Integrand size = 29, antiderivative size = 270 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx=-\frac {2 (b e-a f)^3 (f g-e h)}{f^4 (d e-c f) \sqrt {e+f x}}+\frac {2 b \left (3 a^2 d^2 f^2 h+3 a b d f (d f g-2 d e h-c f h)+b^2 \left (c^2 f^2 h-d^2 e (2 f g-3 e h)-c d f (f g-2 e h)\right )\right ) \sqrt {e+f x}}{d^3 f^4}+\frac {2 b^2 (3 a d f h+b (d f g-3 d e h-c f h)) (e+f x)^{3/2}}{3 d^2 f^4}+\frac {2 b^3 h (e+f x)^{5/2}}{5 d f^4}+\frac {2 (b c-a d)^3 (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{7/2} (d e-c f)^{3/2}} \] Output:
-2*(-a*f+b*e)^3*(-e*h+f*g)/f^4/(-c*f+d*e)/(f*x+e)^(1/2)+2*b*(3*a^2*d^2*f^2 *h+3*a*b*d*f*(-c*f*h-2*d*e*h+d*f*g)+b^2*(c^2*f^2*h-d^2*e*(-3*e*h+2*f*g)-c* d*f*(-2*e*h+f*g)))*(f*x+e)^(1/2)/d^3/f^4+2/3*b^2*(3*a*d*f*h+b*(-c*f*h-3*d* e*h+d*f*g))*(f*x+e)^(3/2)/d^2/f^4+2/5*b^3*h*(f*x+e)^(5/2)/d/f^4+2*(-a*d+b* c)^3*(-c*h+d*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(7/2)/(- c*f+d*e)^(3/2)
Time = 0.79 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx=\frac {30 a^3 d^3 f^3 (-f g+e h)+90 a^2 b d^2 f^2 (c f h (e+f x)+d e (-2 e h+f (g-h x)))-30 a b^2 d f \left (3 c^2 f^2 h (e+f x)-c d f (e+f x) (3 f g-2 e h+f h x)+d^2 e \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )\right )+2 b^3 \left (15 c^3 f^3 h (e+f x)-5 c^2 d f^2 (e+f x) (3 f g-2 e h+f h x)+c d^2 f (e+f x) \left (8 e^2 h-2 e f (5 g+2 h x)+f^2 x (5 g+3 h x)\right )+d^3 e \left (-48 e^3 h+8 e^2 f (5 g-3 h x)-f^3 x^2 (5 g+3 h x)+2 e f^2 x (10 g+3 h x)\right )\right )}{15 d^3 f^4 (-d e+c f) \sqrt {e+f x}}+\frac {2 (b c-a d)^3 (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{7/2} (-d e+c f)^{3/2}} \] Input:
Integrate[((a + b*x)^3*(g + h*x))/((c + d*x)*(e + f*x)^(3/2)),x]
Output:
(30*a^3*d^3*f^3*(-(f*g) + e*h) + 90*a^2*b*d^2*f^2*(c*f*h*(e + f*x) + d*e*( -2*e*h + f*(g - h*x))) - 30*a*b^2*d*f*(3*c^2*f^2*h*(e + f*x) - c*d*f*(e + f*x)*(3*f*g - 2*e*h + f*h*x) + d^2*e*(-8*e^2*h + e*f*(6*g - 4*h*x) + f^2*x *(3*g + h*x))) + 2*b^3*(15*c^3*f^3*h*(e + f*x) - 5*c^2*d*f^2*(e + f*x)*(3* f*g - 2*e*h + f*h*x) + c*d^2*f*(e + f*x)*(8*e^2*h - 2*e*f*(5*g + 2*h*x) + f^2*x*(5*g + 3*h*x)) + d^3*e*(-48*e^3*h + 8*e^2*f*(5*g - 3*h*x) - f^3*x^2* (5*g + 3*h*x) + 2*e*f^2*x*(10*g + 3*h*x))))/(15*d^3*f^4*(-(d*e) + c*f)*Sqr t[e + f*x]) + (2*(b*c - a*d)^3*(d*g - c*h)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/ Sqrt[-(d*e) + c*f]])/(d^(7/2)*(-(d*e) + c*f)^(3/2))
Time = 0.83 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.49, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {167, 27, 170, 27, 164, 73, 221}
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.
