Integrand size = 27, antiderivative size = 422 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx=-\frac {(b c-a d) (b g-a h) \sqrt {e+f x}}{4 b^3 (a+b x)^4}-\frac {\left (17 a^2 d f h+b^2 (8 d e g+c f g+8 c e h)-a b (9 d f g+16 d e h+9 c f h)\right ) \sqrt {e+f x}}{24 b^3 (b e-a f) (a+b x)^3}-\frac {\left (59 a^2 d f^2 h-a b f (3 d f g+112 d e h+3 c f h)-b^2 (c f (5 f g-8 e h)-8 d e (f g+6 e h))\right ) \sqrt {e+f x}}{96 b^3 (b e-a f)^2 (a+b x)^2}-\frac {f \left (5 a^2 d f^2 h+a b f (3 d f g-16 d e h+3 c f h)+b^2 (c f (5 f g-8 e h)-8 d e (f g-2 e h))\right ) \sqrt {e+f x}}{64 b^3 (b e-a f)^3 (a+b x)}+\frac {f^2 \left (5 a^2 d f^2 h+a b f (3 d f g-16 d e h+3 c f h)+b^2 (c f (5 f g-8 e h)-8 d e (f g-2 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{64 b^{7/2} (b e-a f)^{7/2}} \] Output:
-1/4*(-a*d+b*c)*(-a*h+b*g)*(f*x+e)^(1/2)/b^3/(b*x+a)^4-1/24*(17*a^2*d*f*h+ b^2*(8*c*e*h+c*f*g+8*d*e*g)-a*b*(9*c*f*h+16*d*e*h+9*d*f*g))*(f*x+e)^(1/2)/ b^3/(-a*f+b*e)/(b*x+a)^3-1/96*(59*a^2*d*f^2*h-a*b*f*(3*c*f*h+112*d*e*h+3*d *f*g)-b^2*(c*f*(-8*e*h+5*f*g)-8*d*e*(6*e*h+f*g)))*(f*x+e)^(1/2)/b^3/(-a*f+ b*e)^2/(b*x+a)^2-1/64*f*(5*a^2*d*f^2*h+a*b*f*(3*c*f*h-16*d*e*h+3*d*f*g)+b^ 2*(c*f*(-8*e*h+5*f*g)-8*d*e*(-2*e*h+f*g)))*(f*x+e)^(1/2)/b^3/(-a*f+b*e)^3/ (b*x+a)+1/64*f^2*(5*a^2*d*f^2*h+a*b*f*(3*c*f*h-16*d*e*h+3*d*f*g)+b^2*(c*f* (-8*e*h+5*f*g)-8*d*e*(-2*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b* e)^(1/2))/b^(7/2)/(-a*f+b*e)^(7/2)
Time = 3.36 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.33 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx=\frac {\frac {\sqrt {b} \sqrt {e+f x} \left (-15 a^5 d f^3 h-a^4 b f^2 (9 c f h+d (9 f g-38 e h+55 f h x))+b^5 \left (8 d e x \left (-3 f^2 g x^2+2 e f x (g+3 h x)+4 e^2 (2 g+3 h x)\right )+c \left (15 f^3 g x^3+8 e^2 f x (g+2 h x)+16 e^3 (3 g+4 h x)-2 e f^2 x^2 (5 g+12 h x)\right )\right )+a b^4 \left (d \left (9 f^3 g x^3+16 e^3 (g+4 h x)-8 e^2 f x (21 g+22 h x)-2 e f^2 x^2 (47 g+24 h x)\right )+c \left (16 e^3 h+f^3 x^2 (55 g+9 h x)-8 e^2 f (17 g+21 h x)-2 e f^2 x (18 g+47 h x)\right )\right )-a^3 b^2 f \left (3 c f (5 f g-6 e h+11 f h x)+d \left (24 e^2 h-2 e f (9 g+70 h x)+f^2 x (33 g+73 h x)\right )\right )+a^2 b^3 \left (c f \left (-40 e^2 h+f^2 x (73 g+33 h x)+2 e f (59 g+46 h x)\right )+d \left (16 e^3 h+3 f^3 x^2 (11 g+5 h x)-8 e^2 f (5 g+13 h x)+2 e f^2 x (46 g+99 h x)\right )\right )\right )}{(-b e+a f)^3 (a+b x)^4}+\frac {3 f^2 \left (5 a^2 d f^2 h+a b f (3 d f g-16 d e h+3 c f h)+b^2 (c f (5 f g-8 e h)+8 d e (-f g+2 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(-b e+a f)^{7/2}}}{192 b^{7/2}} \] Input:
Integrate[((c + d*x)*Sqrt[e + f*x]*(g + h*x))/(a + b*x)^5,x]
