105 \, a^{3} x} + \frac {4 \, \sqrt {b x^{2} + a x} b^{2} d^{2}}{15 \, a^{2} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{3} c^{2}}{315 \, a^{3} x^{2}} + \frac {16 \, \sqrt {b x^{2} + a x} b^{2} c d}{105 \, a^{2} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} b d^{2}}{15 \, a x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} b^{2} c^{2}}{105 \, a^{2} x^{3}} - \frac {4 \, \sqrt {b x^{2} + a x} b c d}{35 \, a x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} d^{2}}{5 \, x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} b c^{2}}{63 \, a x^{4}} - \frac {4 \, \sqrt {b x^{2} + a x} c d}{7 \, x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} c^{2}}{9 \, x^{5}} \] Input:
integrate((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^6,x, algorithm="maxima")
Output:
32/315*sqrt(b*x^2 + a*x)*b^4*c^2/(a^4*x) - 32/105*sqrt(b*x^2 + a*x)*b^3*c* d/(a^3*x) + 4/15*sqrt(b*x^2 + a*x)*b^2*d^2/(a^2*x) - 16/315*sqrt(b*x^2 + a *x)*b^3*c^2/(a^3*x^2) + 16/105*sqrt(b*x^2 + a*x)*b^2*c*d/(a^2*x^2) - 2/15* sqrt(b*x^2 + a*x)*b*d^2/(a*x^2) + 4/105*sqrt(b*x^2 + a*x)*b^2*c^2/(a^2*x^3 ) - 4/35*sqrt(b*x^2 + a*x)*b*c*d/(a*x^3) - 2/5*sqrt(b*x^2 + a*x)*d^2/x^3 - 2/63*sqrt(b*x^2 + a*x)*b*c^2/(a*x^4) - 4/7*sqrt(b*x^2 + a*x)*c*d/x^4 - 2/ 9*sqrt(b*x^2 + a*x)*c^2/x^5
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (149) = 298\).
Time = 0.16 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.70 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=\frac {2 \, {\left (315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} b^{\frac {3}{2}} d^{2} + 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b^{2} c d + 525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a b d^{2} + 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} b^{\frac {5}{2}} c^{2} + 1890 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a b^{\frac {3}{2}} c d + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{2} \sqrt {b} d^{2} + 1764 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a b^{2} c^{2} + 1638 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} b c d + 63 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{3} d^{2} + 1995 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} b^{\frac {3}{2}} c^{2} + 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{3} \sqrt {b} c d + 1125 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{3} b c^{2} + 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{4} c d + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{4} \sqrt {b} c^{2} + 35 \, a^{5} c^{2}\right )}}{315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9}} \] Input:
integrate((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^6,x, algorithm="giac")
Output:
2/315*(315*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*b^(3/2)*d^2 + 840*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b^2*c*d + 525*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a*b *d^2 + 630*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*b^(5/2)*c^2 + 1890*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a*b^(3/2)*c*d + 315*(sqrt(b)*x - sqrt(b*x^2 + a*x) )^5*a^2*sqrt(b)*d^2 + 1764*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a*b^2*c^2 + 1 638*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^2*b*c*d + 63*(sqrt(b)*x - sqrt(b*x ^2 + a*x))^4*a^3*d^2 + 1995*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^2*b^(3/2)* c^2 + 630*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^3*sqrt(b)*c*d + 1125*(sqrt(b )*x - sqrt(b*x^2 + a*x))^2*a^3*b*c^2 + 90*(sqrt(b)*x - sqrt(b*x^2 + a*x))^ 2*a^4*c*d + 315*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^4*sqrt(b)*c^2 + 35*a^5*c ^2)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^9
