Integrand size = 27, antiderivative size = 374 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\frac {2 f (d e-c f) (f g-e h)}{(b e-a f)^4 \sqrt {e+f x}}-\frac {(b c-a d) (b g-a h) \sqrt {e+f x}}{3 b (b e-a f)^2 (a+b x)^3}-\frac {\left (a^2 d f h+b^2 (6 d e g-11 c f g+6 c e h)+a b (5 d f g-12 d e h+5 c f h)\right ) \sqrt {e+f x}}{12 b (b e-a f)^3 (a+b x)^2}+\frac {\left (a^2 d f^2 h+a b f (5 d f g-12 d e h+5 c f h)-b^2 (c f (19 f g-14 e h)-2 d e (7 f g-4 e h))\right ) \sqrt {e+f x}}{8 b (b e-a f)^4 (a+b x)}-\frac {f \left (a^2 d f^2 h+a b f (5 d f g-12 d e h+5 c f h)-b^2 (5 c f (7 f g-6 e h)-6 d e (5 f g-4 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{3/2} (b e-a f)^{9/2}} \] Output:
2*f*(-c*f+d*e)*(-e*h+f*g)/(-a*f+b*e)^4/(f*x+e)^(1/2)-1/3*(-a*d+b*c)*(-a*h+ b*g)*(f*x+e)^(1/2)/b/(-a*f+b*e)^2/(b*x+a)^3-1/12*(a^2*d*f*h+b^2*(6*c*e*h-1 1*c*f*g+6*d*e*g)+a*b*(5*c*f*h-12*d*e*h+5*d*f*g))*(f*x+e)^(1/2)/b/(-a*f+b*e )^3/(b*x+a)^2+1/8*(a^2*d*f^2*h+a*b*f*(5*c*f*h-12*d*e*h+5*d*f*g)-b^2*(c*f*( -14*e*h+19*f*g)-2*d*e*(-4*e*h+7*f*g)))*(f*x+e)^(1/2)/b/(-a*f+b*e)^4/(b*x+a )-1/8*f*(a^2*d*f^2*h+a*b*f*(5*c*f*h-12*d*e*h+5*d*f*g)-b^2*(5*c*f*(-6*e*h+7 *f*g)-6*d*e*(-4*e*h+5*f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2 ))/b^(3/2)/(-a*f+b*e)^(9/2)
Time = 2.35 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.41 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\frac {-3 a^4 d f^2 h (e+f x)+a^3 b f \left (3 c f (-16 f g+27 e h+11 f h x)+d \left (-94 e^2 h+e f (81 g-38 h x)+f^2 x (33 g+8 h x)\right )\right )-b^4 \left (6 d e x \left (-15 f^2 g x^2+2 e^2 (g+2 h x)+e f x (-5 g+12 h x)\right )+c \left (105 f^3 g x^3+5 e f^2 x^2 (7 g-18 h x)+4 e^3 (2 g+3 h x)-2 e^2 f x (7 g+15 h x)\right )\right )+a b^3 \left (d \left (15 f^3 g x^3+2 e^2 f x (41 g-102 h x)+e f^2 x^2 (245 g-36 h x)-4 e^3 (g+6 h x)\right )+c \left (-4 e^3 h+5 f^3 x^2 (-56 g+3 h x)+49 e f^2 x (-2 g+5 h x)+e^2 f (38 g+82 h x)\right )\right )+a^2 b^2 \left (d \left (-8 e^3 h+e^2 f (28 g-250 h x)+e f^2 x (212 g-95 h x)+f^3 x^2 (40 g+3 h x)\right )+c f \left (28 e^2 h+f^2 x (-231 g+40 h x)+e f (-87 g+212 h x)\right )\right )}{24 b (b e-a f)^4 (a+b x)^3 \sqrt {e+f x}}-\frac {f \left (-a^2 d f^2 h+a b f (-5 d f g+12 d e h-5 c f h)+b^2 (5 c f (7 f g-6 e h)+6 d e (-5 f g+4 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{8 b^{3/2} (-b e+a f)^{9/2}} \] Input:
Integrate[((c + d*x)*(g + h*x))/((a + b*x)^4*(e + f*x)^(3/2)),x]
Output:
(-3*a^4*d*f^2*h*(e + f*x) + a^3*b*f*(3*c*f*(-16*f*g + 27*e*h + 11*f*h*x) + d*(-94*e^2*h + e*f*(81*g - 38*h*x) + f^2*x*(33*g + 8*h*x))) - b^4*(6*d*e* x*(-15*f^2*g*x^2 + 2*e^2*(g + 2*h*x) + e*f*x*(-5*g + 12*h*x)) + c*(105*f^3 *g*x^3 + 5*e*f^2*x^2*(7*g - 18*h*x) + 4*e^3*(2*g + 3*h*x) - 2*e^2*f*x*(7*g + 15*h*x))) + a*b^3*(d*(15*f^3*g*x^3 + 2*e^2*f*x*(41*g - 102*h*x) + e*f^2 *x^2*(245*g - 36*h*x) - 4*e^3*(g + 6*h*x)) + c*(-4*e^3*h + 5*f^3*x^2*(-56* g + 3*h*x) + 49*e*f^2*x*(-2*g + 5*h*x) + e^2*f*(38*g + 82*h*x))) + a^2*b^2 *(d*(-8*e^3*h + e^2*f*(28*g - 250*h*x) + e*f^2*x*(212*g - 95*h*x) + f^3*x^ 2*(40*g + 3*h*x)) + c*f*(28*e^2*h + f^2*x*(-231*g + 40*h*x) + e*f*(-87*g + 212*h*x))))/(24*b*(b*e - a*f)^4*(a + b*x)^3*Sqrt[e + f*x]) - (f*(-(a^2*d* f^2*h) + a*b*f*(-5*d*f*g + 12*d*e*h - 5*c*f*h) + b^2*(5*c*f*(7*f*g - 6*e*h ) + 6*d*e*(-5*f*g + 4*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(8*b^(3/2)*(-(b*e) + a*f)^(9/2))
Time = 0.48 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {161, 52, 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.
