Integrand size = 27, antiderivative size = 331 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\frac {2 d f h \sqrt {e+f x}}{b^4}-\frac {\left (5 a^2 d f h+b^2 (2 d e g+c f g+2 c e h)-a b (3 d f g+4 d e h+3 c f h)\right ) \sqrt {e+f x}}{4 b^4 (a+b x)^2}-\frac {\left (29 a^2 d f^2 h-a b f (11 d f g+36 d e h+11 c f h)+b^2 (2 d e (5 f g+4 e h)+c f (f g+10 e h))\right ) \sqrt {e+f x}}{8 b^4 (b e-a f) (a+b x)}-\frac {(b c-a d) (b g-a h) (e+f x)^{3/2}}{3 b^3 (a+b x)^3}-\frac {f \left (35 a^2 d f^2 h-5 a b f (d f g+12 d e h+c f h)-b^2 (c f (f g-6 e h)-6 d e (f g+4 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{9/2} (b e-a f)^{3/2}} \] Output:
2*d*f*h*(f*x+e)^(1/2)/b^4-1/4*(5*a^2*d*f*h+b^2*(2*c*e*h+c*f*g+2*d*e*g)-a*b *(3*c*f*h+4*d*e*h+3*d*f*g))*(f*x+e)^(1/2)/b^4/(b*x+a)^2-1/8*(29*a^2*d*f^2* h-a*b*f*(11*c*f*h+36*d*e*h+11*d*f*g)+b^2*(2*d*e*(4*e*h+5*f*g)+c*f*(10*e*h+ f*g)))*(f*x+e)^(1/2)/b^4/(-a*f+b*e)/(b*x+a)-1/3*(-a*d+b*c)*(-a*h+b*g)*(f*x +e)^(3/2)/b^3/(b*x+a)^3-1/8*f*(35*a^2*d*f^2*h-5*a*b*f*(c*f*h+12*d*e*h+d*f* g)-b^2*(c*f*(-6*e*h+f*g)-6*d*e*(4*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2) /(-a*f+b*e)^(1/2))/b^(9/2)/(-a*f+b*e)^(3/2)
Time = 1.56 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.18 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=-\frac {\sqrt {e+f x} \left (105 a^4 d f^2 h-5 a^3 b f (3 c f h+d (3 f g+22 e h-56 f h x))+b^4 \left (6 d e x (f x (5 g-8 h x)+2 e (g+2 h x))+c \left (3 f^2 g x^2+4 e^2 (2 g+3 h x)+2 e f x (7 g+15 h x)\right )\right )+a b^3 \left (d \left (2 e f x (11 g-126 h x)+4 e^2 (g+6 h x)+3 f^2 x^2 (-11 g+16 h x)\right )+c \left (4 e^2 h-2 e f (g-11 h x)-f^2 x (8 g+33 h x)\right )\right )+a^2 b^2 \left (c f (-3 f g+8 e h-40 f h x)+d \left (8 e^2 h+e f (8 g-298 h x)+f^2 x (-40 g+231 h x)\right )\right )\right )}{24 b^4 (b e-a f) (a+b x)^3}+\frac {f \left (-35 a^2 d f^2 h+5 a b f (d f g+12 d e h+c f h)+b^2 (c f (f g-6 e h)-6 d e (f g+4 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{8 b^{9/2} (-b e+a f)^{3/2}} \] Input:
Integrate[((c + d*x)*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^4,x]
Output:
-1/24*(Sqrt[e + f*x]*(105*a^4*d*f^2*h - 5*a^3*b*f*(3*c*f*h + d*(3*f*g + 22 *e*h - 56*f*h*x)) + b^4*(6*d*e*x*(f*x*(5*g - 8*h*x) + 2*e*(g + 2*h*x)) + c *(3*f^2*g*x^2 + 4*e^2*(2*g + 3*h*x) + 2*e*f*x*(7*g + 15*h*x))) + a*b^3*(d* (2*e*f*x*(11*g - 126*h*x) + 4*e^2*(g + 6*h*x) + 3*f^2*x^2*(-11*g + 16*h*x) ) + c*(4*e^2*h - 2*e*f*(g - 11*h*x) - f^2*x*(8*g + 33*h*x))) + a^2*b^2*(c* f*(-3*f*g + 8*e*h - 40*f*h*x) + d*(8*e^2*h + e*f*(8*g - 298*h*x) + f^2*x*( -40*g + 231*h*x)))))/(b^4*(b*e - a*f)*(a + b*x)^3) + (f*(-35*a^2*d*f^2*h + 5*a*b*f*(d*f*g + 12*d*e*h + c*f*h) + b^2*(c*f*(f*g - 6*e*h) - 6*d*e*(f*g + 4*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(8*b^(9/2)* (-(b*e) + a*f)^(3/2))
Time = 0.49 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {162, 51, 60, 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.
