Integrand size = 24, antiderivative size = 199 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=-\frac {2 c^2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}+\frac {4 c (4 b c-11 a d) \left (a x+b x^2\right )^{3/2}}{99 a^2 x^6}-\frac {2 \left (33 d^2+\frac {4 b c (4 b c-11 a d)}{a^2}\right ) \left (a x+b x^2\right )^{3/2}}{231 a x^5}+\frac {8 b \left (33 a^2 d^2+4 b c (4 b c-11 a d)\right ) \left (a x+b x^2\right )^{3/2}}{1155 a^4 x^4}-\frac {16 b^2 \left (33 a^2 d^2+4 b c (4 b c-11 a d)\right ) \left (a x+b x^2\right )^{3/2}}{3465 a^5 x^3} \] Output:
-2/11*c^2*(b*x^2+a*x)^(3/2)/a/x^7+4/99*c*(-11*a*d+4*b*c)*(b*x^2+a*x)^(3/2) /a^2/x^6-2/231*(33*d^2+4*b*c*(-11*a*d+4*b*c)/a^2)*(b*x^2+a*x)^(3/2)/a/x^5+ 8/1155*b*(33*a^2*d^2+4*b*c*(-11*a*d+4*b*c))*(b*x^2+a*x)^(3/2)/a^4/x^4-16/3 465*b^2*(33*a^2*d^2+4*b*c*(-11*a*d+4*b*c))*(b*x^2+a*x)^(3/2)/a^5/x^3
Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=-\frac {2 (x (a+b x))^{3/2} \left (128 b^4 c^2 x^4-32 a b^3 c x^3 (6 c+11 d x)+24 a^2 b^2 x^2 \left (10 c^2+22 c d x+11 d^2 x^2\right )+5 a^4 \left (63 c^2+154 c d x+99 d^2 x^2\right )-4 a^3 b x \left (70 c^2+165 c d x+99 d^2 x^2\right )\right )}{3465 a^5 x^7} \] Input:
Integrate[((c + d*x)^2*Sqrt[a*x + b*x^2])/x^7,x]
Output:
(-2*(x*(a + b*x))^(3/2)*(128*b^4*c^2*x^4 - 32*a*b^3*c*x^3*(6*c + 11*d*x) + 24*a^2*b^2*x^2*(10*c^2 + 22*c*d*x + 11*d^2*x^2) + 5*a^4*(63*c^2 + 154*c*d *x + 99*d^2*x^2) - 4*a^3*b*x*(70*c^2 + 165*c*d*x + 99*d^2*x^2)))/(3465*a^5 *x^7)
Time = 0.67 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1262, 27, 1220, 1129, 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.
| \(\displaystyle \int \frac {\sqrt {a x+b x^2} (c+d x)^2}{x^7} \, dx\) |
| \(\Big \downarrow \) 1262 |
| \(\displaystyle -\frac {\int -\frac {3 \left (2 b c^2+d (4 b c-3 a d) x\right ) \sqrt {b x^2+a x}}{2 x^7}dx}{3 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (2 b c^2+d (4 b c-3 a d) x\right ) \sqrt {b x^2+a x}}{x^7}dx}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {-\frac {\left (33 a^2 d^2-44 a b c d+16 b^2 c^2\right ) \int \frac {\sqrt {b x^2+a x}}{x^6}dx}{11 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {-\frac {\left (33 a^2 d^2-44 a b c d+16 b^2 c^2\right ) \left (-\frac {2 b \int \frac {\sqrt {b x^2+a x}}{x^5}dx}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right )}{11 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {-\frac {\left (33 a^2 d^2-44 a b c d+16 b^2 c^2\right ) \left (-\frac {2 b \left (-\frac {4 b \int \frac {\sqrt {b x^2+a x}}{x^4}dx}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right )}{11 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {-\frac {\left (33 a^2 d^2-44 a b c d+16 b^2 c^2\right ) \left (-\frac {2 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\sqrt {b x^2+a x}}{x^3}dx}{5 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right )}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right )}{11 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {-\frac {\left (-\frac {2 b \left (-\frac {4 b \left (\frac {4 b \left (a x+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {2 \left (a x+b x^2\right )^{3/2}}{5 a x^4}\right )}{7 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{7 a x^5}\right )}{3 a}-\frac {2 \left (a x+b x^2\right )^{3/2}}{9 a x^6}\right ) \left (33 a^2 d^2-44 a b c d+16 b^2 c^2\right )}{11 a}-\frac {4 b c^2 \left (a x+b x^2\right )^{3/2}}{11 a x^7}}{2 b}-\frac {d^2 \left (a x+b x^2\right )^{3/2}}{3 b x^6}\) |
Input:
Int[((c + d*x)^2*Sqrt[a*x + b*x^2])/x^7,x]
Output:
-1/3*(d^2*(a*x + b*x^2)^(3/2))/(b*x^6) + ((-4*b*c^2*(a*x + b*x^2)^(3/2))/( 11*a*x^7) - ((16*b^2*c^2 - 44*a*b*c*d + 33*a^2*d^2)*((-2*(a*x + b*x^2)^(3/ 2))/(9*a*x^6) - (2*b*((-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)))/(11*a))/(2*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 = 0.57 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.62
| method | result | size |
| pseudoelliptic | \(-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right ) \left (\left (\frac {11}{7} d^{2} x^{2}+\frac {22}{9} c d x +c^{2}\right ) a^{4}-\frac {8 \left (\frac {99}{70} d^{2} x^{2}+\frac {33}{14} c d x +c^{2}\right ) x b \,a^{3}}{9}+\frac {16 x^{2} b^{2} \left (\frac {11}{10} d^{2} x^{2}+\frac {11}{5} c d x +c^{2}\right ) a^{2}}{21}-\frac {64 x^{3} b^{3} \left (\frac {11 d x}{6}+c \right ) c a}{105}+\frac {128 b^{4} c^{2} x^{4}}{315}\right )}{11 x^{6} a^{5}}\) | \(124\) |
| gosper | \(-\frac {2 \left (b x +a \right ) \left (264 a^{2} b^{2} d^{2} x^{4}-352 a \,b^{3} c d \,x^{4}+128 b^{4} c^{2} x^{4}-396 a^{3} b \,d^{2} x^{3}+528 a^{2} b^{2} c d \,x^{3}-192 a \,b^{3} c^{2} x^{3}+495 a^{4} d^{2} x^{2}-660 a^{3} d c b \,x^{2}+240 a^{2} b^{2} c^{2} x^{2}+770 a^{4} c d x -280 a^{3} b \,c^{2} x +315 a^{4} c^{2}\right ) \sqrt {b \,x^{2}+a x}}{3465 x^{6} a^{5}}\) | \(161\) |
| orering | \(-\frac {2 \left (b x +a \right ) \left (264 a^{2} b^{2} d^{2} x^{4}-352 a \,b^{3} c d \,x^{4}+128 b^{4} c^{2} x^{4}-396 a^{3} b \,d^{2} x^{3}+528 a^{2} b^{2} c d \,x^{3}-192 a \,b^{3} c^{2} x^{3}+495 a^{4} d^{2} x^{2}-660 a^{3} d c b \,x^{2}+240 a^{2} b^{2} c^{2} x^{2}+770 a^{4} c d x -280 a^{3} b \,c^{2} x +315 a^{4} c^{2}\right ) \sqrt {b \,x^{2}+a x}}{3465 x^{6} a^{5}}\) | \(161\) |