Output:
((Sqrt[b]*Sqrt[e + f*x]*(-15*a^5*d*f^3*h - a^4*b*f^2*(9*c*f*h + d*(9*f*g - 38*e*h + 55*f*h*x)) + b^5*(8*d*e*x*(-3*f^2*g*x^2 + 2*e*f*x*(g + 3*h*x) + 4*e^2*(2*g + 3*h*x)) + c*(15*f^3*g*x^3 + 8*e^2*f*x*(g + 2*h*x) + 16*e^3*(3 *g + 4*h*x) - 2*e*f^2*x^2*(5*g + 12*h*x))) + a*b^4*(d*(9*f^3*g*x^3 + 16*e^ 3*(g + 4*h*x) - 8*e^2*f*x*(21*g + 22*h*x) - 2*e*f^2*x^2*(47*g + 24*h*x)) + c*(16*e^3*h + f^3*x^2*(55*g + 9*h*x) - 8*e^2*f*(17*g + 21*h*x) - 2*e*f^2* x*(18*g + 47*h*x))) - a^3*b^2*f*(3*c*f*(5*f*g - 6*e*h + 11*f*h*x) + d*(24* e^2*h - 2*e*f*(9*g + 70*h*x) + f^2*x*(33*g + 73*h*x))) + a^2*b^3*(c*f*(-40 *e^2*h + f^2*x*(73*g + 33*h*x) + 2*e*f*(59*g + 46*h*x)) + d*(16*e^3*h + 3* f^3*x^2*(11*g + 5*h*x) - 8*e^2*f*(5*g + 13*h*x) + 2*e*f^2*x*(46*g + 99*h*x )))))/((-(b*e) + a*f)^3*(a + b*x)^4) + (3*f^2*(5*a^2*d*f^2*h + a*b*f*(3*d* f*g - 16*d*e*h + 3*c*f*h) + b^2*(c*f*(5*f*g - 8*e*h) + 8*d*e*(-(f*g) + 2*e *h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(-(b*e) + a*f)^( 7/2))/(192*b^(7/2))
Time = 0.52 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {162, 51, 52, 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.
| \(\displaystyle \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {\left (5 a^2 d f^2 h+a b f (3 c f h-16 d e h+3 d f g)+b^2 (c f (5 f g-8 e h)-8 d e (f g-2 e h))\right ) \int \frac {\sqrt {e+f x}}{(a+b x)^3}dx}{16 b^2 (b e-a f)^2}-\frac {(e+f x)^{3/2} \left (5 a^3 d f h+b x \left (11 a^2 d f h-a b (3 c f h+16 d e h+3 d f g)+b^2 (8 c e h-5 c f g+8 d e g)\right )-2 a^2 b \left (5 d e h-\frac {3}{2} f (c h+d g)\right )+2 a b^2 \left (c e h-\frac {11 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\left (5 a^2 d f^2 h+a b f (3 c f h-16 d e h+3 d f g)+b^2 (c f (5 f g-8 e h)-8 d e (f g-2 e h))\right ) \left (\frac {f \int \frac {1}{(a+b x)^2 \sqrt {e+f x}}dx}{4 b}-\frac {\sqrt {e+f x}}{2 b (a+b x)^2}\right )}{16 b^2 (b e-a f)^2}-\frac {(e+f x)^{3/2} \left (5 a^3 d f h+b x \left (11 a^2 d f h-a b (3 c f h+16 d e h+3 d f g)+b^2 (8 c e h-5 c f g+8 d e g)\right )-2 a^2 b \left (5 d e h-\frac {3}{2} f (c h+d g)\right )+2 a b^2 \left (c e h-\frac {11 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\left (5 a^2 d f^2 h+a b f (3 c f h-16 d e h+3 d f g)+b^2 (c f (5 f g-8 e h)-8 d e (f g-2 e h))\right ) \left (\frac {f \left (-\frac {f \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 (b e-a f)}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 b}-\frac {\sqrt {e+f x}}{2 b (a+b x)^2}\right )}{16 b^2 (b e-a f)^2}-\frac {(e+f x)^{3/2} \left (5 a^3 d f h+b x \left (11 a^2 d f h-a b (3 c f h+16 d e h+3 d f g)+b^2 (8 c e h-5 c f g+8 d e g)\right )-2 a^2 b \left (5 d e h-\frac {3}{2} f (c h+d g)\right )+2 a