| \(\displaystyle \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2 (a+b x)^3 (f g-e h)}{f \sqrt {e+f x} (d e-c f)}-\frac {2 \int -\frac {(a+b x)^2 (a f (d g-c h)-6 b c (f g-e h)-b (5 d f g-6 d e h+c f h) x)}{2 (c+d x) \sqrt {e+f x}}dx}{f (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b x)^2 (a f (d g-c h)-b c (6 f g-6 e h)-b (5 d f g-6 d e h+c f h) x)}{(c+d x) \sqrt {e+f x}}dx}{f (d e-c f)}+\frac {2 (a+b x)^3 (f g-e h)}{f \sqrt {e+f x} (d e-c f)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\frac {2 \int \frac {(a+b x) (b c (4 b e+a f) (5 d f g-6 d e h+c f h)+5 a d f (a f (d g-c h)-b c (6 f g-6 e h))+b ((4 b d e+5 b c f-4 a d f) (5 d f g-6 d e h+c f h)+5 d f (a f (d g-c h)-b c (6 f g-6 e h))) x)}{2 (c+d x) \sqrt {e+f x}}dx}{5 d f}-\frac {2 b (a+b x)^2 \sqrt {e+f x} (c f h-6 d e h+5 d f g)}{5 d f}}{f (d e-c f)}+\frac {2 (a+b x)^3 (f g-e h)}{f \sqrt {e+f x} (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {(a+b x) (b c (4 b e+a f) (5 d f g-6 d e h+c f h)+5 a d f (a f (d g-c h)-b c (6 f g-6 e h))+b ((4 b d e+5 b c f-4 a d f) (5 d f g-6 d e h+c f h)+5 d f (a f (d g-c h)-b c (6 f g-6 e h))) x)}{(c+d x) \sqrt {e+f x}}dx}{5 d f}-\frac {2 b (a+b x)^2 \sqrt {e+f x} (c f h-6 d e h+5 d f g)}{5 d f}}{f (d e-c f)}+\frac {2 (a+b x)^3 (f g-e h)}{f \sqrt {e+f x} (d e-c f)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {-\frac {5 f^2 (b c-a d)^3 (d g-c h) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d^2}-\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (7 c f h-12 d e h+5 d f g)-15 a b d f \left (3 c^2 f^2 h-c d f (3 f g-2 e h)+2 d^2 e (3 f g-4 e h)\right )-b d f x ((-4 a d f+5 b c f+4 b d e) (c f h-6 d e h+5 d f g)+5 d f (a f (d g-c h)-b c (6 f g-6 e h)))+b^2 \left (15 c^3 f^3 h-5 c^2 d f^2 (3 f g-2 e h)-2 c d^2 e f (5 f g-4 e h)+8 d^3 e^2 (5 f g-6 e h)\right )\right )}{3 d^2 f^2}}{5 d f}-\frac {2 b (a+b x)^2 \sqrt {e+f x} (c f h-6 d e h+5 d f g)}{5 d f}}{f (d e-c f)}+\frac {2 (a+b x)^3 (f g-e h)}{f \sqrt {e+f x} (d e-c f)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {10 f (b c-a d)^3 (d g-c h) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d^2}-\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (7 c f h-12 d e h+5 d f g)-15 a b d f \left (3 c^2 f^2 h-c d f (3 f g-2 e h)+2 d^2 e (3 f g-4 e h)\right )-b d f x ((-4 a d f+5 b c f+4 b d e) (c f h-6 d e h+5 d f g)+5 d f (a f (d g-c h)-b c (6 f g-6 e h)))+b^2 \left (15 c^3 f^3 h-5 c^2 d f^2 (3 f g-2 e h)-2 c d^2 e f (5 f g-4 e h)+8 d^3 e^2 (5 f g-6 e h)\right )\right )}{3 d^2 f^2}}{5 d f}-\frac {2 b (a+b x)^2 \sqrt {e+f x} (c f h-6 d e h+5 d f g)}{5 d f}}{f (d e-c f)}+\frac {2 (a+b x)^3 (f g-e h)}{f \sqrt {e+f x} (d e-c f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {10 f^2 (b c-a d)^3 (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} \sqrt {d e-c f}}-\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (7 c f h-12 d e h+5 d f g)-15 a b d f \left (3 c^2 f^2 h-c d f (3 f g-2 e h)+2 d^2 e (3 f g-4 e h)\right )-b d f x ((-4 a d f+5 b c f+4 b d e) (c f h-6 d e h+5 d f g)+5 d f (a f (d g-c h)-b c (6 f g-6 e h)))+b^2 \left (15 c^3 f^3 h-5 c^2 d f^2 (3 f g-2 e h)-2 c d^2 e f (5 f g-4 e h)+8 d^3 e^2 (5 f g-6 e h)\right )\right )}{3 d^2 f^2}}{5 d f}-\frac {2 b (a+b x)^2 \sqrt {e+f x} (c f h-6 d e h+5 d f g)}{5 d f}}{f (d e-c f)}+\frac {2 (a+b x)^3 (f g-e h)}{f \sqrt {e+f x} (d e-c f)}\) |