| \(\displaystyle \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {\left (35 a^2 d f^2 h-5 a b f (c f h+12 d e h+d f g)-\left (b^2 (c f (f g-6 e h)-6 d e (4 e h+f g))\right )\right ) \int \frac {(e+f x)^{3/2}}{(a+b x)^2}dx}{24 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (11 a^2 d f h-a b (5 c f h+12 d e h+5 d f g)+b^2 (6 c e h-c f g+6 d e g)\right )-a^2 b (c f h+8 d e h+d f g)+a b^2 (2 c e h-5 c f g+2 d e g)+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\left (35 a^2 d f^2 h-5 a b f (c f h+12 d e h+d f g)-\left (b^2 (c f (f g-6 e h)-6 d e (4 e h+f g))\right )\right ) \left (\frac {3 f \int \frac {\sqrt {e+f x}}{a+b x}dx}{2 b}-\frac {(e+f x)^{3/2}}{b (a+b x)}\right )}{24 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (11 a^2 d f h-a b (5 c f h+12 d e h+5 d f g)+b^2 (6 c e h-c f g+6 d e g)\right )-a^2 b (c f h+8 d e h+d f g)+a b^2 (2 c e h-5 c f g+2 d e g)+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (35 a^2 d f^2 h-5 a b f (c f h+12 d e h+d f g)-\left (b^2 (c f (f g-6 e h)-6 d e (4 e h+f g))\right )\right ) \left (\frac {3 f \left (\frac {(b e-a f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b}+\frac {2 \sqrt {e+f x}}{b}\right )}{2 b}-\frac {(e+f x)^{3/2}}{b (a+b x)}\right )}{24 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (11 a^2 d f h-a b (5 c f h+12 d e h+5 d f g)+b^2 (6 c e h-c f g+6 d e g)\right )-a^2 b (c f h+8 d e h+d f g)+a b^2 (2 c e h-5 c f g+2 d e g)+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (35 a^2 d f^2 h-5 a b f (c f h+12 d e h+d f g)-\left (b^2 (c f (f g-6 e h)-6 d e (4 e h+f g))\right )\right ) \left (\frac {3 f \left (\frac {2 (b e-a f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b f}+\frac {2 \sqrt {e+f x}}{b}\right )}{2 b}-\frac {(e+f x)^{3/2}}{b (a+b x)}\right )}{24 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (11 a^2 d f h-a b (5 c f h+12 d e h+5 d f g)+b^2 (6 c e h-c f g+6 d e g)\right )-a^2 b (c f h+8 d e h+d f g)+a b^2 (2 c e h-5 c f g+2 d e g)+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {3 f \left (\frac {2 \sqrt {e+f x}}{b}-\frac {2 \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2}}\right )}{2 b}-\frac {(e+f x)^{3/2}}{b (a+b x)}\right ) \left (35 a^2 d f^2 h-5 a b f (c f h+12 d e h+d f g)-\left (b^2 (c f (f g-6 e h)-6 d e (4 e h+f g))\right )\right )}{24 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (11 a^2 d f h-a b (5 c f h+12 d e h+5 d f g)+b^2 (6 c e h-c f g+6 d e g)\right )-a^2 b (c f h+8 d e h+d f g)+a b^2 (2 c e h-5 c f g+2 d e g)+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\) |
Input:
Int[((c + d*x)*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^4,x]
Output:
-1/12*((e + f*x)^(5/2)*(4*b^3*c*e*g + 7*a^3*d*f*h + a*b^2*(2*d*e*g - 5*c*f *g + 2*c*e*h) - a^2*b*(d*f*g + 8*d*e*h + c*f*h) + b*(11*a^2*d*f*h + b^2*(6 *d*e*g - c*f*g + 6*c*e*h) - a*b*(5*d*f*g + 12*d*e*h + 5*c*f*h))*x))/(b^2*( b*e - a*f)^2*(a + b*x)^3) + ((35*a^2*d*f^2*h - 5*a*b*f*(d*f*g + 12*d*e*h + c*f*h) - b^2*(c*f*(f*g - 6*e*h) - 6*d*e*(f*g + 4*e*h)))*(-((e + f*x)^(3/2 )/(b*(a + b*x))) + (3*f*((2*Sqrt[e + f*x])/b - (2*Sqrt[b*e - a*f]*ArcTanh[ (Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/b^(3/2)))/(2*b)))/(24*b^2*(b*e - a*f)^2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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 = 0.53 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.18