b^2 \left (c e h-\frac {11 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (5 a^2 d f^2 h+a b f (3 c f h-16 d e h+3 d f g)+b^2 (c f (5 f g-8 e h)-8 d e (f g-2 e h))\right ) \left (\frac {f \left (-\frac {\int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b e-a f}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 b}-\frac {\sqrt {e+f x}}{2 b (a+b x)^2}\right )}{16 b^2 (b e-a f)^2}-\frac {(e+f x)^{3/2} \left (5 a^3 d f h+b x \left (11 a^2 d f h-a b (3 c f h+16 d e h+3 d f g)+b^2 (8 c e h-5 c f g+8 d e g)\right )-2 a^2 b \left (5 d e h-\frac {3}{2} f (c h+d g)\right )+2 a b^2 \left (c e h-\frac {11 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {f \left (\frac {f \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b e-a f)^{3/2}}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 b}-\frac {\sqrt {e+f x}}{2 b (a+b x)^2}\right ) \left (5 a^2 d f^2 h+a b f (3 c f h-16 d e h+3 d f g)+b^2 (c f (5 f g-8 e h)-8 d e (f g-2 e h))\right )}{16 b^2 (b e-a f)^2}-\frac {(e+f x)^{3/2} \left (5 a^3 d f h+b x \left (11 a^2 d f h-a b (3 c f h+16 d e h+3 d f g)+b^2 (8 c e h-5 c f g+8 d e g)\right )-2 a^2 b \left (5 d e h-\frac {3}{2} f (c h+d g)\right )+2 a b^2 \left (c e h-\frac {11 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\) |
Input:
Int[((c + d*x)*Sqrt[e + f*x]*(g + h*x))/(a + b*x)^5,x]
Output:
-1/24*((e + f*x)^(3/2)*(6*b^3*c*e*g + 5*a^3*d*f*h + 2*a*b^2*(d*e*g - (11*c *f*g)/2 + c*e*h) - 2*a^2*b*(5*d*e*h - (3*f*(d*g + c*h))/2) + b*(11*a^2*d*f *h + b^2*(8*d*e*g - 5*c*f*g + 8*c*e*h) - a*b*(3*d*f*g + 16*d*e*h + 3*c*f*h ))*x))/(b^2*(b*e - a*f)^2*(a + b*x)^4) + ((5*a^2*d*f^2*h + a*b*f*(3*d*f*g - 16*d*e*h + 3*c*f*h) + b^2*(c*f*(5*f*g - 8*e*h) - 8*d*e*(f*g - 2*e*h)))*( -1/2*Sqrt[e + f*x]/(b*(a + b*x)^2) + (f*(-(Sqrt[e + f*x]/((b*e - a*f)*(a + b*x))) + (f*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b *e - a*f)^(3/2))))/(4*b)))/(16*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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 0.58 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.27
| method | result | size |
| pseudoelliptic | \(\frac {\frac {5 \left (b x +a \right )^{4} f^{2} \left (\left (c g \,f^{2}-\frac {8 e \left (c h +d g \right ) f}{5}+\frac {16 d \,e^{2} h}{5}\right ) b^{2}+\frac {3 a \left (f \left (c h +d g \right )-\frac {16 d e h}{3}\right ) f b}{5}+a^{2} d \,f^{2} h \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{64}-\frac {5 \sqrt {f x +e}\, \sqrt {\left (a f -b e \right ) b}\, \left (\left (-c \,f^{3} g \,x^{3}+\frac {2 x^{2} \left (\frac {12 \left (c h +d g \right ) x}{5}+c g \right ) e \,f^{2}}{3}-\frac {8 x \left (6 d h \,x^{2}+2 \left (c h +d g \right ) x +c g \right ) e^{2} f}{15}-\frac {16 \left (2 d h \,x^{2}+\frac {4 \left (c h +d g \right ) x}{3}+c g \right ) e^{3}}{5}\right ) b^{5}-\frac {16 a \left (\frac {\left (9 \left (c h +d g \right ) x^{3}+55 c g \,x^{2}\right ) f^{3}}{16}-\frac {9 \left (\frac {4 d h \,x^{2}}{3}+\frac {47 \left (c h +d g \right ) x}{18}+c g \right ) x e \,f^{2}}{4}-\frac {17 \left (\frac {22 d h \,x^{2}}{17}+\frac {21 \left (c h +d g \right ) x}{17}+c g \right ) e^{2} f}{2}+e^{3} \left (4 d h x +c h +d g \right )\right ) b^{4}}{15}+\frac {8 a^{2} \left (\frac {\left (-3 d h \,x^{3}+\frac {33 \left (-c h -d g \right ) x^{2}}{5}-\frac {73 c g x}{5}\right ) f^{3}}{8}-\frac {59 e \left (\frac {99 d h \,x^{2}}{59}+\frac {46 \left (c h +d g \right ) x}{59}+c g \right ) f^{2}}{20}+e^{2} \left (\frac {13}{5} d h x +c h +d g \right ) f -\frac {2 d \,e^{3} h}{5}\right ) b^{3}}{3}-\frac {6 a^{3} \left (\frac {\left (-\frac {73 d h \,x^{2}}{3}+11 \left (-c h -d g \right ) x -5 c g \right ) f^{2}}{6}+e \left (\frac {70}{9} d h x +c h +d g \right ) f -\frac {4 d \,e^{2} h}{3}\right ) f \,b^{2}}{5}+\frac {3 a^{4} \left (\left (\frac {55}{9} d h x +c h +d g \right ) f -\frac {38 d e h}{9}\right ) f^{2} b}{5}+a^{5} d \,f^{3} h \right )}{64}}{\sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{4} \left (a f -b e \right )^{3} b^{3}}\) | \(536\) |
| derivativedivides | \(2 f^{2} \left (-\frac {-\frac {\left (5 a^{2} d \,f^{2} h +3 a b c \,f^{2} h -16 a b d e f h +3 a b d \,f^{2} g -8 b^{2} c e f h +5 b^{2} c \,f^{2} g +16 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {7}{2}}}{128 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (73 a^{2} d \,f^{2} h -33 a b c \,f^{2} h -80 a b d e f h -33 a b d \,f^{2} g +88 b^{2} c e f h -55 b^{2} c \,f^{2} g -48 b^{2} d \,e^{2} h +88 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{384 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (55 a^{2} d \,f^{2} h +33 a b c \,f^{2} h -176 a b d e f h +33 a b d \,f^{2} g +40 b^{2} c e f h -73 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h +40 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{384 b^{2} \left (a f -b e \right )}+\frac {\left (5 a^{2} d \,f^{2} h +3 a b c \,f^{2} h -16 a b d e f h +3 a b d \,f^{2} g -8 b^{2} c e f h +5 b^{2} c \,f^{2} g +16 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \sqrt {f x +e}}{128 b^{3}}}{\left (\left (f x +e \right ) b +a f -b e \right )^{4}}+\frac {\left (5 a^{2} d \,f^{2} h +3 a b c \,f^{2} h -16 a b d e f h +3 a b d \,f^{2} g -8 b^{2} c e f h +5 b^{2} c \,f^{2} g +16 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{128 b^{3} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\) | \(595\) |
