Integrand size = 24, antiderivative size = 170 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=b d^3 \sqrt {a x+b x^2}-\frac {2 d^2 (3 b c+a d) \sqrt {a x+b x^2}}{x}-\frac {2 c d^2 \left (a x+b x^2\right )^{3/2}}{x^3}-\frac {2 c^3 \left (a x+b x^2\right )^{5/2}}{7 a x^6}+\frac {2 c^2 (2 b c-21 a d) \left (a x+b x^2\right )^{5/2}}{35 a^2 x^5}+3 \sqrt {b} d^2 (2 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \] Output:
b*d^3*(b*x^2+a*x)^(1/2)-2*d^2*(a*d+3*b*c)*(b*x^2+a*x)^(1/2)/x-2*c*d^2*(b*x ^2+a*x)^(3/2)/x^3-2/7*c^3*(b*x^2+a*x)^(5/2)/a/x^6+2/35*c^2*(-21*a*d+2*b*c) *(b*x^2+a*x)^(5/2)/a^2/x^5+3*b^(1/2)*d^2*(a*d+2*b*c)*arctanh(b^(1/2)*x/(b* x^2+a*x)^(1/2))
Time = 0.73 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=\frac {\sqrt {x (a+b x)} \left (\frac {4 b^3 c^3 x^3}{a^2}-\frac {2 b^2 c^2 x^2 (c+21 d x)}{a}+b x \left (-16 c^3-84 c^2 d x-280 c d^2 x^2+35 d^3 x^3\right )-2 a \left (5 c^3+21 c^2 d x+35 c d^2 x^2+35 d^3 x^3\right )-\frac {105 \sqrt {b} d^2 (2 b c+a d) x^{7/2} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {a+b x}}\right )}{35 x^4} \] Input:
Integrate[((c + d*x)^3*(a*x + b*x^2)^(3/2))/x^6,x]
Output:
(Sqrt[x*(a + b*x)]*((4*b^3*c^3*x^3)/a^2 - (2*b^2*c^2*x^2*(c + 21*d*x))/a + b*x*(-16*c^3 - 84*c^2*d*x - 280*c*d^2*x^2 + 35*d^3*x^3) - 2*a*(5*c^3 + 21 *c^2*d*x + 35*c*d^2*x^2 + 35*d^3*x^3) - (105*Sqrt[b]*d^2*(2*b*c + a*d)*x^( 7/2)*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/Sqrt[a + b*x]))/(35*x^4)
Time = 0.82 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1262, 27, 2167, 2009}
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.
| \(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2} (c+d x)^3}{x^6} \, dx\) |
| \(\Big \downarrow \) 1262 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a x\right )^{3/2} \left (2 b c^3+6 b d x c^2+3 d^2 (2 b c+a d) x^2\right )}{2 x^6}dx}{b}+\frac {d^3 \left (a x+b x^2\right )^{5/2}}{b x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a x\right )^{3/2} \left (2 b c^3+6 b d x c^2+3 d^2 (2 b c+a d) x^2\right )}{x^6}dx}{2 b}+\frac {d^3 \left (a x+b x^2\right )^{5/2}}{b x^4}\) |
| \(\Big \downarrow \) 2167 |
| \(\displaystyle \frac {\int \left (\frac {2 b \left (b x^2+a x\right )^{3/2} c^3}{x^6}+\frac {6 b d \left (b x^2+a x\right )^{3/2} c^2}{x^5}+\frac {3 d^2 (2 b c+a d) \left (b x^2+a x\right )^{3/2}}{x^4}\right )dx}{2 b}+\frac {d^3 \left (a x+b x^2\right )^{5/2}}{b x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {8 b^2 c^3 \left (a x+b x^2\right )^{5/2}}{35 a^2 x^5}+6 b^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (a d+2 b c)-\frac {4 b c^3 \left (a x+b x^2\right )^{5/2}}{7 a x^6}-\frac {12 b c^2 d \left (a x+b x^2\right )^{5/2}}{5 a x^5}-\frac {6 b d^2 \sqrt {a x+b x^2} (a d+2 b c)}{x}-\frac {2 d^2 \left (a x+b x^2\right )^{3/2} (a d+2 b c)}{x^3}}{2 b}+\frac {d^3 \left (a x+b x^2\right )^{5/2}}{b x^4}\) |
Input:
Int[((c + d*x)^3*(a*x + b*x^2)^(3/2))/x^6,x]
Output:
(d^3*(a*x + b*x^2)^(5/2))/(b*x^4) + ((-6*b*d^2*(2*b*c + a*d)*Sqrt[a*x + b* x^2])/x - (2*d^2*(2*b*c + a*d)*(a*x + b*x^2)^(3/2))/x^3 - (4*b*c^3*(a*x + b*x^2)^(5/2))/(7*a*x^6) + (8*b^2*c^3*(a*x + b*x^2)^(5/2))/(35*a^2*x^5) - ( 12*b*c^2*d*(a*x + b*x^2)^(5/2))/(5*a*x^5) + 6*b^(3/2)*d^2*(2*b*c + a*d)*Ar cTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(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_.)*((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]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, (d + e*x)^m*Pq, x] , x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + Expon[Pq, x] + 2*p + 1, 0] && ILt Q[m, 0]