Input:
Int[((a + b*x)^3*(g + h*x))/((c + d*x)*(e + f*x)^(3/2)),x]
Output:
(2*(f*g - e*h)*(a + b*x)^3)/(f*(d*e - c*f)*Sqrt[e + f*x]) + ((-2*b*(5*d*f* g - 6*d*e*h + c*f*h)*(a + b*x)^2*Sqrt[e + f*x])/(5*d*f) + ((-2*b*Sqrt[e + f*x]*(6*a^2*d^2*f^2*(5*d*f*g - 12*d*e*h + 7*c*f*h) - 15*a*b*d*f*(3*c^2*f^2 *h + 2*d^2*e*(3*f*g - 4*e*h) - c*d*f*(3*f*g - 2*e*h)) + b^2*(15*c^3*f^3*h + 8*d^3*e^2*(5*f*g - 6*e*h) - 2*c*d^2*e*f*(5*f*g - 4*e*h) - 5*c^2*d*f^2*(3 *f*g - 2*e*h)) - b*d*f*((4*b*d*e + 5*b*c*f - 4*a*d*f)*(5*d*f*g - 6*d*e*h + c*f*h) + 5*d*f*(a*f*(d*g - c*h) - b*c*(6*f*g - 6*e*h)))*x))/(3*d^2*f^2) + (10*(b*c - a*d)^3*f^2*(d*g - c*h)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d* e - c*f]])/(d^(5/2)*Sqrt[d*e - c*f]))/(5*d*f))/(f*(d*e - c*f))
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.55 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.55
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {f x +e}\, f^{4} \left (a d -b c \right )^{3} \left (c h -d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+2 \left (\left (-a^{3} d^{3} g +3 x c b \left (\frac {x \left (\frac {3 h x}{5}+g \right ) b^{2}}{9}+a \left (\frac {h x}{3}+g \right ) b +a^{2} h \right ) d^{2}-3 x \,c^{2} b^{2} \left (\frac {\left (\frac {h x}{3}+g \right ) b}{3}+a h \right ) d +b^{3} c^{3} h x \right ) f^{4}+\left (\left (-\frac {x^{2} \left (\frac {3 h x}{5}+g \right ) b^{3}}{3}-3 a x \left (\frac {h x}{3}+g \right ) b^{2}+3 a^{2} \left (-h x +g \right ) b +h \,a^{3}\right ) d^{3}+3 c \left (-\frac {x \left (\frac {h x}{5}+g \right ) b^{2}}{9}+a \left (-\frac {h x}{3}+g \right ) b +a^{2} h \right ) b \,d^{2}-3 c^{2} \left (\frac {\left (-\frac {h x}{3}+g \right ) b}{3}+a h \right ) b^{2} d +b^{3} c^{3} h \right ) e \,f^{3}-6 d \left (\left (-\frac {2 x \left (\frac {3 h x}{10}+g \right ) b^{2}}{9}+a \left (-\frac {2 h x}{3}+g \right ) b +a^{2} h \right ) d^{2}+\frac {\left (\frac {\left (-\frac {2 h x}{5}+g \right ) b}{3}+a h \right ) c b d}{3}-\frac {b^{2} c^{2} h}{9}\right ) b \,e^{2} f^{2}+8 d^{2} \left (\left (\left (-\frac {h x}{5}+\frac {g}{3}\right ) b +a h \right ) d +\frac {b c h}{15}\right ) b^{2} e^{3} f -\frac {16 b^{3} d^{3} e^{4} h}{5}\right ) \sqrt {\left (c f -d e \right ) d}}{f^{4} d^{3} \left (c f -d e \right ) \sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}}\) | \(419\) |
