Integrand size = 24, antiderivative size = 196 \[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 c^2}{7 a x^3 \sqrt {a x+b x^2}}+\frac {4 c (4 b c-7 a d)}{35 a^2 x^2 \sqrt {a x+b x^2}}+\frac {2 \left (35 d^2+\frac {12 b c (4 b c-7 a d)}{a^2}\right )}{35 a x \sqrt {a x+b x^2}}-\frac {8 \left (35 a^2 d^2+12 b c (4 b c-7 a d)\right ) \sqrt {a x+b x^2}}{105 a^4 x^2}+\frac {16 b \left (35 a^2 d^2+12 b c (4 b c-7 a d)\right ) \sqrt {a x+b x^2}}{105 a^5 x} \] Output:
-2/7*c^2/a/x^3/(b*x^2+a*x)^(1/2)+4/35*c*(-7*a*d+4*b*c)/a^2/x^2/(b*x^2+a*x) ^(1/2)+2/35*(35*d^2+12*b*c*(-7*a*d+4*b*c)/a^2)/a/x/(b*x^2+a*x)^(1/2)-8/105 *(35*a^2*d^2+12*b*c*(-7*a*d+4*b*c))*(b*x^2+a*x)^(1/2)/a^4/x^2+16/105*b*(35 *a^2*d^2+12*b*c*(-7*a*d+4*b*c))*(b*x^2+a*x)^(1/2)/a^5/x
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \left (384 b^4 c^2 x^4+96 a b^3 c x^3 (2 c-7 d x)+8 a^2 b^2 x^2 \left (-6 c^2-42 c d x+35 d^2 x^2\right )+4 a^3 b x \left (6 c^2+21 c d x+35 d^2 x^2\right )-a^4 \left (15 c^2+42 c d x+35 d^2 x^2\right )\right )}{105 a^5 x^3 \sqrt {x (a+b x)}} \] Input:
Integrate[(c + d*x)^2/(x^3*(a*x + b*x^2)^(3/2)),x]
Output:
(2*(384*b^4*c^2*x^4 + 96*a*b^3*c*x^3*(2*c - 7*d*x) + 8*a^2*b^2*x^2*(-6*c^2 - 42*c*d*x + 35*d^2*x^2) + 4*a^3*b*x*(6*c^2 + 21*c*d*x + 35*d^2*x^2) - a^ 4*(15*c^2 + 42*c*d*x + 35*d^2*x^2)))/(105*a^5*x^3*Sqrt[x*(a + b*x)])
Time = 0.59 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.89, 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, 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.
| \(\displaystyle \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1262 |
| \(\displaystyle -\frac {\int -\frac {6 b c^2+d (12 b c-5 a d) x}{2 x^3 \left (b x^2+a x\right )^{3/2}}dx}{3 b}-\frac {d^2}{3 b x^2 \sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 b c^2+d (12 b c-5 a d) x}{x^3 \left (b x^2+a x\right )^{3/2}}dx}{6 b}-\frac {d^2}{3 b x^2 \sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {-\frac {\left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \int \frac {1}{x^2 \left (b x^2+a x\right )^{3/2}}dx}{7 a}-\frac {12 b c^2}{7 a x^3 \sqrt {a x+b x^2}}}{6 b}-\frac {d^2}{3 b x^2 \sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {-\frac {\left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \left (-\frac {6 b \int \frac {1}{x \left (b x^2+a x\right )^{3/2}}dx}{5 a}-\frac {2}{5 a x^2 \sqrt {a x+b x^2}}\right )}{7 a}-\frac {12 b c^2}{7 a x^3 \sqrt {a x+b x^2}}}{6 