Time = 0.68 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\frac {21 a^{2} b \,d^{2} x^{4} \left (a d +2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{2}+\left (\frac {8 \left (-\frac {35}{16} d^{3} x^{3}+\frac {35}{2} c \,d^{2} x^{2}+\frac {21}{4} c^{2} d x +c^{3}\right ) x \,a^{2} b^{\frac {3}{2}}}{5}+\frac {a \,c^{2} x^{2} \left (21 d x +c \right ) b^{\frac {5}{2}}}{5}-\frac {2 b^{\frac {7}{2}} c^{3} x^{3}}{5}+a^{3} \sqrt {b}\, \left (7 d^{3} x^{3}+7 c \,d^{2} x^{2}+\frac {21}{5} c^{2} d x +c^{3}\right )\right ) \sqrt {x \left (b x +a \right )}\right )}{7 \sqrt {b}\, x^{4} a^{2}}\) | \(162\) |
| risch | \(-\frac {\left (b x +a \right ) \left (-35 a^{2} b \,d^{3} x^{4}+70 a^{3} d^{3} x^{3}+280 a^{2} b c \,d^{2} x^{3}+42 a \,b^{2} c^{2} d \,x^{3}-4 b^{3} c^{3} x^{3}+70 a^{3} c \,d^{2} x^{2}+84 a^{2} b \,c^{2} x^{2} d +2 a \,b^{2} c^{3} x^{2}+42 a^{3} c^{2} d x +16 a^{2} b \,c^{3} x +10 a^{3} c^{3}\right )}{35 x^{3} \sqrt {x \left (b x +a \right )}\, a^{2}}+\frac {3 \left (a d +2 b c \right ) \sqrt {b}\, d^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2}\) | \(190\) |
| default | \(c^{3} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 a \,x^{6}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )+d^{3} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{3}}+\frac {4 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{2}}-\frac {6 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{a}\right )}{a}\right )+3 c \,d^{2} \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{3 a \,x^{4}}+\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{3}}+\frac {4 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{a \,x^{2}}-\frac {6 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2}\right )}{a}\right )}{a}\right )}{3 a}\right )-\frac {6 c^{2} d \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 a \,x^{5}}\) | \(353\) |
Input:
int((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-2/7/b^(1/2)*(-21/2*a^2*b*d^2*x^4*(a*d+2*b*c)*arctanh((x*(b*x+a))^(1/2)/x/ b^(1/2))+(8/5*(-35/16*d^3*x^3+35/2*c*d^2*x^2+21/4*c^2*d*x+c^3)*x*a^2*b^(3/ 2)+1/5*a*c^2*x^2*(21*d*x+c)*b^(5/2)-2/5*b^(7/2)*c^3*x^3+a^3*b^(1/2)*(7*d^3 *x^3+7*c*d^2*x^2+21/5*c^2*d*x+c^3))*(x*(b*x+a))^(1/2))/x^4/a^2
Time = 0.11 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.25 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=\left [\frac {105 \, {\left (2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b} x^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (35 \, a^{2} b d^{3} x^{4} - 10 \, a^{3} c^{3} + 2 \, {\left (2 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d - 140 \, a^{2} b c d^{2} - 35 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (a b^{2} c^{3} + 42 \, a^{2} b c^{2} d + 35 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (8 \, a^{2} b c^{3} + 21 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{70 \, a^{2} x^{4}}, -\frac {105 \, {\left (2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b} x^{4} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (35 \, a^{2} b d^{3} x^{4} - 10 \, a^{3} c^{3} + 2 \, {\left (2 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d - 140 \, a^{2} b c d^{2} - 35 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (a b^{2} c^{3} + 42 \, a^{2} b c^{2} d + 35 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (8 \, a^{2} b c^{3} + 21 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{35 \, a^{2} x^{4}}\right ] \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^6,x, algorithm="fricas")