| risch | \(\frac {2 b \left (3 x^{2} h \,b^{2} d^{2} f^{2}+15 a b \,d^{2} f^{2} h x -5 b^{2} c d \,f^{2} h x -9 b^{2} d^{2} e f h x +5 b^{2} d^{2} f^{2} g x +45 a^{2} d^{2} f^{2} h -45 a b c d \,f^{2} h -75 a b \,d^{2} e f h +45 a b \,d^{2} f^{2} g +15 c^{2} b^{2} f^{2} h +25 b^{2} c d e f h -15 b^{2} c d \,f^{2} g +33 b^{2} d^{2} e^{2} h -25 b^{2} d^{2} e f g \right ) \sqrt {f x +e}}{15 f^{4} d^{3}}+\frac {\frac {2 \left (a^{3} e \,f^{3} h -a^{3} f^{4} g -3 a^{2} b \,e^{2} f^{2} h +3 a^{2} b e \,f^{3} g +3 a \,b^{2} e^{3} f h -3 a \,b^{2} e^{2} f^{2} g -b^{3} e^{4} h +b^{3} e^{3} f g \right ) d^{3}}{\left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 f^{4} \left (a^{3} c \,d^{3} h -a^{3} d^{4} g -3 a^{2} b \,c^{2} d^{2} h +3 a^{2} b c \,d^{3} g +3 a \,b^{2} c^{3} d h -3 a \,b^{2} c^{2} d^{2} g -c^{4} h \,b^{3}+b^{3} c^{3} d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}}{f^{4} d^{3}}\) | \(430\) |
| derivativedivides | \(\frac {\frac {2 b \left (\frac {d^{2} h \left (f x +e \right )^{\frac {5}{2}} b^{2}}{5}+a b \,d^{2} f h \left (f x +e \right )^{\frac {3}{2}}-\frac {b^{2} c d f h \left (f x +e \right )^{\frac {3}{2}}}{3}-b^{2} d^{2} e h \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{2} d^{2} f g \left (f x +e \right )^{\frac {3}{2}}}{3}+3 a^{2} d^{2} f^{2} h \sqrt {f x +e}-3 a b c d \,f^{2} h \sqrt {f x +e}-6 a b \,d^{2} e f h \sqrt {f x +e}+3 a b \,d^{2} f^{2} g \sqrt {f x +e}+b^{2} c^{2} f^{2} h \sqrt {f x +e}+2 b^{2} c d e f h \sqrt {f x +e}-b^{2} c d \,f^{2} g \sqrt {f x +e}+3 b^{2} d^{2} e^{2} h \sqrt {f x +e}-2 b^{2} d^{2} e f g \sqrt {f x +e}\right )}{d^{3}}-\frac {2 \left (-a^{3} e \,f^{3} h +a^{3} f^{4} g +3 a^{2} b \,e^{2} f^{2} h -3 a^{2} b e \,f^{3} g -3 a \,b^{2} e^{3} f h +3 a \,b^{2} e^{2} f^{2} g +b^{3} e^{4} h -b^{3} e^{3} f g \right )}{\left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 \left (a^{3} c \,d^{3} h -a^{3} d^{4} g -3 a^{2} b \,c^{2} d^{2} h +3 a^{2} b c \,d^{3} g +3 a \,b^{2} c^{3} d h -3 a \,b^{2} c^{2} d^{2} g -c^{4} h \,b^{3}+b^{3} c^{3} d g \right ) f^{4} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) d^{3} \sqrt {\left (c f -d e \right ) d}}}{f^{4}}\) | \(495\) |
| default | \(\frac {\frac {2 b \left (\frac {d^{2} h \left (f x +e \right )^{\frac {5}{2}} b^{2}}{5}+a b \,d^{2} f h \left (f x +e \right )^{\frac {3}{2}}-\frac {b^{2} c d f h \left (f x +e \right )^{\frac {3}{2}}}{3}-b^{2} d^{2} e h \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{2} d^{2} f g \left (f x +e \right )^{\frac {3}{2}}}{3}+3 a^{2} d^{2} f^{2} h \sqrt {f x +e}-3 a b c d \,f^{2} h \sqrt {f x +e}-6 a b \,d^{2} e f h \sqrt {f x +e}+3 a b \,d^{2} f^{2} g \sqrt {f x +e}+b^{2} c^{2} f^{2} h \sqrt {f x +e}+2 b^{2} c d e f h \sqrt {f x +e}-b^{2} c d \,f^{2} g \sqrt {f x +e}+3 b^{2} d^{2} e^{2} h \sqrt {f x +e}-2 b^{2} d^{2} e f g \sqrt {f x +e}\right )}{d^{3}}-\frac {2 \left (-a^{3} e \,f^{3} h +a^{3} f^{4} g +3 a^{2} b \,e^{2} f^{2} h -3 a^{2} b e \,f^{3} g -3 a \,b^{2} e^{3} f h +3 a \,b^{2} e^{2} f^{2} g +b^{3} e^{4} h -b^{3} e^{3} f g \right )}{\left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 \left (a^{3} c \,d^{3} h -a^{3} d^{4} g -3 a^{2} b \,c^{2} d^{2} h +3 a^{2} b c \,d^{3} g +3 a \,b^{2} c^{3} d h -3 a \,b^{2} c^{2} d^{2} g -c^{4} h \,b^{3}+b^{3} c^{3} d g \right ) f^{4} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) d^{3} \sqrt {\left (c f -d e \right ) d}}}{f^{4}}\) | \(495\) |