| default | \(2 f^{2} \left (-\frac {-\frac {\left (5 a^{2} d \,f^{2} h +3 a b c \,f^{2} h -16 a b d e f h +3 a b d \,f^{2} g -8 b^{2} c e f h +5 b^{2} c \,f^{2} g +16 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {7}{2}}}{128 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (73 a^{2} d \,f^{2} h -33 a b c \,f^{2} h -80 a b d e f h -33 a b d \,f^{2} g +88 b^{2} c e f h -55 b^{2} c \,f^{2} g -48 b^{2} d \,e^{2} h +88 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{384 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (55 a^{2} d \,f^{2} h +33 a b c \,f^{2} h -176 a b d e f h +33 a b d \,f^{2} g +40 b^{2} c e f h -73 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h +40 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{384 b^{2} \left (a f -b e \right )}+\frac {\left (5 a^{2} d \,f^{2} h +3 a b c \,f^{2} h -16 a b d e f h +3 a b d \,f^{2} g -8 b^{2} c e f h +5 b^{2} c \,f^{2} g +16 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \sqrt {f x +e}}{128 b^{3}}}{\left (\left (f x +e \right ) b +a f -b e \right )^{4}}+\frac {\left (5 a^{2} d \,f^{2} h +3 a b c \,f^{2} h -16 a b d e f h +3 a b d \,f^{2} g -8 b^{2} c e f h +5 b^{2} c \,f^{2} g +16 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{128 b^{3} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\) | \(595\) |
Input:
int((d*x+c)*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
5/64/((a*f-b*e)*b)^(1/2)*((b*x+a)^4*f^2*((c*g*f^2-8/5*e*(c*h+d*g)*f+16/5*d *e^2*h)*b^2+3/5*a*(f*(c*h+d*g)-16/3*d*e*h)*f*b+a^2*d*f^2*h)*arctan(b*(f*x+ e)^(1/2)/((a*f-b*e)*b)^(1/2))-(f*x+e)^(1/2)*((a*f-b*e)*b)^(1/2)*((-c*f^3*g *x^3+2/3*x^2*(12/5*(c*h+d*g)*x+c*g)*e*f^2-8/15*x*(6*d*h*x^2+2*(c*h+d*g)*x+ c*g)*e^2*f-16/5*(2*d*h*x^2+4/3*(c*h+d*g)*x+c*g)*e^3)*b^5-16/15*a*(1/16*(9* (c*h+d*g)*x^3+55*c*g*x^2)*f^3-9/4*(4/3*d*h*x^2+47/18*(c*h+d*g)*x+c*g)*x*e* f^2-17/2*(22/17*d*h*x^2+21/17*(c*h+d*g)*x+c*g)*e^2*f+e^3*(4*d*h*x+c*h+d*g) )*b^4+8/3*a^2*(1/8*(-3*d*h*x^3+33/5*(-c*h-d*g)*x^2-73/5*c*g*x)*f^3-59/20*e *(99/59*d*h*x^2+46/59*(c*h+d*g)*x+c*g)*f^2+e^2*(13/5*d*h*x+c*h+d*g)*f-2/5* d*e^3*h)*b^3-6/5*a^3*(1/6*(-73/3*d*h*x^2+11*(-c*h-d*g)*x-5*c*g)*f^2+e*(70/ 9*d*h*x+c*h+d*g)*f-4/3*d*e^2*h)*f*b^2+3/5*a^4*((55/9*d*h*x+c*h+d*g)*f-38/9 *d*e*h)*f^2*b+a^5*d*f^3*h))/(b*x+a)^4/(a*f-b*e)^3/b^3
Leaf count of result is larger than twice the leaf count of optimal. 1685 vs. \(2 (398) = 796\).
Time = 0.34 (sec) , antiderivative size = 3384, normalized size of antiderivative = 8.02 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)*(f*x+e)**(1/2)*(h*x+g)/(b*x+a)**5,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5,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. 1350 vs. \(2 (398) = 796\).