b}-\frac {d^2}{3 b x^2 \sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {-\frac {\left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \left (-\frac {6 b \left (-\frac {4 b \int \frac {1}{\left (b x^2+a x\right )^{3/2}}dx}{3 a}-\frac {2}{3 a x \sqrt {a x+b x^2}}\right )}{5 a}-\frac {2}{5 a x^2 \sqrt {a x+b x^2}}\right )}{7 a}-\frac {12 b c^2}{7 a x^3 \sqrt {a x+b x^2}}}{6 b}-\frac {d^2}{3 b x^2 \sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {-\frac {\left (-\frac {6 b \left (\frac {8 b (a+2 b x)}{3 a^3 \sqrt {a x+b x^2}}-\frac {2}{3 a x \sqrt {a x+b x^2}}\right )}{5 a}-\frac {2}{5 a x^2 \sqrt {a x+b x^2}}\right ) \left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right )}{7 a}-\frac {12 b c^2}{7 a x^3 \sqrt {a x+b x^2}}}{6 b}-\frac {d^2}{3 b x^2 \sqrt {a x+b x^2}}\) |
Input:
Int[(c + d*x)^2/(x^3*(a*x + b*x^2)^(3/2)),x]
Output:
-1/3*d^2/(b*x^2*Sqrt[a*x + b*x^2]) + ((-12*b*c^2)/(7*a*x^3*Sqrt[a*x + b*x^ 2]) - ((48*b^2*c^2 - 84*a*b*c*d + 35*a^2*d^2)*(-2/(5*a*x^2*Sqrt[a*x + b*x^ 2]) - (6*b*(-2/(3*a*x*Sqrt[a*x + b*x^2]) + (8*b*(a + 2*b*x))/(3*a^3*Sqrt[a *x + b*x^2])))/(5*a)))/(7*a))/(6*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_))^(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 = 121, normalized size of antiderivative = 0.62
| method | result | size |
| pseudoelliptic | \(\frac {\left (-70 d^{2} x^{2}-84 c d x -30 c^{2}\right ) a^{4}+48 x \left (\frac {35}{6} d^{2} x^{2}+\frac {7}{2} c d x +c^{2}\right ) b \,a^{3}-96 x^{2} \left (-\frac {35}{6} d^{2} x^{2}+7 c d x +c^{2}\right ) b^{2} a^{2}+384 \left (-\frac {7 d x}{2}+c \right ) x^{3} b^{3} c a +768 b^{4} c^{2} x^{4}}{105 \sqrt {x \left (b x +a \right )}\, x^{3} a^{5}}\) | \(121\) |
| risch | \(-\frac {2 \left (b x +a \right ) \left (-175 d^{2} x^{3} a^{2} b +462 a \,b^{2} c d \,x^{3}-279 b^{3} c^{2} x^{3}+35 a^{3} d^{2} x^{2}-126 x^{2} a^{2} b c d +87 a \,b^{2} c^{2} x^{2}+42 a^{3} c d x -39 a^{2} b \,c^{2} x +15 c^{2} a^{3}\right )}{105 a^{5} x^{3} \sqrt {x \left (b x +a \right )}}+\frac {2 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{\sqrt {x \left (b x +a \right )}\, a^{5}}\) | \(158\) |
| gosper | \(-\frac {2 \left (b x +a \right ) \left (-280 a^{2} b^{2} d^{2} x^{4}+672 a \,b^{3} c d \,x^{4}-384 b^{4} c^{2} x^{4}-140 a^{3} b \,d^{2} x^{3}+336 a^{2} b^{2} c d \,x^{3}-192 a \,b^{3} c^{2} x^{3}+35 a^{4} d^{2} x^{2}-84 a^{3} d c b \,x^{2}+48 a^{2} b^{2} c^{2} x^{2}+42 a^{4} c d x -24 a^{3} b \,c^{2} x +15 a^{4} c^{2}\right )}{105 x^{2} a^{5} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}\) | \(161\) |