| trager | \(-\frac {2 \left (264 a^{2} b^{3} d^{2} x^{5}-352 a \,b^{4} c d \,x^{5}+128 b^{5} c^{2} x^{5}-132 a^{3} b^{2} d^{2} x^{4}+176 a^{2} b^{3} c d \,x^{4}-64 a \,b^{4} c^{2} x^{4}+99 a^{4} b \,d^{2} x^{3}-132 a^{3} b^{2} c d \,x^{3}+48 a^{2} b^{3} c^{2} x^{3}+495 a^{5} d^{2} x^{2}+110 a^{4} b c d \,x^{2}-40 a^{3} b^{2} c^{2} x^{2}+770 a^{5} c d x +35 a^{4} b \,c^{2} x +315 a^{5} c^{2}\right ) \sqrt {b \,x^{2}+a x}}{3465 x^{6} a^{5}}\) | \(197\) |
| risch | \(-\frac {2 \left (b x +a \right ) \left (264 a^{2} b^{3} d^{2} x^{5}-352 a \,b^{4} c d \,x^{5}+128 b^{5} c^{2} x^{5}-132 a^{3} b^{2} d^{2} x^{4}+176 a^{2} b^{3} c d \,x^{4}-64 a \,b^{4} c^{2} x^{4}+99 a^{4} b \,d^{2} x^{3}-132 a^{3} b^{2} c d \,x^{3}+48 a^{2} b^{3} c^{2} x^{3}+495 a^{5} d^{2} x^{2}+110 a^{4} b c d \,x^{2}-40 a^{3} b^{2} c^{2} x^{2}+770 a^{5} c d x +35 a^{4} b \,c^{2} x +315 a^{5} c^{2}\right )}{3465 x^{5} \sqrt {x \left (b x +a \right )}\, a^{5}}\) | \(200\) |
| default | \(c^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{11 a \,x^{7}}-\frac {8 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{9 a \,x^{6}}-\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )+d^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )+2 c d \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{9 a \,x^{6}}-\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{7 a \,x^{5}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 a \,x^{4}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )\) | \(290\) |
Input:
int((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^7,x,method=_RETURNVERBOSE)
Output:
-2/11*(x*(b*x+a))^(1/2)*(b*x+a)*((11/7*d^2*x^2+22/9*c*d*x+c^2)*a^4-8/9*(99 /70*d^2*x^2+33/14*c*d*x+c^2)*x*b*a^3+16/21*x^2*b^2*(11/10*d^2*x^2+11/5*c*d *x+c^2)*a^2-64/105*x^3*b^3*(11/6*d*x+c)*c*a+128/315*b^4*c^2*x^4)/x^6/a^5
Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=-\frac {2 \, {\left (315 \, a^{5} c^{2} + 8 \, {\left (16 \, b^{5} c^{2} - 44 \, a b^{4} c d + 33 \, a^{2} b^{3} d^{2}\right )} x^{5} - 4 \, {\left (16 \, a b^{4} c^{2} - 44 \, a^{2} b^{3} c d + 33 \, a^{3} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (16 \, a^{2} b^{3} c^{2} - 44 \, a^{3} b^{2} c d + 33 \, a^{4} b d^{2}\right )} x^{3} - 5 \, {\left (8 \, a^{3} b^{2} c^{2} - 22 \, a^{4} b c d - 99 \, a^{5} d^{2}\right )} x^{2} + 35 \, {\left (a^{4} b c^{2} + 22 \, a^{5} c d\right )} x\right )} \sqrt {b x^{2} + a x}}{3465 \, a^{5} x^{6}} \] Input:
integrate((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^7,x, algorithm="fricas")
Output:
-2/3465*(315*a^5*c^2 + 8*(16*b^5*c^2 - 44*a*b^4*c*d + 33*a^2*b^3*d^2)*x^5 - 4*(16*a*b^4*c^2 - 44*a^2*b^3*c*d + 33*a^3*b^2*d^2)*x^4 + 3*(16*a^2*b^3*c ^2 - 44*a^3*b^2*c*d + 33*a^4*b*d^2)*x^3 - 5*(8*a^3*b^2*c^2 - 22*a^4*b*c*d - 99*a^5*d^2)*x^2 + 35*(a^4*b*c^2 + 22*a^5*c*d)*x)*sqrt(b*x^2 + a*x)/(a^5* x^6)
\[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=\int \frac {\sqrt {x \left (a + b x\right )} \left (c + d x\right )^{2}}{x^{7}}\, dx \] Input:
integrate((d*x+c)**2*(b*x**2+a*x)**(1/2)/x**7,x)
Output:
Integral(sqrt(x*(a + b*x))*(c + d*x)**2/x**7, x)
Time = 0.04 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.74 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} b^{5} c^{2}}{3465 \, a^{5} x} + \frac {64 \, \sqrt {b x^{2} + a x} b^{4} c d}{315 \, a^{4} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{3} d^{2}}{105 \, a^{3} x} + \frac {128 \, \sqrt {b x^{2} + a x} b^{4} c^{2}}{3465 \, a^{4} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} b^{3} c d}{315 \, a^{3} x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} b^{2} d^{2}}{105 \, a^{2} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} b^{3} c^{2}}{1155 \, a^{3} x^{3}} + \frac {8 \, \sqrt {b x^{2} + a x} b^{2} c d}{105 \, a^{2} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} b d^{2}}{35 \, a x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} b^{2} c^{2}}{693 \, a^{2} x^{4}} - \frac {4 \, \sqrt {b x^{2} + a x} b c d}{63 \, a x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} d^{2}}{7 \, x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} b c^{2}}{99 \, a x^{5}} - \frac {4 \, \sqrt {b x^{2} + a x} c d}{9 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} c^{2}}{11 \, x^{6}} \] Input:
integrate((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^7,x, algorithm="maxima")
Output:
-256/3465*sqrt(b*x^2 + a*x)*b^5*c^2/(a^5*x) + 64/315*sqrt(b*x^2 + a*x)*b^4 *c*d/(a^4*x) - 16/105*sqrt(b*x^2 + a*x)*b^3*d^2/(a^3*x) + 128/3465*sqrt(b* x^2 + a*x)*b^4*c^2/(a^4*x^2) - 32/315*sqrt(b*x^2 + a*x)*b^3*c*d/(a^3*x^2) + 8/105*sqrt(b*x^2 + a*x)*b^2*d^2/(a^2*x^2) - 32/1155*sqrt(b*x^2 + a*x)*b^ 3*c^2/(a^3*x^3) + 8/105*sqrt(b*x^2 + a*x)*b^2*c*d/(a^2*x^3) - 2/35*sqrt(b* x^2 + a*x)*b*d^2/(a*x^3) + 16/693*sqrt(b*x^2 + a*x)*b^2*c^2/(a^2*x^4) - 4/ 63*sqrt(b*x^2 + a*x)*b*c*d/(a*x^4) - 2/7*sqrt(b*x^2 + a*x)*d^2/x^4 - 2/99* sqrt(b*x^2 + a*x)*b*c^2/(a*x^5) - 4/9*sqrt(b*x^2 + a*x)*c*d/x^5 - 2/11*sqr t(b*x^2 + a*x)*c^2/x^6
Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (179) = 358\).