Output:
[1/70*(105*(2*a^2*b*c*d^2 + a^3*d^3)*sqrt(b)*x^4*log(2*b*x + a + 2*sqrt(b* x^2 + a*x)*sqrt(b)) + 2*(35*a^2*b*d^3*x^4 - 10*a^3*c^3 + 2*(2*b^3*c^3 - 21 *a*b^2*c^2*d - 140*a^2*b*c*d^2 - 35*a^3*d^3)*x^3 - 2*(a*b^2*c^3 + 42*a^2*b *c^2*d + 35*a^3*c*d^2)*x^2 - 2*(8*a^2*b*c^3 + 21*a^3*c^2*d)*x)*sqrt(b*x^2 + a*x))/(a^2*x^4), -1/35*(105*(2*a^2*b*c*d^2 + a^3*d^3)*sqrt(-b)*x^4*arcta n(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (35*a^2*b*d^3*x^4 - 10*a^3*c^3 + 2*(2*b^3*c^3 - 21*a*b^2*c^2*d - 140*a^2*b*c*d^2 - 35*a^3*d^3)*x^3 - 2*(a* b^2*c^3 + 42*a^2*b*c^2*d + 35*a^3*c*d^2)*x^2 - 2*(8*a^2*b*c^3 + 21*a^3*c^2 *d)*x)*sqrt(b*x^2 + a*x))/(a^2*x^4)]
\[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )^{3}}{x^{6}}\, dx \] Input:
integrate((d*x+c)**3*(b*x**2+a*x)**(3/2)/x**6,x)
Output:
Integral((x*(a + b*x))**(3/2)*(c + d*x)**3/x**6, x)
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (150) = 300\).
Time = 0.04 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.13 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=3 \, b^{\frac {3}{2}} c d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \frac {3}{2} \, a \sqrt {b} d^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \frac {4 \, \sqrt {b x^{2} + a x} b^{3} c^{3}}{35 \, a^{2} x} - \frac {6 \, \sqrt {b x^{2} + a x} b^{2} c^{2} d}{5 \, a x} - \frac {7 \, \sqrt {b x^{2} + a x} b c d^{2}}{x} - \frac {3 \, \sqrt {b x^{2} + a x} a d^{3}}{x} - \frac {2 \, \sqrt {b x^{2} + a x} b^{2} c^{3}}{35 \, a x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} b c^{2} d}{5 \, x^{2}} - \frac {\sqrt {b x^{2} + a x} a c d^{2}}{x^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} d^{3}}{x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} b c^{3}}{70 \, x^{3}} + \frac {9 \, \sqrt {b x^{2} + a x} a c^{2} d}{5 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} c d^{2}}{x^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} a c^{3}}{14 \, x^{4}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{2} d}{x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{3}}{2 \, x^{5}} \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^6,x, algorithm="maxima")
Output:
3*b^(3/2)*c*d^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 3/2*a*sqrt( b)*d^3*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 4/35*sqrt(b*x^2 + a* x)*b^3*c^3/(a^2*x) - 6/5*sqrt(b*x^2 + a*x)*b^2*c^2*d/(a*x) - 7*sqrt(b*x^2 + a*x)*b*c*d^2/x - 3*sqrt(b*x^2 + a*x)*a*d^3/x - 2/35*sqrt(b*x^2 + a*x)*b^ 2*c^3/(a*x^2) + 3/5*sqrt(b*x^2 + a*x)*b*c^2*d/x^2 - sqrt(b*x^2 + a*x)*a*c* d^2/x^2 + (b*x^2 + a*x)^(3/2)*d^3/x^2 + 3/70*sqrt(b*x^2 + a*x)*b*c^3/x^3 + 9/5*sqrt(b*x^2 + a*x)*a*c^2*d/x^3 - (b*x^2 + a*x)^(3/2)*c*d^2/x^3 + 3/14* sqrt(b*x^2 + a*x)*a*c^3/x^4 - 3*(b*x^2 + a*x)^(3/2)*c^2*d/x^4 - 1/2*(b*x^2 + a*x)^(3/2)*c^3/x^5
Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (150) = 300\).