Time = 0.17 (sec) , antiderivative size = 1350, normalized size of antiderivative = 3.20 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5,x, algorithm="giac")
Output:
1/64*(8*b^2*d*e*f^3*g - 5*b^2*c*f^4*g - 3*a*b*d*f^4*g - 16*b^2*d*e^2*f^2*h + 8*b^2*c*e*f^3*h + 16*a*b*d*e*f^3*h - 3*a*b*c*f^4*h - 5*a^2*d*f^4*h)*arc tan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^6*e^3 - 3*a*b^5*e^2*f + 3*a^ 2*b^4*e*f^2 - a^3*b^3*f^3)*sqrt(-b^2*e + a*b*f)) + 1/192*(24*(f*x + e)^(7/ 2)*b^5*d*e*f^3*g - 88*(f*x + e)^(5/2)*b^5*d*e^2*f^3*g + 40*(f*x + e)^(3/2) *b^5*d*e^3*f^3*g + 24*sqrt(f*x + e)*b^5*d*e^4*f^3*g - 15*(f*x + e)^(7/2)*b ^5*c*f^4*g - 9*(f*x + e)^(7/2)*a*b^4*d*f^4*g + 55*(f*x + e)^(5/2)*b^5*c*e* f^4*g + 121*(f*x + e)^(5/2)*a*b^4*d*e*f^4*g - 73*(f*x + e)^(3/2)*b^5*c*e^2 *f^4*g - 47*(f*x + e)^(3/2)*a*b^4*d*e^2*f^4*g - 15*sqrt(f*x + e)*b^5*c*e^3 *f^4*g - 81*sqrt(f*x + e)*a*b^4*d*e^3*f^4*g - 55*(f*x + e)^(5/2)*a*b^4*c*f ^5*g - 33*(f*x + e)^(5/2)*a^2*b^3*d*f^5*g + 146*(f*x + e)^(3/2)*a*b^4*c*e* f^5*g - 26*(f*x + e)^(3/2)*a^2*b^3*d*e*f^5*g + 45*sqrt(f*x + e)*a*b^4*c*e^ 2*f^5*g + 99*sqrt(f*x + e)*a^2*b^3*d*e^2*f^5*g - 73*(f*x + e)^(3/2)*a^2*b^ 3*c*f^6*g + 33*(f*x + e)^(3/2)*a^3*b^2*d*f^6*g - 45*sqrt(f*x + e)*a^2*b^3* c*e*f^6*g - 51*sqrt(f*x + e)*a^3*b^2*d*e*f^6*g + 15*sqrt(f*x + e)*a^3*b^2* c*f^7*g + 9*sqrt(f*x + e)*a^4*b*d*f^7*g - 48*(f*x + e)^(7/2)*b^5*d*e^2*f^2 *h + 48*(f*x + e)^(5/2)*b^5*d*e^3*f^2*h + 48*(f*x + e)^(3/2)*b^5*d*e^4*f^2 *h - 48*sqrt(f*x + e)*b^5*d*e^5*f^2*h + 24*(f*x + e)^(7/2)*b^5*c*e*f^3*h + 48*(f*x + e)^(7/2)*a*b^4*d*e*f^3*h - 88*(f*x + e)^(5/2)*b^5*c*e^2*f^3*h + 32*(f*x + e)^(5/2)*a*b^4*d*e^2*f^3*h + 40*(f*x + e)^(3/2)*b^5*c*e^3*f^...