Time = 0.14 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.72 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=\frac {2 \, {\left (4620 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} b^{2} d^{2} + 13860 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} b^{\frac {5}{2}} c d + 10395 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a b^{\frac {3}{2}} d^{2} + 11088 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b^{3} c^{2} + 38808 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a b^{2} c d + 9009 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{2} b d^{2} + 36960 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a b^{\frac {5}{2}} c^{2} + 43890 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{2} b^{\frac {3}{2}} c d + 3465 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{3} \sqrt {b} d^{2} + 51480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} b^{2} c^{2} + 24750 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{3} b c d + 495 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{4} d^{2} + 38115 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{3} b^{\frac {3}{2}} c^{2} + 6930 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{4} \sqrt {b} c d + 15785 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{4} b c^{2} + 770 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{5} c d + 3465 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{5} \sqrt {b} c^{2} + 315 \, a^{6} c^{2}\right )}}{3465 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{11}} \] Input:
integrate((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^7,x, algorithm="giac")
Output:
2/3465*(4620*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*b^2*d^2 + 13860*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*b^(5/2)*c*d + 10395*(sqrt(b)*x - sqrt(b*x^2 + a*x)) ^7*a*b^(3/2)*d^2 + 11088*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b^3*c^2 + 38808 *(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a*b^2*c*d + 9009*(sqrt(b)*x - sqrt(b*x^ 2 + a*x))^6*a^2*b*d^2 + 36960*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a*b^(5/2)* c^2 + 43890*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^2*b^(3/2)*c*d + 3465*(sqrt (b)*x - sqrt(b*x^2 + a*x))^5*a^3*sqrt(b)*d^2 + 51480*(sqrt(b)*x - sqrt(b*x ^2 + a*x))^4*a^2*b^2*c^2 + 24750*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^3*b*c *d + 495*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^4*d^2 + 38115*(sqrt(b)*x - sq rt(b*x^2 + a*x))^3*a^3*b^(3/2)*c^2 + 6930*(sqrt(b)*x - sqrt(b*x^2 + a*x))^ 3*a^4*sqrt(b)*c*d + 15785*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^4*b*c^2 + 77 0*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^5*c*d + 3465*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^5*sqrt(b)*c^2 + 315*a^6*c^2)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^11
Time = 11.07 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.74 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=\frac {16\,b^2\,c^2\,\sqrt {b\,x^2+a\,x}}{693\,a^2\,x^4}-\frac {2\,d^2\,\sqrt {b\,x^2+a\,x}}{7\,x^4}-\frac {4\,c\,d\,\sqrt {b\,x^2+a\,x}}{9\,x^5}-\frac {2\,c^2\,\sqrt {b\,x^2+a\,x}}{11\,x^6}-\frac {32\,b^3\,c^2\,\sqrt {b\,x^2+a\,x}}{1155\,a^3\,x^3}+\frac {128\,b^4\,c^2\,\sqrt {b\,x^2+a\,x}}{3465\,a^4\,x^2}-\frac {256\,b^5\,c^2\,\sqrt {b\,x^2+a\,x}}{3465\,a^5\,x}+\frac {8\,b^2\,d^2\,\sqrt {b\,x^2+a\,x}}{105\,a^2\,x^2}-\frac {16\,b^3\,d^2\,\sqrt {b\,x^2+a\,x}}{105\,a^3\,x}-\frac {2\,b\,c^2\,\sqrt {b\,x^2+a\,x}}{99\,a\,x^5}-\frac {2\,b\,d^2\,\sqrt {b\,x^2+a\,x}}{35\,a\,x^3}+\frac {8\,b^2\,c\,d\,\sqrt {b\,x^2+a\,x}}{105\,a^2\,x^3}-\frac {32\,b^3\,c\,d\,\sqrt {b\,x^2+a\,x}}{315\,a^3\,x^2}+\frac {64\,b^4\,c\,d\,\sqrt {b\,x^2+a\,x}}{315\,a^4\,x}-\frac {4\,b\,c\,d\,\sqrt {b\,x^2+a\,x}}{63\,a\,x^4} \] Input:
int(((a*x + b*x^2)^(1/2)*(c + d*x)^2)/x^7,x)
Output:
(16*b^2*c^2*(a*x + b*x^2)^(1/2))/(693*a^2*x^4) - (2*d^2*(a*x + b*x^2)^(1/2 ))/(7*x^4) - (4*c*d*(a*x + b*x^2)^(1/2))/(9*x^5) - (2*c^2*(a*x + b*x^2)^(1 /2))/(11*x^6) - (32*b^3*c^2*(a*x + b*x^2)^(1/2))/(1155*a^3*x^3) + (128*b^4 *c^2*(a*x + b*x^2)^(1/2))/(3465*a^4*x^2) - (256*b^5*c^2*(a*x + b*x^2)^(1/2 ))/(3465*a^5*x) + (8*b^2*d^2*(a*x + b*x^2)^(1/2))/(105*a^2*x^2) - (16*b^3* d^2*(a*x + b*x^2)^(1/2))/(105*a^3*x) - (2*b*c^2*(a*x + b*x^2)^(1/2))/(99*a *x^5) - (2*b*d^2*(a*x + b*x^2)^(1/2))/(35*a*x^3) + (8*b^2*c*d*(a*x + b*x^2 )^(1/2))/(105*a^2*x^3) - (32*b^3*c*d*(a*x + b*x^2)^(1/2))/(315*a^3*x^2) + (64*b^4*c*d*(a*x + b*x^2)^(1/2))/(315*a^4*x) - (4*b*c*d*(a*x + b*x^2)^(1/2 ))/(63*a*x^4)
Time = 0.23 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.74 \[ \int \frac {(c+d x)^2 \sqrt {a x+b x^2}}{x^7} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} c^{2}}{11}-\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{5} c d x}{9}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5} d^{2} x^{2}}{7}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b \,c^{2} x}{99}-\frac {4 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b c d \,x^{2}}{63}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b \,d^{2} x^{3}}{35}+\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} c^{2} x^{2}}{693}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} c d \,x^{3}}{105}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} d^{2} x^{4}}{105}-\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} c^{2} x^{3}}{1155}-\frac {32 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} c d \,x^{4}}{315}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} d^{2} x^{5}}{105}+\frac {128 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c^{2} x^{4}}{3465}+\frac {64 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c d \,x^{5}}{315}-\frac {256 \sqrt {x}\, \sqrt {b x +a}\, b^{5} c^{2} x^{5}}{3465}+\frac {16 \sqrt {b}\, a^{2} b^{3} d^{2} x^{6}}{105}-\frac {64 \sqrt {b}\, a \,b^{4} c d \,x^{6}}{315}+\frac {256 \sqrt {b}\, b^{5} c^{2} x^{6}}{3465}}{a^{5} x^{6}} \] Input:
int((d*x+c)^2*(b*x^2+a*x)^(1/2)/x^7,x)
Output:
(2*( - 315*sqrt(x)*sqrt(a + b*x)*a**5*c**2 - 770*sqrt(x)*sqrt(a + b*x)*a** 5*c*d*x - 495*sqrt(x)*sqrt(a + b*x)*a**5*d**2*x**2 - 35*sqrt(x)*sqrt(a + b *x)*a**4*b*c**2*x - 110*sqrt(x)*sqrt(a + b*x)*a**4*b*c*d*x**2 - 99*sqrt(x) *sqrt(a + b*x)*a**4*b*d**2*x**3 + 40*sqrt(x)*sqrt(a + b*x)*a**3*b**2*c**2* x**2 + 132*sqrt(x)*sqrt(a + b*x)*a**3*b**2*c*d*x**3 + 132*sqrt(x)*sqrt(a + b*x)*a**3*b**2*d**2*x**4 - 48*sqrt(x)*sqrt(a + b*x)*a**2*b**3*c**2*x**3 - 176*sqrt(x)*sqrt(a + b*x)*a**2*b**3*c*d*x**4 - 264*sqrt(x)*sqrt(a + b*x)* a**2*b**3*d**2*x**5 + 64*sqrt(x)*sqrt(a + b*x)*a*b**4*c**2*x**4 + 352*sqrt (x)*sqrt(a + b*x)*a*b**4*c*d*x**5 - 128*sqrt(x)*sqrt(a + b*x)*b**5*c**2*x* *5 + 264*sqrt(b)*a**2*b**3*d**2*x**6 - 352*sqrt(b)*a*b**4*c*d*x**6 + 128*s qrt(b)*b**5*c**2*x**6))/(3465*a**5*x**6)