| \(\displaystyle \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 161 |
| \(\displaystyle \frac {\left (a^2 d f^2 h+a b f (5 c f h-12 d e h+5 d f g)-\left (b^2 (5 c f (7 f g-6 e h)-6 d e (5 f g-4 e h))\right )\right ) \int \frac {1}{(a+b x)^3 \sqrt {e+f x}}dx}{6 b f (b e-a f)^2}-\frac {x \left (a^2 d f^2 h-a b f^2 (c h+d g)+b^2 (c f (7 f g-6 e h)-6 d e (f g-e h))\right )+a^2 d e f h+2 a b \left (-\frac {7}{2} e f (c h+d g)+3 c f^2 g+3 d e^2 h\right )+b^2 c e f g}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\left (a^2 d f^2 h+a b f (5 c f h-12 d e h+5 d f g)-\left (b^2 (5 c f (7 f g-6 e h)-6 d e (5 f g-4 e h))\right )\right ) \left (-\frac {3 f \int \frac {1}{(a+b x)^2 \sqrt {e+f x}}dx}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{6 b f (b e-a f)^2}-\frac {x \left (a^2 d f^2 h-a b f^2 (c h+d g)+b^2 (c f (7 f g-6 e h)-6 d e (f g-e h))\right )+a^2 d e f h+2 a b \left (-\frac {7}{2} e f (c h+d g)+3 c f^2 g+3 d e^2 h\right )+b^2 c e f g}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\left (a^2 d f^2 h+a b f (5 c f h-12 d e h+5 d f g)-\left (b^2 (5 c f (7 f g-6 e h)-6 d e (5 f g-4 e h))\right )\right ) \left (-\frac {3 f \left (-\frac {f \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 (b e-a f)}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{6 b f (b e-a f)^2}-\frac {x \left (a^2 d f^2 h-a b f^2 (c h+d g)+b^2 (c f (7 f g-6 e h)-6 d e (f g-e h))\right )+a^2 d e f h+2 a b \left (-\frac {7}{2} e f (c h+d g)+3 c f^2 g+3 d e^2 h\right )+b^2 c e f g}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (a^2 d f^2 h+a b f (5 c f h-12 d e h+5 d f g)-\left (b^2 (5 c f (7 f g-6 e h)-6 d e (5 f g-4 e h))\right )\right ) \left (-\frac {3 f \left (-\frac {\int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b e-a f}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{6 b f (b e-a f)^2}-\frac {x \left (a^2 d f^2 h-a b f^2 (c h+d g)+b^2 (c f (7 f g-6 e h)-6 d e (f g-e h))\right )+a^2 d e f h+2 a b \left (-\frac {7}{2} e f (c h+d g)+3 c f^2 g+3 d e^2 h\right )+b^2 c e f g}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (-\frac {3 f \left (\frac {f \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b e-a f)^{3/2}}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right ) \left (a^2 d f^2 h+a b f (5 c f h-12 d e h+5 d f g)-\left (b^2 (5 c f (7 f g-6 e h)-6 d e (5 f g-4 e h))\right )\right )}{6 b f (b e-a f)^2}-\frac {x \left (a^2 d f^2 h-a b f^2 (c h+d g)+b^2 (c f (7 f g-6 e h)-6 d e (f g-e h))\right )+a^2 d e f h+2 a b \left (-\frac {7}{2} e f (c h+d g)+3 c f^2 g+3 d e^2 h\right )+b^2 c e f g}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}\) |
Input:
Int[((c + d*x)*(g + h*x))/((a + b*x)^4*(e + f*x)^(3/2)),x]
Output:
-1/3*(b^2*c*e*f*g + a^2*d*e*f*h + 2*a*b*(3*c*f^2*g + 3*d*e^2*h - (7*e*f*(d *g + c*h))/2) + (a^2*d*f^2*h - a*b*f^2*(d*g + c*h) + b^2*(c*f*(7*f*g - 6*e *h) - 6*d*e*(f*g - e*h)))*x)/(b*f*(b*e - a*f)^2*(a + b*x)^3*Sqrt[e + f*x]) + ((a^2*d*f^2*h + a*b*f*(5*d*f*g - 12*d*e*h + 5*c*f*h) - b^2*(5*c*f*(7*f* g - 6*e*h) - 6*d*e*(5*f*g - 4*e*h)))*(-1/2*Sqrt[e + f*x]/((b*e - a*f)*(a + b*x)^2) - (3*f*(-(Sqrt[e + f*x]/((b*e - a*f)*(a + b*x))) + (f*ArcTanh[(Sq rt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*e - a*f)^(3/2))))/(4*(b *e - a*f))))/(6*b*f*(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 + 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^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 ) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* h*(2 + 3*n + 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*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( n + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c , d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
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 = 0.57 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.39
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (-35 c g \,f^{2}+30 e \left (c h +d g \right ) f -24 d \,e^{2} h \right ) b^{2}+5 a \left (f \left (c h +d g \right )-\frac {12 d e h}{5}\right ) f b +a^{2} d \,f^{2} h \right ) \left (b x +a \right )^{3} \sqrt {f x +e}\, f \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )-\left (\left (35 c \,f^{3} g \,x^{3}+\frac {35 x^{2} \left (\frac {18 \left (-c h -d g \right ) x}{7}+c g \right ) e \,f^{2}}{3}-\frac {14 x \left (-\frac {36 d h \,x^{2}}{7}+\frac {15 \left (c h +d g \right ) x}{7}+c g \right ) e^{2} f}{3}+\frac {8 e^{3} \left (3 d h \,x^{2}+\frac {3 \left (c h +d g \right ) x}{2}+c g \right )}{3}\right ) b^{4}+\frac {4 a \left (5 \left (\frac {3 \left (-c h -d g \right ) x^{3}}{4}+14 c g \,x^{2}\right ) f^{3}+\frac {49 x e \left (\frac {18 d h \,x^{2}}{49}+\frac {5 \left (-c h -d g \right ) x}{2}+c g \right ) f^{2}}{2}-\frac {19 \left (-\frac {102 d h \,x^{2}}{19}+\frac {41 \left (c h +d g \right ) x}{19}+c g \right ) e^{2} f}{2}+e^{3} \left (6 d h x +c h +d g \right )\right ) b^{3}}{3}-\frac {28 a^{2} \left (\left (\frac {3 d h \,x^{3}}{28}+\frac {10 \left (c h +d g \right ) x^{2}}{7}-\frac {33 c g x}{4}\right ) f^{3}-\frac {87 e \left (\frac {95 d h \,x^{2}}{87}+\frac {212 \left (-c h -d g \right ) x}{87}+c g \right ) f^{2}}{28}+e^{2} \left (-\frac {125}{14} d h x +c h +d g \right ) f -\frac {2 d \,e^{3} h}{7}\right ) b^{2}}{3}-27 a^{3} \left (\frac {\left (\frac {8 d h \,x^{2}}{3}+11 \left (c h +d g \right ) x -16 c g \right ) f^{2}}{27}+e \left (-\frac {38}{81} d h x +c h +d g \right ) f -\frac {94 d \,e^{2} h}{81}\right ) f b +a^{4} d \,f^{2} h \left (f x +e \right )\right ) \sqrt {\left (a f -b e \right ) b}}{8 \sqrt {f x +e}\, \sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{3} \left (a f -b e \right )^{4} b}\) | \(519\) |