Time = 0.15 (sec) , antiderivative size = 525, normalized size of antiderivative = 3.09 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=\sqrt {b x^{2} + a x} b d^{3} - \frac {3 \, {\left (2 \, b^{2} c d^{2} + a b d^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{2 \, \sqrt {b}} + \frac {2 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b^{2} c^{2} d + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a b c d^{2} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{2} d^{3} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} b^{\frac {5}{2}} c^{3} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a b^{\frac {3}{2}} c^{2} d + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{2} \sqrt {b} c d^{2} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a b^{2} c^{3} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} b c^{2} d + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{3} c d^{2} + 140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} b^{\frac {3}{2}} c^{3} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{3} \sqrt {b} c^{2} d + 98 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{3} b c^{3} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{4} c^{2} d + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{4} \sqrt {b} c^{3} + 5 \, a^{5} c^{3}\right )}}{35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7}} \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^6,x, algorithm="giac")
Output:
sqrt(b*x^2 + a*x)*b*d^3 - 3/2*(2*b^2*c*d^2 + a*b*d^3)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/sqrt(b) + 2/35*(105*(sqrt(b)*x - sqrt( b*x^2 + a*x))^6*b^2*c^2*d + 210*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a*b*c*d^ 2 + 35*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^2*d^3 + 35*(sqrt(b)*x - sqrt(b* x^2 + a*x))^5*b^(5/2)*c^3 + 210*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a*b^(3/2 )*c^2*d + 105*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^2*sqrt(b)*c*d^2 + 105*(s qrt(b)*x - sqrt(b*x^2 + a*x))^4*a*b^2*c^3 + 210*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^2*b*c^2*d + 35*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^3*c*d^2 + 140 *(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^2*b^(3/2)*c^3 + 105*(sqrt(b)*x - sqrt (b*x^2 + a*x))^3*a^3*sqrt(b)*c^2*d + 98*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2* a^3*b*c^3 + 21*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^4*c^2*d + 35*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^4*sqrt(b)*c^3 + 5*a^5*c^3)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^7
Timed out. \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^3}{x^6} \,d x \] Input:
int(((a*x + b*x^2)^(3/2)*(c + d*x)^3)/x^6,x)
Output:
int(((a*x + b*x^2)^(3/2)*(c + d*x)^3)/x^6, x)
Time = 0.26 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.01 \[ \int \frac {(c+d x)^3 \left (a x+b x^2\right )^{3/2}}{x^6} \, dx=\frac {-10 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c^{3}-42 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c^{2} d x -70 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c \,d^{2} x^{2}-70 \sqrt {x}\, \sqrt {b x +a}\, a^{3} d^{3} x^{3}-16 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,c^{3} x -84 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,c^{2} d \,x^{2}-280 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b c \,d^{2} x^{3}+35 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,d^{3} x^{4}-2 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c^{3} x^{2}-42 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c^{2} d \,x^{3}+4 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c^{3} x^{3}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} d^{3} x^{4}+210 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b c \,d^{2} x^{4}+75 \sqrt {b}\, a^{3} d^{3} x^{4}+160 \sqrt {b}\, a^{2} b c \,d^{2} x^{4}-18 \sqrt {b}\, a \,b^{2} c^{2} d \,x^{4}-4 \sqrt {b}\, b^{3} c^{3} x^{4}}{35 a^{2} x^{4}} \] Input:
int((d*x+c)^3*(b*x^2+a*x)^(3/2)/x^6,x)
Output:
( - 10*sqrt(x)*sqrt(a + b*x)*a**3*c**3 - 42*sqrt(x)*sqrt(a + b*x)*a**3*c** 2*d*x - 70*sqrt(x)*sqrt(a + b*x)*a**3*c*d**2*x**2 - 70*sqrt(x)*sqrt(a + b* x)*a**3*d**3*x**3 - 16*sqrt(x)*sqrt(a + b*x)*a**2*b*c**3*x - 84*sqrt(x)*sq rt(a + b*x)*a**2*b*c**2*d*x**2 - 280*sqrt(x)*sqrt(a + b*x)*a**2*b*c*d**2*x **3 + 35*sqrt(x)*sqrt(a + b*x)*a**2*b*d**3*x**4 - 2*sqrt(x)*sqrt(a + b*x)* a*b**2*c**3*x**2 - 42*sqrt(x)*sqrt(a + b*x)*a*b**2*c**2*d*x**3 + 4*sqrt(x) *sqrt(a + b*x)*b**3*c**3*x**3 + 105*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*s qrt(b))/sqrt(a))*a**3*d**3*x**4 + 210*sqrt(b)*log((sqrt(a + b*x) + sqrt(x) *sqrt(b))/sqrt(a))*a**2*b*c*d**2*x**4 + 75*sqrt(b)*a**3*d**3*x**4 + 160*sq rt(b)*a**2*b*c*d**2*x**4 - 18*sqrt(b)*a*b**2*c**2*d*x**4 - 4*sqrt(b)*b**3* c**3*x**4)/(35*a**2*x**4)