Input:
int((b*x+a)^3*(h*x+g)/(d*x+c)/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/(f*x+e)^(1/2)*((f*x+e)^(1/2)*f^4*(a*d-b*c)^3*(c*h-d*g)*arctan(d*(f*x+e)^ (1/2)/((c*f-d*e)*d)^(1/2))+((-a^3*d^3*g+3*x*c*b*(1/9*x*(3/5*h*x+g)*b^2+a*( 1/3*h*x+g)*b+a^2*h)*d^2-3*x*c^2*b^2*(1/3*(1/3*h*x+g)*b+a*h)*d+b^3*c^3*h*x) *f^4+((-1/3*x^2*(3/5*h*x+g)*b^3-3*a*x*(1/3*h*x+g)*b^2+3*a^2*(-h*x+g)*b+h*a ^3)*d^3+3*c*(-1/9*x*(1/5*h*x+g)*b^2+a*(-1/3*h*x+g)*b+a^2*h)*b*d^2-3*c^2*(1 /3*(-1/3*h*x+g)*b+a*h)*b^2*d+b^3*c^3*h)*e*f^3-6*d*((-2/9*x*(3/10*h*x+g)*b^ 2+a*(-2/3*h*x+g)*b+a^2*h)*d^2+1/3*(1/3*(-2/5*h*x+g)*b+a*h)*c*b*d-1/9*b^2*c ^2*h)*b*e^2*f^2+8*d^2*(((-1/5*h*x+1/3*g)*b+a*h)*d+1/15*b*c*h)*b^2*e^3*f-16 /5*b^3*d^3*e^4*h)*((c*f-d*e)*d)^(1/2))/((c*f-d*e)*d)^(1/2)/f^4/d^3/(c*f-d* e)
Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (248) = 496\).
Time = 0.16 (sec) , antiderivative size = 2179, normalized size of antiderivative = 8.07 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(h*x+g)/(d*x+c)/(f*x+e)^(3/2),x, algorithm="fricas")
Output:
[-1/15*(15*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^4* g - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*e*f^4*h + ((b^ 3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^5*g - (b^3*c^4 - 3* a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^5*h)*x)*sqrt(d^2*e - c*d*f)*l og((d*f*x + 2*d*e - c*f - 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)) - 2*(3*(b^3*d^5*e^2*f^3 - 2*b^3*c*d^4*e*f^4 + b^3*c^2*d^3*f^5)*h*x^3 + (5* (b^3*d^5*e^2*f^3 - 2*b^3*c*d^4*e*f^4 + b^3*c^2*d^3*f^5)*g - (6*b^3*d^5*e^3 *f^2 - (7*b^3*c*d^4 + 15*a*b^2*d^5)*e^2*f^3 - 2*(2*b^3*c^2*d^3 - 15*a*b^2* c*d^4)*e*f^4 + 5*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3)*f^5)*h)*x^2 - 5*(8*b^3*d^ 5*e^4*f + 3*a^3*c*d^4*f^5 - 2*(5*b^3*c*d^4 + 9*a*b^2*d^5)*e^3*f^2 - (b^3*c ^2*d^3 - 27*a*b^2*c*d^4 - 9*a^2*b*d^5)*e^2*f^3 + 3*(b^3*c^3*d^2 - 3*a*b^2* c^2*d^3 - 3*a^2*b*c*d^4 - a^3*d^5)*e*f^4)*g + (48*b^3*d^5*e^5 - 8*(7*b^3*c *d^4 + 15*a*b^2*d^5)*e^4*f - 2*(b^3*c^2*d^3 - 75*a*b^2*c*d^4 - 45*a^2*b*d^ 5)*e^3*f^2 - 5*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 27*a^2*b*c*d^4 + 3*a^3*d^5 )*e^2*f^3 + 15*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + a^3*c*d^4) *e*f^4)*h - (5*(4*b^3*d^5*e^3*f^2 - (5*b^3*c*d^4 + 9*a*b^2*d^5)*e^2*f^3 - 2*(b^3*c^2*d^3 - 9*a*b^2*c*d^4)*e*f^4 + 3*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3)* f^5)*g - (24*b^3*d^5*e^4*f - 4*(7*b^3*c*d^4 + 15*a*b^2*d^5)*e^3*f^2 - (b^3 *c^2*d^3 - 75*a*b^2*c*d^4 - 45*a^2*b*d^5)*e^2*f^3 - 10*(b^3*c^3*d^2 - 3*a* b^2*c^2*d^3 + 9*a^2*b*c*d^4)*e*f^4 + 15*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + ...
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (262) = 524\).