Time = 10.39 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=\frac {4\,b^2\,c^2\,\sqrt {b\,x^2+a\,x}}{105\,a^2\,x^3}-\frac {2\,d^2\,\sqrt {b\,x^2+a\,x}}{5\,x^3}-\frac {4\,c\,d\,\sqrt {b\,x^2+a\,x}}{7\,x^4}-\frac {2\,c^2\,\sqrt {b\,x^2+a\,x}}{9\,x^5}-\frac {16\,b^3\,c^2\,\sqrt {b\,x^2+a\,x}}{315\,a^3\,x^2}+\frac {32\,b^4\,c^2\,\sqrt {b\,x^2+a\,x}}{315\,a^4\,x}+\frac {4\,b^2\,d^2\,\sqrt {b\,x^2+a\,x}}{15\,a^2\,x}-\frac {2\,b\,c^2\,\sqrt {b\,x^2+a\,x}}{63\,a\,x^4}-\frac {2\,b\,d^2\,\sqrt {b\,x^2+a\,x}}{15\,a\,x^2}+\frac {16\,b^2\,c\,d\,\sqrt {b\,x^2+a\,x}}{105\,a^2\,x^2}-\frac {32\,b^3\,c\,d\,\sqrt {b\,x^2+a\,x}}{105\,a^3\,x}-\frac {4\,b\,c\,d\,\sqrt {b\,x^2+a\,x}}{35\,a\,x^3} \] Input:
int(((a*x + b*x^2)^(1/2)*(c + d*x)^2)/x^6,x)
Output:
(4*b^2*c^2*(a*x + b*x^2)^(1/2))/(105*a^2*x^3) - (2*d^2*(a*x + b*x^2)^(1/2) )/(5*x^3) - (4*c*d*(a*x + b*x^2)^(1/2))/(7*x^4) - (2*c^2*(a*x + b*x^2)^(1/ 2))/(9*x^5) - (16*b^3*c^2*(a*x + b*x^2)^(1/2))/(315*a^3*x^2) + (32*b^4*c^2 *(a*x + b*x^2)^(1/2))/(315*a^4*x) + (4*b^2*d^2*(a*x + b*x^2)^(1/2))/(15*a^ 2*x) - (2*b*c^2*(a*x + b*x^2)^(1/2))/(63*a*x^4) - (2*b*d^2*(a*x + b*x^2)^( 1/2))/(15*a*x^2) + (16*b^2*c*d*(a*x + b*x^2)^(1/2))/(105*a^2*x^2) - (32*b^ 3*c*d*(a*x + b*x^2)^(1/2))/(105*a^3*x) - (4*b*c*d*(a*x + b*x^2)^(1/2))/(35 *a*x^3)
Time = 0.24 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^6} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{4} c^{2}}{9}-\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{4} c d x}{7}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{4} d^{2} x^{2}}{5}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b \,c^{2} x}{63}-\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b c d \,x^{2}}{35}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b \,d^{2} x^{3}}{15}+\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} c^{2} x^{2}}{105}+\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} c d \,x^{3}}{105}+\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} d^{2} x^{4}}{15}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} c^{2} x^{3}}{315}-\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} c d \,x^{4}}{105}+\frac {32 \sqrt {x}\, \sqrt {b x +a}\, b^{4} c^{2} x^{4}}{315}-\frac {4 \sqrt {b}\, a^{2} b^{2} d^{2} x^{5}}{15}+\frac {32 \sqrt {b}\, a \,b^{3} c d \,x^{5}}{105}-\frac {32 \sqrt {b}\, b^{4} c^{2} x^{5}}{315}}{a^{4} x^{5}} \] Input:
int((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^6,x)
Output:
(2*( - 35*sqrt(x)*sqrt(a + b*x)*a**4*c**2 - 90*sqrt(x)*sqrt(a + b*x)*a**4* c*d*x - 63*sqrt(x)*sqrt(a + b*x)*a**4*d**2*x**2 - 5*sqrt(x)*sqrt(a + b*x)* a**3*b*c**2*x - 18*sqrt(x)*sqrt(a + b*x)*a**3*b*c*d*x**2 - 21*sqrt(x)*sqrt (a + b*x)*a**3*b*d**2*x**3 + 6*sqrt(x)*sqrt(a + b*x)*a**2*b**2*c**2*x**2 + 24*sqrt(x)*sqrt(a + b*x)*a**2*b**2*c*d*x**3 + 42*sqrt(x)*sqrt(a + b*x)*a* *2*b**2*d**2*x**4 - 8*sqrt(x)*sqrt(a + b*x)*a*b**3*c**2*x**3 - 48*sqrt(x)* sqrt(a + b*x)*a*b**3*c*d*x**4 + 16*sqrt(x)*sqrt(a + b*x)*b**4*c**2*x**4 - 42*sqrt(b)*a**2*b**2*d**2*x**5 + 48*sqrt(b)*a*b**3*c*d*x**5 - 16*sqrt(b)*b **4*c**2*x**5))/(315*a**4*x**5)