| method | result | size |
| pseudoelliptic | \(-\frac {35 \left (\left (\frac {\left (-c g \,f^{2}+6 e \left (c h +d g \right ) f +24 d \,e^{2} h \right ) b^{2}}{35}-\frac {a \left (f \left (c h +d g \right )+12 d e h \right ) f b}{7}+a^{2} d \,f^{2} h \right ) \left (b x +a \right )^{3} f \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )-\sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}\, \left (\frac {\left (\frac {c \,f^{2} g \,x^{2}}{7}+\frac {2 x \left (-\frac {24 d h \,x^{2}}{7}+\frac {15 \left (c h +d g \right ) x}{7}+c g \right ) e f}{3}+\frac {8 \left (3 d h \,x^{2}+\frac {3 \left (c h +d g \right ) x}{2}+c g \right ) e^{2}}{21}\right ) b^{4}}{5}+\frac {4 \left (-2 x \left (-6 d h \,x^{2}+\frac {33 \left (c h +d g \right ) x}{8}+c g \right ) f^{2}-\frac {\left (126 d h \,x^{2}+11 \left (-c h -d g \right ) x +c g \right ) e f}{2}+e^{2} \left (6 d h x +c h +d g \right )\right ) a \,b^{3}}{105}+\frac {8 a^{2} \left (\left (\frac {231 d h \,x^{2}}{8}+5 \left (-c h -d g \right ) x -\frac {3 c g}{8}\right ) f^{2}+e \left (-\frac {149}{4} d h x +c h +d g \right ) f +d \,e^{2} h \right ) b^{2}}{105}-\frac {a^{3} \left (\left (-\frac {56}{3} d h x +c h +d g \right ) f +\frac {22 d e h}{3}\right ) f b}{7}+a^{4} d \,f^{2} h \right )\right )}{8 \sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{3} b^{4} \left (a f -b e \right )}\) | \(389\) |
| derivativedivides | \(2 f \left (\frac {d h \sqrt {f x +e}}{b^{4}}-\frac {\frac {-\frac {b^{2} \left (29 a^{2} d \,f^{2} h -11 a b c \,f^{2} h -36 a b d e f h -11 a b d \,f^{2} g +10 b^{2} c e f h +b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +10 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 \left (a f -b e \right )}-\frac {b \left (17 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -24 a b d e f h -5 a b d \,f^{2} g +6 b^{2} c e f h -b^{2} c \,f^{2} g +6 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} a^{3} d \,f^{3} h +\frac {5}{16} a^{2} b c \,f^{3} h +\frac {47}{16} a^{2} b d e \,f^{2} h +\frac {5}{16} a^{2} b d \,f^{3} g -\frac {11}{16} a \,b^{2} c e \,f^{2} h +\frac {1}{16} a \,b^{2} c \,f^{3} g -\frac {9}{4} a \,b^{2} d \,e^{2} f h -\frac {11}{16} a \,b^{2} d e \,f^{2} g +\frac {3}{8} b^{3} c \,e^{2} f h -\frac {1}{16} b^{3} c e \,f^{2} g +\frac {1}{2} b^{3} d \,e^{3} h +\frac {3}{8} b^{3} d \,e^{2} f g \right ) \sqrt {f x +e}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (35 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -60 a b d e f h -5 a b d \,f^{2} g +6 b^{2} c e f h -b^{2} c \,f^{2} g +24 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}}{b^{4}}\right )\) | \(488\) |