Time = 2.29 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.04 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx=\frac {f^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,f^2\,\sqrt {e+f\,x}\,\left (5\,b^2\,c\,f^2\,g+5\,a^2\,d\,f^2\,h+16\,b^2\,d\,e^2\,h+3\,a\,b\,c\,f^2\,h+3\,a\,b\,d\,f^2\,g-8\,b^2\,c\,e\,f\,h-8\,b^2\,d\,e\,f\,g-16\,a\,b\,d\,e\,f\,h\right )}{\sqrt {a\,f-b\,e}\,\left (5\,b^2\,c\,f^4\,g+5\,a^2\,d\,f^4\,h-8\,b^2\,c\,e\,f^3\,h-8\,b^2\,d\,e\,f^3\,g+16\,b^2\,d\,e^2\,f^2\,h+3\,a\,b\,c\,f^4\,h+3\,a\,b\,d\,f^4\,g-16\,a\,b\,d\,e\,f^3\,h\right )}\right )\,\left (5\,b^2\,c\,f^2\,g+5\,a^2\,d\,f^2\,h+16\,b^2\,d\,e^2\,h+3\,a\,b\,c\,f^2\,h+3\,a\,b\,d\,f^2\,g-8\,b^2\,c\,e\,f\,h-8\,b^2\,d\,e\,f\,g-16\,a\,b\,d\,e\,f\,h\right )}{64\,b^{7/2}\,{\left (a\,f-b\,e\right )}^{7/2}}-\frac {\frac {\sqrt {e+f\,x}\,\left (5\,b^2\,c\,f^4\,g+5\,a^2\,d\,f^4\,h-8\,b^2\,c\,e\,f^3\,h-8\,b^2\,d\,e\,f^3\,g+16\,b^2\,d\,e^2\,f^2\,h+3\,a\,b\,c\,f^4\,h+3\,a\,b\,d\,f^4\,g-16\,a\,b\,d\,e\,f^3\,h\right )}{64\,b^3}-\frac {{\left (e+f\,x\right )}^{7/2}\,\left (5\,b^2\,c\,f^4\,g+5\,a^2\,d\,f^4\,h-8\,b^2\,c\,e\,f^3\,h-8\,b^2\,d\,e\,f^3\,g+16\,b^2\,d\,e^2\,f^2\,h+3\,a\,b\,c\,f^4\,h+3\,a\,b\,d\,f^4\,g-16\,a\,b\,d\,e\,f^3\,h\right )}{64\,{\left (a\,f-b\,e\right )}^3}+\frac {{\left (e+f\,x\right )}^{3/2}\,\left (55\,a^2\,d\,f^4\,h-73\,b^2\,c\,f^4\,g+40\,b^2\,c\,e\,f^3\,h+40\,b^2\,d\,e\,f^3\,g+48\,b^2\,d\,e^2\,f^2\,h+33\,a\,b\,c\,f^4\,h+33\,a\,b\,d\,f^4\,g-176\,a\,b\,d\,e\,f^3\,h\right )}{192\,b^2\,\left (a\,f-b\,e\right )}-\frac {{\left (e+f\,x\right )}^{5/2}\,\left (55\,b^2\,c\,f^4\,g-73\,a^2\,d\,f^4\,h-88\,b^2\,c\,e\,f^3\,h-88\,b^2\,d\,e\,f^3\,g+48\,b^2\,d\,e^2\,f^2\,h+33\,a\,b\,c\,f^4\,h+33\,a\,b\,d\,f^4\,g+80\,a\,b\,d\,e\,f^3\,h\right )}{192\,b\,{\left (a\,f-b\,e\right )}^2}}{b^4\,{\left (e+f\,x\right )}^4-{\left (e+f\,x\right )}^3\,\left (4\,b^4\,e-4\,a\,b^3\,f\right )-\left (e+f\,x\right )\,\left (-4\,a^3\,b\,f^3+12\,a^2\,b^2\,e\,f^2-12\,a\,b^3\,e^2\,f+4\,b^4\,e^3\right )+a^4\,f^4+b^4\,e^4+{\left (e+f\,x\right )}^2\,\left (6\,a^2\,b^2\,f^2-12\,a\,b^3\,e\,f+6\,b^4\,e^2\right )+6\,a^2\,b^2\,e^2\,f^2-4\,a\,b^3\,e^3\,f-4\,a^3\,b\,e\,f^3} \] Input:
int(((e + f*x)^(1/2)*(g + h*x)*(c + d*x))/(a + b*x)^5,x)
Output:
(f^2*atan((b^(1/2)*f^2*(e + f*x)^(1/2)*(5*b^2*c*f^2*g + 5*a^2*d*f^2*h + 16 *b^2*d*e^2*h + 3*a*b*c*f^2*h + 3*a*b*d*f^2*g - 8*b^2*c*e*f*h - 8*b^2*d*e*f *g - 16*a*b*d*e*f*h))/((a*f - b*e)^(1/2)*(5*b^2*c*f^4*g + 5*a^2*d*f^4*h - 8*b^2*c*e*f^3*h - 8*b^2*d*e*f^3*g + 16*b^2*d*e^2*f^2*h + 3*a*b*c*f^4*h + 3 *a*b*d*f^4*g - 16*a*b*d*e*f^3*h)))*(5*b^2*c*f^2*g + 5*a^2*d*f^2*h + 16*b^2 *d*e^2*h + 3*a*b*c*f^2*h + 3*a*b*d*f^2*g - 8*b^2*c*e*f*h - 8*b^2*d*e*f*g - 16*a*b*d*e*f*h))/(64*b^(7/2)*(a*f - b*e)^(7/2)) - (((e + f*x)^(1/2)*(5*b^ 2*c*f^4*g + 5*a^2*d*f^4*h - 8*b^2*c*e*f^3*h - 8*b^2*d*e*f^3*g + 16*b^2*d*e ^2*f^2*h + 3*a*b*c*f^4*h + 3*a*b*d*f^4*g - 16*a*b*d*e*f^3*h))/(64*b^3) - ( (e + f*x)^(7/2)*(5*b^2*c*f^4*g + 5*a^2*d*f^4*h - 8*b^2*c*e*f^3*h - 8*b^2*d *e*f^3*g + 16*b^2*d*e^2*f^2*h + 3*a*b*c*f^4*h + 3*a*b*d*f^4*g - 16*a*b*d*e *f^3*h))/(64*(a*f - b*e)^3) + ((e + f*x)^(3/2)*(55*a^2*d*f^4*h - 73*b^2*c* f^4*g + 40*b^2*c*e*f^3*h + 40*b^2*d*e*f^3*g + 48*b^2*d*e^2*f^2*h + 33*a*b* c*f^4*h + 33*a*b*d*f^4*g - 176*a*b*d*e*f^3*h))/(192*b^2*(a*f - b*e)) - ((e + f*x)^(5/2)*(55*b^2*c*f^4*g - 73*a^2*d*f^4*h - 88*b^2*c*e*f^3*h - 88*b^2 *d*e*f^3*g + 48*b^2*d*e^2*f^2*h + 33*a*b*c*f^4*h + 33*a*b*d*f^4*g + 80*a*b *d*e*f^3*h))/(192*b*(a*f - b*e)^2))/(b^4*(e + f*x)^4 - (e + f*x)^3*(4*b^4* e - 4*a*b^3*f) - (e + f*x)*(4*b^4*e^3 - 4*a^3*b*f^3 + 12*a^2*b^2*e*f^2 - 1 2*a*b^3*e^2*f) + a^4*f^4 + b^4*e^4 + (e + f*x)^2*(6*b^4*e^2 + 6*a^2*b^2*f^ 2 - 12*a*b^3*e*f) + 6*a^2*b^2*e^2*f^2 - 4*a*b^3*e^3*f - 4*a^3*b*e*f^3)
Time = 0.18 (sec) , antiderivative size = 3715, normalized size of antiderivative = 8.80 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:
int((d*x+c)*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5,x)
Output:
(15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e )))*a**6*d*f**4*h + 9*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt (b)*sqrt(a*f - b*e)))*a**5*b*c*f**4*h - 48*sqrt(b)*sqrt(a*f - b*e)*atan((s qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*d*e*f**3*h + 9*sqrt(b)*s qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*d* f**4*g + 60*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a *f - b*e)))*a**5*b*d*f**4*h*x - 24*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*e*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**2*c*f **4*g + 36*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a* f - b*e)))*a**4*b**2*c*f**4*h*x + 48*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d*e**2*f**2*h - 24*sqrt(b)* sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b** 2*d*e*f**3*g - 192*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b) *sqrt(a*f - b*e)))*a**4*b**2*d*e*f**3*h*x + 36*sqrt(b)*sqrt(a*f - b*e)*ata n((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d*f**4*g*x + 90*s qrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a **4*b**2*d*f**4*h*x**2 - 96*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) /(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**3*c*e*f**3*h*x + 60*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**3*c*f**...