| orering | \(-\frac {2 \left (b x +a \right ) \left (-280 a^{2} b^{2} d^{2} x^{4}+672 a \,b^{3} c d \,x^{4}-384 b^{4} c^{2} x^{4}-140 a^{3} b \,d^{2} x^{3}+336 a^{2} b^{2} c d \,x^{3}-192 a \,b^{3} c^{2} x^{3}+35 a^{4} d^{2} x^{2}-84 a^{3} d c b \,x^{2}+48 a^{2} b^{2} c^{2} x^{2}+42 a^{4} c d x -24 a^{3} b \,c^{2} x +15 a^{4} c^{2}\right )}{105 x^{2} a^{5} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}\) | \(161\) |
| trager | \(-\frac {2 \left (-280 a^{2} b^{2} d^{2} x^{4}+672 a \,b^{3} c d \,x^{4}-384 b^{4} c^{2} x^{4}-140 a^{3} b \,d^{2} x^{3}+336 a^{2} b^{2} c d \,x^{3}-192 a \,b^{3} c^{2} x^{3}+35 a^{4} d^{2} x^{2}-84 a^{3} d c b \,x^{2}+48 a^{2} b^{2} c^{2} x^{2}+42 a^{4} c d x -24 a^{3} b \,c^{2} x +15 a^{4} c^{2}\right ) \sqrt {b \,x^{2}+a x}}{105 \left (b x +a \right ) a^{5} x^{4}}\) | \(163\) |
| default | \(c^{2} \left (-\frac {2}{7 a \,x^{3} \sqrt {b \,x^{2}+a x}}-\frac {8 b \left (-\frac {2}{5 a \,x^{2} \sqrt {b \,x^{2}+a x}}-\frac {6 b \left (-\frac {2}{3 a x \sqrt {b \,x^{2}+a x}}+\frac {8 b \left (2 b x +a \right )}{3 a^{3} \sqrt {b \,x^{2}+a x}}\right )}{5 a}\right )}{7 a}\right )+d^{2} \left (-\frac {2}{3 a x \sqrt {b \,x^{2}+a x}}+\frac {8 b \left (2 b x +a \right )}{3 a^{3} \sqrt {b \,x^{2}+a x}}\right )+2 c d \left (-\frac {2}{5 a \,x^{2} \sqrt {b \,x^{2}+a x}}-\frac {6 b \left (-\frac {2}{3 a x \sqrt {b \,x^{2}+a x}}+\frac {8 b \left (2 b x +a \right )}{3 a^{3} \sqrt {b \,x^{2}+a x}}\right )}{5 a}\right )\) | \(221\) |
Input:
int((d*x+c)^2/x^3/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/105*((-70*d^2*x^2-84*c*d*x-30*c^2)*a^4+48*x*(35/6*d^2*x^2+7/2*c*d*x+c^2) *b*a^3-96*x^2*(-35/6*d^2*x^2+7*c*d*x+c^2)*b^2*a^2+384*(-7/2*d*x+c)*x^3*b^3 *c*a+768*b^4*c^2*x^4)/(x*(b*x+a))^(1/2)/x^3/a^5
Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.81 \[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (15 \, a^{4} c^{2} - 8 \, {\left (48 \, b^{4} c^{2} - 84 \, a b^{3} c d + 35 \, a^{2} b^{2} d^{2}\right )} x^{4} - 4 \, {\left (48 \, a b^{3} c^{2} - 84 \, a^{2} b^{2} c d + 35 \, a^{3} b d^{2}\right )} x^{3} + {\left (48 \, a^{2} b^{2} c^{2} - 84 \, a^{3} b c d + 35 \, a^{4} d^{2}\right )} x^{2} - 6 \, {\left (4 \, a^{3} b c^{2} - 7 \, a^{4} c d\right )} x\right )} \sqrt {b x^{2} + a x}}{105 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} \] Input:
integrate((d*x+c)^2/x^3/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
-2/105*(15*a^4*c^2 - 8*(48*b^4*c^2 - 84*a*b^3*c*d + 35*a^2*b^2*d^2)*x^4 - 4*(48*a*b^3*c^2 - 84*a^2*b^2*c*d + 35*a^3*b*d^2)*x^3 + (48*a^2*b^2*c^2 - 8 4*a^3*b*c*d + 35*a^4*d^2)*x^2 - 6*(4*a^3*b*c^2 - 7*a^4*c*d)*x)*sqrt(b*x^2 + a*x)/(a^5*b*x^5 + a^6*x^4)
\[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{2}}{x^{3} \left (x \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**2/x**3/(b*x**2+a*x)**(3/2),x)
Output:
Integral((c + d*x)**2/(x**3*(x*(a + b*x))**(3/2)), x)
Time = 0.03 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.36 \[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=\frac {256 \, b^{4} c^{2} x}{35 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {64 \, b^{3} c d x}{5 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {16 \, b^{2} d^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {128 \, b^{3} c^{2}}{35 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {32 \, b^{2} c d}{5 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {8 \, b d^{2}}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {32 \, b^{2} c^{2}}{35 \, \sqrt {b x^{2} + a x} a^{3} x} + \frac {8 \, b c d}{5 \, \sqrt {b x^{2} + a x} a^{2} x} - \frac {2 \, d^{2}}{3 \, \sqrt {b x^{2} + a x} a x} + \frac {16 \, b c^{2}}{35 \, \sqrt {b x^{2} + a x} a^{2} x^{2}} - \frac {4 \, c d}{5 \, \sqrt {b x^{2} + a x} a x^{2}} - \frac {2 \, c^{2}}{7 \, \sqrt {b x^{2} + a x} a x^{3}} \] Input:
integrate((d*x+c)^2/x^3/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
256/35*b^4*c^2*x/(sqrt(b*x^2 + a*x)*a^5) - 64/5*b^3*c*d*x/(sqrt(b*x^2 + a* x)*a^4) + 16/3*b^2*d^2*x/(sqrt(b*x^2 + a*x)*a^3) + 128/35*b^3*c^2/(sqrt(b* x^2 + a*x)*a^4) - 32/5*b^2*c*d/(sqrt(b*x^2 + a*x)*a^3) + 8/3*b*d^2/(sqrt(b *x^2 + a*x)*a^2) - 32/35*b^2*c^2/(sqrt(b*x^2 + a*x)*a^3*x) + 8/5*b*c*d/(sq rt(b*x^2 + a*x)*a^2*x) - 2/3*d^2/(sqrt(b*x^2 + a*x)*a*x) + 16/35*b*c^2/(sq rt(b*x^2 + a*x)*a^2*x^2) - 4/5*c*d/(sqrt(b*x^2 + a*x)*a*x^2) - 2/7*c^2/(sq rt(b*x^2 + a*x)*a*x^3)
\[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:
integrate((d*x+c)^2/x^3/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^2/((b*x^2 + a*x)^(3/2)*x^3), x)