| default | \(2 f \left (\frac {d h \sqrt {f x +e}}{b^{4}}-\frac {\frac {-\frac {b^{2} \left (29 a^{2} d \,f^{2} h -11 a b c \,f^{2} h -36 a b d e f h -11 a b d \,f^{2} g +10 b^{2} c e f h +b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +10 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 \left (a f -b e \right )}-\frac {b \left (17 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -24 a b d e f h -5 a b d \,f^{2} g +6 b^{2} c e f h -b^{2} c \,f^{2} g +6 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} a^{3} d \,f^{3} h +\frac {5}{16} a^{2} b c \,f^{3} h +\frac {47}{16} a^{2} b d e \,f^{2} h +\frac {5}{16} a^{2} b d \,f^{3} g -\frac {11}{16} a \,b^{2} c e \,f^{2} h +\frac {1}{16} a \,b^{2} c \,f^{3} g -\frac {9}{4} a \,b^{2} d \,e^{2} f h -\frac {11}{16} a \,b^{2} d e \,f^{2} g +\frac {3}{8} b^{3} c \,e^{2} f h -\frac {1}{16} b^{3} c e \,f^{2} g +\frac {1}{2} b^{3} d \,e^{3} h +\frac {3}{8} b^{3} d \,e^{2} f g \right ) \sqrt {f x +e}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (35 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -60 a b d e f h -5 a b d \,f^{2} g +6 b^{2} c e f h -b^{2} c \,f^{2} g +24 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}}{b^{4}}\right )\) | \(488\) |
| risch | \(\frac {2 d f h \sqrt {f x +e}}{b^{4}}-\frac {2 f \left (\frac {-\frac {b^{2} \left (29 a^{2} d \,f^{2} h -11 a b c \,f^{2} h -36 a b d e f h -11 a b d \,f^{2} g +10 b^{2} c e f h +b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +10 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 \left (a f -b e \right )}-\frac {b \left (17 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -24 a b d e f h -5 a b d \,f^{2} g +6 b^{2} c e f h -b^{2} c \,f^{2} g +6 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} a^{3} d \,f^{3} h +\frac {5}{16} a^{2} b c \,f^{3} h +\frac {47}{16} a^{2} b d e \,f^{2} h +\frac {5}{16} a^{2} b d \,f^{3} g -\frac {11}{16} a \,b^{2} c e \,f^{2} h +\frac {1}{16} a \,b^{2} c \,f^{3} g -\frac {9}{4} a \,b^{2} d \,e^{2} f h -\frac {11}{16} a \,b^{2} d e \,f^{2} g +\frac {3}{8} b^{3} c \,e^{2} f h -\frac {1}{16} b^{3} c e \,f^{2} g +\frac {1}{2} b^{3} d \,e^{3} h +\frac {3}{8} b^{3} d \,e^{2} f g \right ) \sqrt {f x +e}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (35 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -60 a b d e f h -5 a b d \,f^{2} g +6 b^{2} c e f h -b^{2} c \,f^{2} g +24 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 \left (a f -b e \right ) \sqrt {\left (a f -b e \right ) b}}\right )}{b^{4}}\) | \(488\) |
Input:
int((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
-35/8/((a*f-b*e)*b)^(1/2)*((1/35*(-c*g*f^2+6*e*(c*h+d*g)*f+24*d*e^2*h)*b^2 -1/7*a*(f*(c*h+d*g)+12*d*e*h)*f*b+a^2*d*f^2*h)*(b*x+a)^3*f*arctan(b*(f*x+e )^(1/2)/((a*f-b*e)*b)^(1/2))-((a*f-b*e)*b)^(1/2)*(f*x+e)^(1/2)*(1/5*(1/7*c *f^2*g*x^2+2/3*x*(-24/7*d*h*x^2+15/7*(c*h+d*g)*x+c*g)*e*f+8/21*(3*d*h*x^2+ 3/2*(c*h+d*g)*x+c*g)*e^2)*b^4+4/105*(-2*x*(-6*d*h*x^2+33/8*(c*h+d*g)*x+c*g )*f^2-1/2*(126*d*h*x^2+11*(-c*h-d*g)*x+c*g)*e*f+e^2*(6*d*h*x+c*h+d*g))*a*b ^3+8/105*a^2*((231/8*d*h*x^2+5*(-c*h-d*g)*x-3/8*c*g)*f^2+e*(-149/4*d*h*x+c *h+d*g)*f+d*e^2*h)*b^2-1/7*a^3*((-56/3*d*h*x+c*h+d*g)*f+22/3*d*e*h)*f*b+a^ 4*d*f^2*h))/(b*x+a)^3/b^4/(a*f-b*e)
Leaf count of result is larger than twice the leaf count of optimal. 1111 vs. \(2 (305) = 610\).