Time = 25.13 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{3} h \left (e + f x\right )^{\frac {5}{2}}}{5 d f^{3}} + \frac {\left (a f - b e\right )^{3} \left (e h - f g\right )}{f^{3} \sqrt {e + f x} \left (c f - d e\right )} + \frac {\left (e + f x\right )^{\frac {3}{2}} \cdot \left (3 a b^{2} d f h - b^{3} c f h - 3 b^{3} d e h + b^{3} d f g\right )}{3 d^{2} f^{3}} + \frac {\sqrt {e + f x} \left (3 a^{2} b d^{2} f^{2} h - 3 a b^{2} c d f^{2} h - 6 a b^{2} d^{2} e f h + 3 a b^{2} d^{2} f^{2} g + b^{3} c^{2} f^{2} h + 2 b^{3} c d e f h - b^{3} c d f^{2} g + 3 b^{3} d^{2} e^{2} h - 2 b^{3} d^{2} e f g\right )}{d^{3} f^{3}} + \frac {f \left (a d - b c\right )^{3} \left (c h - d g\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{4} \sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {\frac {b^{3} h x^{4}}{4 d} + \frac {x^{3} \cdot \left (3 a b^{2} d h - b^{3} c h + b^{3} d g\right )}{3 d^{2}} + \frac {x^{2} \cdot \left (3 a^{2} b d^{2} h - 3 a b^{2} c d h + 3 a b^{2} d^{2} g + b^{3} c^{2} h - b^{3} c d g\right )}{2 d^{3}} + \frac {x \left (a^{3} d^{3} h - 3 a^{2} b c d^{2} h + 3 a^{2} b d^{3} g + 3 a b^{2} c^{2} d h - 3 a b^{2} c d^{2} g - b^{3} c^{3} h + b^{3} c^{2} d g\right )}{d^{4}} - \frac {\left (a d - b c\right )^{3} \left (c h - d g\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{4}}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(h*x+g)/(d*x+c)/(f*x+e)**(3/2),x)
Output:
Piecewise((2*(b**3*h*(e + f*x)**(5/2)/(5*d*f**3) + (a*f - b*e)**3*(e*h - f *g)/(f**3*sqrt(e + f*x)*(c*f - d*e)) + (e + f*x)**(3/2)*(3*a*b**2*d*f*h - b**3*c*f*h - 3*b**3*d*e*h + b**3*d*f*g)/(3*d**2*f**3) + sqrt(e + f*x)*(3*a **2*b*d**2*f**2*h - 3*a*b**2*c*d*f**2*h - 6*a*b**2*d**2*e*f*h + 3*a*b**2*d **2*f**2*g + b**3*c**2*f**2*h + 2*b**3*c*d*e*f*h - b**3*c*d*f**2*g + 3*b** 3*d**2*e**2*h - 2*b**3*d**2*e*f*g)/(d**3*f**3) + f*(a*d - b*c)**3*(c*h - d *g)*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**4*sqrt((c*f - d*e)/d)*(c*f - d*e)))/f, Ne(f, 0)), ((b**3*h*x**4/(4*d) + x**3*(3*a*b**2*d*h - b**3*c* h + b**3*d*g)/(3*d**2) + x**2*(3*a**2*b*d**2*h - 3*a*b**2*c*d*h + 3*a*b**2 *d**2*g + b**3*c**2*h - b**3*c*d*g)/(2*d**3) + x*(a**3*d**3*h - 3*a**2*b*c *d**2*h + 3*a**2*b*d**3*g + 3*a*b**2*c**2*d*h - 3*a*b**2*c*d**2*g - b**3*c **3*h + b**3*c**2*d*g)/d**4 - (a*d - b*c)**3*(c*h - d*g)*Piecewise((x/c, E q(d, 0)), (log(c + d*x)/d, True))/d**4)/e**(3/2), True))
Exception generated. \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^3*(h*x+g)/(d*x+c)/(f*x+e)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (248) = 496\).