Time = 9.88 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2\,c^2\,\sqrt {b\,x^2+a\,x}}{7\,a^2\,x^4}-\frac {\sqrt {b\,x^2+a\,x}\,\left (70\,a^3\,d^2-252\,a^2\,b\,c\,d+174\,a\,b^2\,c^2\right )}{105\,a^5\,x^2}-\frac {\sqrt {b\,x^2+a\,x}\,\left (x\,\left (\frac {140\,a^2\,b^2\,d^2-504\,a\,b^3\,c\,d+348\,b^4\,c^2}{105\,a^5}-\frac {4\,b^2\,\left (175\,a^2\,d^2-462\,a\,b\,c\,d+279\,b^2\,c^2\right )}{105\,a^5}\right )-\frac {2\,b\,\left (175\,a^2\,d^2-462\,a\,b\,c\,d+279\,b^2\,c^2\right )}{105\,a^4}\right )}{x\,\left (a+b\,x\right )}-\frac {2\,c\,\sqrt {b\,x^2+a\,x}\,\left (14\,a\,d-13\,b\,c\right )}{35\,a^3\,x^3} \] Input:
int((c + d*x)^2/(x^3*(a*x + b*x^2)^(3/2)),x)
Output:
- (2*c^2*(a*x + b*x^2)^(1/2))/(7*a^2*x^4) - ((a*x + b*x^2)^(1/2)*(70*a^3*d ^2 + 174*a*b^2*c^2 - 252*a^2*b*c*d))/(105*a^5*x^2) - ((a*x + b*x^2)^(1/2)* (x*((348*b^4*c^2 + 140*a^2*b^2*d^2 - 504*a*b^3*c*d)/(105*a^5) - (4*b^2*(17 5*a^2*d^2 + 279*b^2*c^2 - 462*a*b*c*d))/(105*a^5)) - (2*b*(175*a^2*d^2 + 2 79*b^2*c^2 - 462*a*b*c*d))/(105*a^4)))/(x*(a + b*x)) - (2*c*(a*x + b*x^2)^ (1/2)*(14*a*d - 13*b*c))/(35*a^3*x^3)
Time = 0.20 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.19 \[ \int \frac {(c+d x)^2}{x^3 \left (a x+b x^2\right )^{3/2}} \, dx=\frac {-\frac {16 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b \,d^{2} x^{4}}{3}+\frac {64 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c d \,x^{4}}{5}-\frac {256 \sqrt {b}\, \sqrt {b x +a}\, b^{3} c^{2} x^{4}}{35}-\frac {2 \sqrt {x}\, a^{4} c^{2}}{7}-\frac {4 \sqrt {x}\, a^{4} c d x}{5}-\frac {2 \sqrt {x}\, a^{4} d^{2} x^{2}}{3}+\frac {16 \sqrt {x}\, a^{3} b \,c^{2} x}{35}+\frac {8 \sqrt {x}\, a^{3} b c d \,x^{2}}{5}+\frac {8 \sqrt {x}\, a^{3} b \,d^{2} x^{3}}{3}-\frac {32 \sqrt {x}\, a^{2} b^{2} c^{2} x^{2}}{35}-\frac {32 \sqrt {x}\, a^{2} b^{2} c d \,x^{3}}{5}+\frac {16 \sqrt {x}\, a^{2} b^{2} d^{2} x^{4}}{3}+\frac {128 \sqrt {x}\, a \,b^{3} c^{2} x^{3}}{35}-\frac {64 \sqrt {x}\, a \,b^{3} c d \,x^{4}}{5}+\frac {256 \sqrt {x}\, b^{4} c^{2} x^{4}}{35}}{\sqrt {b x +a}\, a^{5} x^{4}} \] Input:
int((d*x+c)^2/x^3/(b*x^2+a*x)^(3/2),x)
Output:
(2*( - 280*sqrt(b)*sqrt(a + b*x)*a**2*b*d**2*x**4 + 672*sqrt(b)*sqrt(a + b *x)*a*b**2*c*d*x**4 - 384*sqrt(b)*sqrt(a + b*x)*b**3*c**2*x**4 - 15*sqrt(x )*a**4*c**2 - 42*sqrt(x)*a**4*c*d*x - 35*sqrt(x)*a**4*d**2*x**2 + 24*sqrt( x)*a**3*b*c**2*x + 84*sqrt(x)*a**3*b*c*d*x**2 + 140*sqrt(x)*a**3*b*d**2*x* *3 - 48*sqrt(x)*a**2*b**2*c**2*x**2 - 336*sqrt(x)*a**2*b**2*c*d*x**3 + 280 *sqrt(x)*a**2*b**2*d**2*x**4 + 192*sqrt(x)*a*b**3*c**2*x**3 - 672*sqrt(x)* a*b**3*c*d*x**4 + 384*sqrt(x)*b**4*c**2*x**4))/(105*sqrt(a + b*x)*a**5*x** 4)