Time = 0.20 (sec) , antiderivative size = 2236, normalized size of antiderivative = 6.76 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x, algorithm="fricas")
Output:
[-1/48*(3*(((6*b^5*d*e*f^2 - (b^5*c + 5*a*b^4*d)*f^3)*g + (24*b^5*d*e^2*f + 6*(b^5*c - 10*a*b^4*d)*e*f^2 - 5*(a*b^4*c - 7*a^2*b^3*d)*f^3)*h)*x^3 + 3 *((6*a*b^4*d*e*f^2 - (a*b^4*c + 5*a^2*b^3*d)*f^3)*g + (24*a*b^4*d*e^2*f + 6*(a*b^4*c - 10*a^2*b^3*d)*e*f^2 - 5*(a^2*b^3*c - 7*a^3*b^2*d)*f^3)*h)*x^2 + (6*a^3*b^2*d*e*f^2 - (a^3*b^2*c + 5*a^4*b*d)*f^3)*g + (24*a^3*b^2*d*e^2 *f + 6*(a^3*b^2*c - 10*a^4*b*d)*e*f^2 - 5*(a^4*b*c - 7*a^5*d)*f^3)*h + 3*( (6*a^2*b^3*d*e*f^2 - (a^2*b^3*c + 5*a^3*b^2*d)*f^3)*g + (24*a^2*b^3*d*e^2* f + 6*(a^2*b^3*c - 10*a^3*b^2*d)*e*f^2 - 5*(a^3*b^2*c - 7*a^4*b*d)*f^3)*h) *x)*sqrt(b^2*e - a*b*f)*log((b*f*x + 2*b*e - a*f + 2*sqrt(b^2*e - a*b*f)*s qrt(f*x + e))/(b*x + a)) - 2*(48*(b^6*d*e^2*f - 2*a*b^5*d*e*f^2 + a^2*b^4* d*f^3)*h*x^3 - 3*((10*b^6*d*e^2*f + (b^6*c - 21*a*b^5*d)*e*f^2 - (a*b^5*c - 11*a^2*b^4*d)*f^3)*g + (8*b^6*d*e^3 + 2*(5*b^6*c - 46*a*b^5*d)*e^2*f - 7 *(3*a*b^5*c - 23*a^2*b^4*d)*e*f^2 + 11*(a^2*b^4*c - 7*a^3*b^3*d)*f^3)*h)*x ^2 - (4*(2*b^6*c + a*b^5*d)*e^3 - 2*(5*a*b^5*c - 2*a^2*b^4*d)*e^2*f - (a^2 *b^4*c + 23*a^3*b^3*d)*e*f^2 + 3*(a^3*b^3*c + 5*a^4*b^2*d)*f^3)*g - (4*(a* b^5*c + 2*a^2*b^4*d)*e^3 + 2*(2*a^2*b^4*c - 59*a^3*b^3*d)*e^2*f - (23*a^3* b^3*c - 215*a^4*b^2*d)*e*f^2 + 15*(a^4*b^2*c - 7*a^5*b*d)*f^3)*h - 2*((6*b ^6*d*e^3 + (7*b^6*c + 5*a*b^5*d)*e^2*f - (11*a*b^5*c + 31*a^2*b^4*d)*e*f^2 + 4*(a^2*b^4*c + 5*a^3*b^3*d)*f^3)*g + (6*(b^6*c + 2*a*b^5*d)*e^3 + (5*a* b^5*c - 161*a^2*b^4*d)*e^2*f - (31*a^2*b^4*c - 289*a^3*b^3*d)*e*f^2 + 2...