Time = 0.15 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.01 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx=-\frac {2 \, {\left (b^{3} c^{3} d g - 3 \, a b^{2} c^{2} d^{2} g + 3 \, a^{2} b c d^{3} g - a^{3} d^{4} g - b^{3} c^{4} h + 3 \, a b^{2} c^{3} d h - 3 \, a^{2} b c^{2} d^{2} h + a^{3} c d^{3} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{4} e - c d^{3} f\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, {\left (b^{3} e^{3} f g - 3 \, a b^{2} e^{2} f^{2} g + 3 \, a^{2} b e f^{3} g - a^{3} f^{4} g - b^{3} e^{4} h + 3 \, a b^{2} e^{3} f h - 3 \, a^{2} b e^{2} f^{2} h + a^{3} e f^{3} h\right )}}{{\left (d e f^{4} - c f^{5}\right )} \sqrt {f x + e}} + \frac {2 \, {\left (5 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d^{4} f^{17} g - 30 \, \sqrt {f x + e} b^{3} d^{4} e f^{17} g - 15 \, \sqrt {f x + e} b^{3} c d^{3} f^{18} g + 45 \, \sqrt {f x + e} a b^{2} d^{4} f^{18} g + 3 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} d^{4} f^{16} h - 15 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d^{4} e f^{16} h + 45 \, \sqrt {f x + e} b^{3} d^{4} e^{2} f^{16} h - 5 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c d^{3} f^{17} h + 15 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d^{4} f^{17} h + 30 \, \sqrt {f x + e} b^{3} c d^{3} e f^{17} h - 90 \, \sqrt {f x + e} a b^{2} d^{4} e f^{17} h + 15 \, \sqrt {f x + e} b^{3} c^{2} d^{2} f^{18} h - 45 \, \sqrt {f x + e} a b^{2} c d^{3} f^{18} h + 45 \, \sqrt {f x + e} a^{2} b d^{4} f^{18} h\right )}}{15 \, d^{5} f^{20}} \] Input:
integrate((b*x+a)^3*(h*x+g)/(d*x+c)/(f*x+e)^(3/2),x, algorithm="giac")
Output:
-2*(b^3*c^3*d*g - 3*a*b^2*c^2*d^2*g + 3*a^2*b*c*d^3*g - a^3*d^4*g - b^3*c^ 4*h + 3*a*b^2*c^3*d*h - 3*a^2*b*c^2*d^2*h + a^3*c*d^3*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((d^4*e - c*d^3*f)*sqrt(-d^2*e + c*d*f)) - 2*( b^3*e^3*f*g - 3*a*b^2*e^2*f^2*g + 3*a^2*b*e*f^3*g - a^3*f^4*g - b^3*e^4*h + 3*a*b^2*e^3*f*h - 3*a^2*b*e^2*f^2*h + a^3*e*f^3*h)/((d*e*f^4 - c*f^5)*sq rt(f*x + e)) + 2/15*(5*(f*x + e)^(3/2)*b^3*d^4*f^17*g - 30*sqrt(f*x + e)*b ^3*d^4*e*f^17*g - 15*sqrt(f*x + e)*b^3*c*d^3*f^18*g + 45*sqrt(f*x + e)*a*b ^2*d^4*f^18*g + 3*(f*x + e)^(5/2)*b^3*d^4*f^16*h - 15*(f*x + e)^(3/2)*b^3* d^4*e*f^16*h + 45*sqrt(f*x + e)*b^3*d^4*e^2*f^16*h - 5*(f*x + e)^(3/2)*b^3 *c*d^3*f^17*h + 15*(f*x + e)^(3/2)*a*b^2*d^4*f^17*h + 30*sqrt(f*x + e)*b^3 *c*d^3*e*f^17*h - 90*sqrt(f*x + e)*a*b^2*d^4*e*f^17*h + 15*sqrt(f*x + e)*b ^3*c^2*d^2*f^18*h - 45*sqrt(f*x + e)*a*b^2*c*d^3*f^18*h + 45*sqrt(f*x + e) *a^2*b*d^4*f^18*h)/(d^5*f^20)
Time = 0.17 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.90 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx={\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,b^3\,f\,g-8\,b^3\,e\,h+6\,a\,b^2\,f\,h}{3\,d\,f^4}-\frac {2\,b^3\,h\,\left (c\,f^5-d\,e\,f^4\right )}{3\,d^2\,f^8}\right )-\sqrt {e+f\,x}\,\left (\frac {\left (\frac {2\,b^3\,f\,g-8\,b^3\,e\,h+6\,a\,b^2\,f\,h}{d\,f^4}-\frac {2\,b^3\,h\,\left (c\,f^5-d\,e\,f^4\right )}{d^2\,f^8}\right )\,\left (c\,f^5-d\,e\,f^4\right )}{d\,f^4}-\frac {6\,b\,\left (a\,f-b\,e\right )\,\left (a\,f\,h-2\,b\,e\,h+b\,f\,g\right )}{d\,f^4}\right )+\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\left (c\,h-d\,g\right )\,\left (d^4\,e-c\,d^3\,f\right )}{d^{5/2}\,{\left (c\,f-d\,e\right )}^{3/2}\,\left (-2\,h\,a^3\,c\,d^3+2\,g\,a^3\,d^4+6\,h\,a^2\,b\,c^2\,d^2-6\,g\,a^2\,b\,c\,d^3-6\,h\,a\,b^2\,c^3\,d+6\,g\,a\,b^2\,c^2\,d^2+2\,h\,b^3\,c^4-2\,g\,b^3\,c^3\,d\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (c\,h-d\,g\right )}{d^{7/2}\,{\left (c\,f-d\,e\right )}^{3/2}}+\frac {2\,b^3\,h\,{\left (e+f\,x\right )}^{5/2}}{5\,d\,f^4}-\frac {2\,\left (-h\,a^3\,d^3\,e\,f^3+g\,a^3\,d^3\,f^4+3\,h\,a^2\,b\,d^3\,e^2\,f^2-3\,g\,a^2\,b\,d^3\,e\,f^3-3\,h\,a\,b^2\,d^3\,e^3\,f+3\,g\,a\,b^2\,d^3\,e^2\,f^2+h\,b^3\,d^3\,e^4-g\,b^3\,d^3\,e^3\,f\right )}{d^3\,f^4\,\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )} \] Input:
int(((g + h*x)*(a + b*x)^3)/((e + f*x)^(3/2)*(c + d*x)),x)
Output:
(e + f*x)^(3/2)*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(3*d*f^4) - (2*b^3* h*(c*f^5 - d*e*f^4))/(3*d^2*f^8)) - (e + f*x)^(1/2)*((((2*b^3*f*g - 8*b^3* e*h + 6*a*b^2*f*h)/(d*f^4) - (2*b^3*h*(c*f^5 - d*e*f^4))/(d^2*f^8))*(c*f^5 - d*e*f^4))/(d*f^4) - (6*b*(a*f - b*e)*(a*f*h - 2*b*e*h + b*f*g))/(d*f^4) ) + (2*atan((2*(e + f*x)^(1/2)*(a*d - b*c)^3*(c*h - d*g)*(d^4*e - c*d^3*f) )/(d^(5/2)*(c*f - d*e)^(3/2)*(2*a^3*d^4*g + 2*b^3*c^4*h - 2*a^3*c*d^3*h - 2*b^3*c^3*d*g - 6*a^2*b*c*d^3*g - 6*a*b^2*c^3*d*h + 6*a*b^2*c^2*d^2*g + 6* a^2*b*c^2*d^2*h)))*(a*d - b*c)^3*(c*h - d*g))/(d^(7/2)*(c*f - d*e)^(3/2)) + (2*b^3*h*(e + f*x)^(5/2))/(5*d*f^4) - (2*(a^3*d^3*f^4*g + b^3*d^3*e^4*h - a^3*d^3*e*f^3*h - b^3*d^3*e^3*f*g + 3*a*b^2*d^3*e^2*f^2*g + 3*a^2*b*d^3* e^2*f^2*h - 3*a^2*b*d^3*e*f^3*g - 3*a*b^2*d^3*e^3*f*h))/(d^3*f^4*(e + f*x) ^(1/2)*(c*f - d*e))
Time = 0.27 (sec) , antiderivative size = 1423, normalized size of antiderivative = 5.27 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x) (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(h*x+g)/(d*x+c)/(f*x+e)^(3/2),x)
Output:
(2*(15*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt( d)*sqrt(c*f - d*e)))*a**3*c*d**3*f**4*h - 15*sqrt(d)*sqrt(e + f*x)*sqrt(c* f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d**4*f**4* g - 45*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt( d)*sqrt(c*f - d*e)))*a**2*b*c**2*d**2*f**4*h + 45*sqrt(d)*sqrt(e + f*x)*sq rt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c*d **3*f**4*g + 45*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)* d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c**3*d*f**4*h - 45*sqrt(d)*sqrt(e + f *x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b* *2*c**2*d**2*f**4*g - 15*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt( e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**3*c**4*f**4*h + 15*sqrt(d)*sqrt( e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e))) *b**3*c**3*d*f**4*g + 15*a**3*c*d**4*e*f**4*h - 15*a**3*c*d**4*f**5*g - 15 *a**3*d**5*e**2*f**3*h + 15*a**3*d**5*e*f**4*g + 45*a**2*b*c**2*d**3*e*f** 4*h + 45*a**2*b*c**2*d**3*f**5*h*x - 135*a**2*b*c*d**4*e**2*f**3*h + 45*a* *2*b*c*d**4*e*f**4*g - 90*a**2*b*c*d**4*e*f**4*h*x + 90*a**2*b*d**5*e**3*f **2*h - 45*a**2*b*d**5*e**2*f**3*g + 45*a**2*b*d**5*e**2*f**3*h*x - 45*a*b **2*c**3*d**2*e*f**4*h - 45*a*b**2*c**3*d**2*f**5*h*x + 15*a*b**2*c**2*d** 3*e**2*f**3*h + 45*a*b**2*c**2*d**3*e*f**4*g + 30*a*b**2*c**2*d**3*e*f**4* h*x + 45*a*b**2*c**2*d**3*f**5*g*x + 15*a*b**2*c**2*d**3*f**5*h*x**2 + ...