Timed out. \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)*(f*x+e)**(3/2)*(h*x+g)/(b*x+a)**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,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(a*f-b*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (305) = 610\).
Time = 0.15 (sec) , antiderivative size = 881, normalized size of antiderivative = 2.66 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x, algorithm="giac")
Output:
1/8*(6*b^2*d*e*f^2*g - b^2*c*f^3*g - 5*a*b*d*f^3*g + 24*b^2*d*e^2*f*h + 6* b^2*c*e*f^2*h - 60*a*b*d*e*f^2*h - 5*a*b*c*f^3*h + 35*a^2*d*f^3*h)*arctan( sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^5*e - a*b^4*f)*sqrt(-b^2*e + a*b *f)) + 2*sqrt(f*x + e)*d*f*h/b^4 - 1/24*(30*(f*x + e)^(5/2)*b^4*d*e*f^2*g - 48*(f*x + e)^(3/2)*b^4*d*e^2*f^2*g + 18*sqrt(f*x + e)*b^4*d*e^3*f^2*g + 3*(f*x + e)^(5/2)*b^4*c*f^3*g - 33*(f*x + e)^(5/2)*a*b^3*d*f^3*g + 8*(f*x + e)^(3/2)*b^4*c*e*f^3*g + 88*(f*x + e)^(3/2)*a*b^3*d*e*f^3*g - 3*sqrt(f*x + e)*b^4*c*e^2*f^3*g - 51*sqrt(f*x + e)*a*b^3*d*e^2*f^3*g - 8*(f*x + e)^( 3/2)*a*b^3*c*f^4*g - 40*(f*x + e)^(3/2)*a^2*b^2*d*f^4*g + 6*sqrt(f*x + e)* a*b^3*c*e*f^4*g + 48*sqrt(f*x + e)*a^2*b^2*d*e*f^4*g - 3*sqrt(f*x + e)*a^2 *b^2*c*f^5*g - 15*sqrt(f*x + e)*a^3*b*d*f^5*g + 24*(f*x + e)^(5/2)*b^4*d*e ^2*f*h - 48*(f*x + e)^(3/2)*b^4*d*e^3*f*h + 24*sqrt(f*x + e)*b^4*d*e^4*f*h + 30*(f*x + e)^(5/2)*b^4*c*e*f^2*h - 108*(f*x + e)^(5/2)*a*b^3*d*e*f^2*h - 48*(f*x + e)^(3/2)*b^4*c*e^2*f^2*h + 240*(f*x + e)^(3/2)*a*b^3*d*e^2*f^2 *h + 18*sqrt(f*x + e)*b^4*c*e^3*f^2*h - 132*sqrt(f*x + e)*a*b^3*d*e^3*f^2* h - 33*(f*x + e)^(5/2)*a*b^3*c*f^3*h + 87*(f*x + e)^(5/2)*a^2*b^2*d*f^3*h + 88*(f*x + e)^(3/2)*a*b^3*c*e*f^3*h - 328*(f*x + e)^(3/2)*a^2*b^2*d*e*f^3 *h - 51*sqrt(f*x + e)*a*b^3*c*e^2*f^3*h + 249*sqrt(f*x + e)*a^2*b^2*d*e^2* f^3*h - 40*(f*x + e)^(3/2)*a^2*b^2*c*f^4*h + 136*(f*x + e)^(3/2)*a^3*b*d*f ^4*h + 48*sqrt(f*x + e)*a^2*b^2*c*e*f^4*h - 198*sqrt(f*x + e)*a^3*b*d*e...
Time = 0.35 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.28 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx=\frac {2\,d\,f\,h\,\sqrt {e+f\,x}}{b^4}-\frac {\sqrt {e+f\,x}\,\left (\frac {a\,b^2\,c\,f^4\,g}{8}-\frac {19\,a^3\,d\,f^4\,h}{8}+\frac {5\,a^2\,b\,c\,f^4\,h}{8}+\frac {5\,a^2\,b\,d\,f^4\,g}{8}-\frac {b^3\,c\,e\,f^3\,g}{8}+b^3\,d\,e^3\,f\,h+\frac {3\,b^3\,c\,e^2\,f^2\,h}{4}+\frac {3\,b^3\,d\,e^2\,f^2\,g}{4}-\frac {11\,a\,b^2\,c\,e\,f^3\,h}{8}-\frac {11\,a\,b^2\,d\,e\,f^3\,g}{8}+\frac {47\,a^2\,b\,d\,e\,f^3\,h}{8}-\frac {9\,a\,b^2\,d\,e^2\,f^2\,h}{2}\right )+{\left (e+f\,x\right )}^{3/2}\,\left (\frac {b^3\,c\,f^3\,g}{3}+\frac {5\,a\,b^2\,c\,f^3\,h}{3}+\frac {5\,a\,b^2\,d\,f^3\,g}{3}-\frac {17\,a^2\,b\,d\,f^3\,h}{3}-2\,b^3\,c\,e\,f^2\,h-2\,b^3\,d\,e\,f^2\,g-2\,b^3\,d\,e^2\,f\,h+8\,a\,b^2\,d\,e\,f^2\,h\right )-\frac {{\left (e+f\,x\right )}^{5/2}\,\left (b^4\,c\,f^3\,g-11\,a\,b^3\,c\,f^3\,h-11\,a\,b^3\,d\,f^3\,g+10\,b^4\,c\,e\,f^2\,h+10\,b^4\,d\,e\,f^2\,g+8\,b^4\,d\,e^2\,f\,h+29\,a^2\,b^2\,d\,f^3\,h-36\,a\,b^3\,d\,e\,f^2\,h\right )}{8\,\left (a\,f-b\,e\right )}}{b^7\,{\left (e+f\,x\right )}^3-{\left (e+f\,x\right )}^2\,\left (3\,b^7\,e-3\,a\,b^6\,f\right )+\left (e+f\,x\right )\,\left (3\,a^2\,b^5\,f^2-6\,a\,b^6\,e\,f+3\,b^7\,e^2\right )-b^7\,e^3+a^3\,b^4\,f^3-3\,a^2\,b^5\,e\,f^2+3\,a\,b^6\,e^2\,f}+\frac {f\,\mathrm {atan}\left (\frac {\sqrt {b}\,f\,\sqrt {e+f\,x}\,\left (b^2\,c\,f^2\,g-35\,a^2\,d\,f^2\,h-24\,b^2\,d\,e^2\,h+5\,a\,b\,c\,f^2\,h+5\,a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g+60\,a\,b\,d\,e\,f\,h\right )}{\sqrt {a\,f-b\,e}\,\left (b^2\,c\,f^3\,g-35\,a^2\,d\,f^3\,h-6\,b^2\,c\,e\,f^2\,h-6\,b^2\,d\,e\,f^2\,g-24\,b^2\,d\,e^2\,f\,h+5\,a\,b\,c\,f^3\,h+5\,a\,b\,d\,f^3\,g+60\,a\,b\,d\,e\,f^2\,h\right )}\right )\,\left (b^2\,c\,f^2\,g-35\,a^2\,d\,f^2\,h-24\,b^2\,d\,e^2\,h+5\,a\,b\,c\,f^2\,h+5\,a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g+60\,a\,b\,d\,e\,f\,h\right )}{8\,b^{9/2}\,{\left (a\,f-b\,e\right )}^{3/2}} \] Input:
int(((e + f*x)^(3/2)*(g + h*x)*(c + d*x))/(a + b*x)^4,x)
Output:
(2*d*f*h*(e + f*x)^(1/2))/b^4 - ((e + f*x)^(1/2)*((a*b^2*c*f^4*g)/8 - (19* a^3*d*f^4*h)/8 + (5*a^2*b*c*f^4*h)/8 + (5*a^2*b*d*f^4*g)/8 - (b^3*c*e*f^3* g)/8 + b^3*d*e^3*f*h + (3*b^3*c*e^2*f^2*h)/4 + (3*b^3*d*e^2*f^2*g)/4 - (11 *a*b^2*c*e*f^3*h)/8 - (11*a*b^2*d*e*f^3*g)/8 + (47*a^2*b*d*e*f^3*h)/8 - (9 *a*b^2*d*e^2*f^2*h)/2) + (e + f*x)^(3/2)*((b^3*c*f^3*g)/3 + (5*a*b^2*c*f^3 *h)/3 + (5*a*b^2*d*f^3*g)/3 - (17*a^2*b*d*f^3*h)/3 - 2*b^3*c*e*f^2*h - 2*b ^3*d*e*f^2*g - 2*b^3*d*e^2*f*h + 8*a*b^2*d*e*f^2*h) - ((e + f*x)^(5/2)*(b^ 4*c*f^3*g - 11*a*b^3*c*f^3*h - 11*a*b^3*d*f^3*g + 10*b^4*c*e*f^2*h + 10*b^ 4*d*e*f^2*g + 8*b^4*d*e^2*f*h + 29*a^2*b^2*d*f^3*h - 36*a*b^3*d*e*f^2*h))/ (8*(a*f - b*e)))/(b^7*(e + f*x)^3 - (e + f*x)^2*(3*b^7*e - 3*a*b^6*f) + (e + f*x)*(3*b^7*e^2 + 3*a^2*b^5*f^2 - 6*a*b^6*e*f) - b^7*e^3 + a^3*b^4*f^3 - 3*a^2*b^5*e*f^2 + 3*a*b^6*e^2*f) + (f*atan((b^(1/2)*f*(e + f*x)^(1/2)*(b ^2*c*f^2*g - 35*a^2*d*f^2*h - 24*b^2*d*e^2*h + 5*a*b*c*f^2*h + 5*a*b*d*f^2 *g - 6*b^2*c*e*f*h - 6*b^2*d*e*f*g + 60*a*b*d*e*f*h))/((a*f - b*e)^(1/2)*( b^2*c*f^3*g - 35*a^2*d*f^3*h - 6*b^2*c*e*f^2*h - 6*b^2*d*e*f^2*g - 24*b^2* d*e^2*f*h + 5*a*b*c*f^3*h + 5*a*b*d*f^3*g + 60*a*b*d*e*f^2*h)))*(b^2*c*f^2 *g - 35*a^2*d*f^2*h - 24*b^2*d*e^2*h + 5*a*b*c*f^2*h + 5*a*b*d*f^2*g - 6*b ^2*c*e*f*h - 6*b^2*d*e*f*g + 60*a*b*d*e*f*h))/(8*b^(9/2)*(a*f - b*e)^(3/2) )
Time = 0.17 (sec) , antiderivative size = 2616, normalized size of antiderivative = 7.90 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:
int((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^4,x)
Output:
( - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*d*f**3*h + 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/ (sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c*f**3*h + 180*sqrt(b)*sqrt(a*f - b*e)*a tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d*e*f**2*h + 15*sq rt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a* *4*b*d*f**3*g - 315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b )*sqrt(a*f - b*e)))*a**4*b*d*f**3*h*x - 18*sqrt(b)*sqrt(a*f - b*e)*atan((s qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*e*f**2*h + 3*sqrt(b )*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b **2*c*f**3*g + 45*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* sqrt(a*f - b*e)))*a**3*b**2*c*f**3*h*x - 72*sqrt(b)*sqrt(a*f - b*e)*atan(( sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d*e**2*f*h - 18*sqrt (b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3 *b**2*d*e*f**2*g + 540*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqr t(b)*sqrt(a*f - b*e)))*a**3*b**2*d*e*f**2*h*x + 45*sqrt(b)*sqrt(a*f - b*e) *atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d*f**3*g*x - 315*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e )))*a**3*b**2*d*f**3*h*x**2 - 54*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f* x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**3*c*e*f**2*h*x + 9*sqrt(b)*sqrt(a *f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**3*c...