| derivativedivides | \(2 f \left (-\frac {-c e f h +c g \,f^{2}+d \,e^{2} h -d e f g}{\left (a f -b e \right )^{4} \sqrt {f x +e}}+\frac {\frac {\left (\frac {7}{8} b^{3} c e f h -\frac {19}{16} b^{3} c \,f^{2} g -\frac {1}{2} b^{3} d \,e^{2} h +\frac {7}{8} b^{3} d e f g +\frac {1}{16} a^{2} b d \,f^{2} h +\frac {5}{16} a \,b^{2} c \,f^{2} h -\frac {3}{4} a \,b^{2} d e f h +\frac {5}{16} a \,b^{2} d \,f^{2} g \right ) \left (f x +e \right )^{\frac {5}{2}}+\left (\frac {1}{6} a^{3} d \,f^{3} h +\frac {5}{6} a^{2} b c \,f^{3} h -\frac {13}{6} a^{2} b d e \,f^{2} h +\frac {5}{6} a^{2} b d \,f^{3} g +\frac {7}{6} a \,b^{2} c e \,f^{2} h -\frac {17}{6} a \,b^{2} c \,f^{3} g +a \,b^{2} d \,e^{2} f h +\frac {7}{6} a \,b^{2} d e \,f^{2} g -2 b^{3} c \,e^{2} f h +\frac {17}{6} b^{3} c e \,f^{2} g +b^{3} d \,e^{3} h -2 b^{3} d \,e^{2} f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {\left (a^{4} d \,f^{4} h -11 a^{3} b c \,f^{4} h +18 a^{3} b d e \,f^{3} h -11 a^{3} b d \,f^{4} g +4 a^{2} b^{2} c e \,f^{3} h +29 a^{2} b^{2} c \,f^{4} g -31 a^{2} b^{2} d \,e^{2} f^{2} h +4 a^{2} b^{2} d e \,f^{3} g +25 a \,b^{3} c \,e^{2} f^{2} h -58 a \,b^{3} c e \,f^{3} g +4 a \,b^{3} d \,e^{3} f h +25 a \,b^{3} d \,e^{2} f^{2} g -18 b^{4} c \,e^{3} f h +29 b^{4} c \,e^{2} f^{2} g +8 b^{4} e^{4} h d -18 b^{4} d \,e^{3} f g \right ) \sqrt {f x +e}}{16 b}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (a^{2} d \,f^{2} h +5 a b c \,f^{2} h -12 a b d e f h +5 a b d \,f^{2} g +30 c e f h \,b^{2}-35 b^{2} c \,f^{2} g -24 d \,e^{2} h \,b^{2}+30 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 b \sqrt {\left (a f -b e \right ) b}}}{\left (a f -b e \right )^{4}}\right )\) | \(634\) |
| default | \(2 f \left (-\frac {-c e f h +c g \,f^{2}+d \,e^{2} h -d e f g}{\left (a f -b e \right )^{4} \sqrt {f x +e}}+\frac {\frac {\left (\frac {7}{8} b^{3} c e f h -\frac {19}{16} b^{3} c \,f^{2} g -\frac {1}{2} b^{3} d \,e^{2} h +\frac {7}{8} b^{3} d e f g +\frac {1}{16} a^{2} b d \,f^{2} h +\frac {5}{16} a \,b^{2} c \,f^{2} h -\frac {3}{4} a \,b^{2} d e f h +\frac {5}{16} a \,b^{2} d \,f^{2} g \right ) \left (f x +e \right )^{\frac {5}{2}}+\left (\frac {1}{6} a^{3} d \,f^{3} h +\frac {5}{6} a^{2} b c \,f^{3} h -\frac {13}{6} a^{2} b d e \,f^{2} h +\frac {5}{6} a^{2} b d \,f^{3} g +\frac {7}{6} a \,b^{2} c e \,f^{2} h -\frac {17}{6} a \,b^{2} c \,f^{3} g +a \,b^{2} d \,e^{2} f h +\frac {7}{6} a \,b^{2} d e \,f^{2} g -2 b^{3} c \,e^{2} f h +\frac {17}{6} b^{3} c e \,f^{2} g +b^{3} d \,e^{3} h -2 b^{3} d \,e^{2} f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {\left (a^{4} d \,f^{4} h -11 a^{3} b c \,f^{4} h +18 a^{3} b d e \,f^{3} h -11 a^{3} b d \,f^{4} g +4 a^{2} b^{2} c e \,f^{3} h +29 a^{2} b^{2} c \,f^{4} g -31 a^{2} b^{2} d \,e^{2} f^{2} h +4 a^{2} b^{2} d e \,f^{3} g +25 a \,b^{3} c \,e^{2} f^{2} h -58 a \,b^{3} c e \,f^{3} g +4 a \,b^{3} d \,e^{3} f h +25 a \,b^{3} d \,e^{2} f^{2} g -18 b^{4} c \,e^{3} f h +29 b^{4} c \,e^{2} f^{2} g +8 b^{4} e^{4} h d -18 b^{4} d \,e^{3} f g \right ) \sqrt {f x +e}}{16 b}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (a^{2} d \,f^{2} h +5 a b c \,f^{2} h -12 a b d e f h +5 a b d \,f^{2} g +30 c e f h \,b^{2}-35 b^{2} c \,f^{2} g -24 d \,e^{2} h \,b^{2}+30 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 b \sqrt {\left (a f -b e \right ) b}}}{\left (a f -b e \right )^{4}}\right )\) | \(634\) |
Input:
int((d*x+c)*(h*x+g)/(b*x+a)^4/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/8*(((-35*c*g*f^2+30*e*(c*h+d*g)*f-24*d*e^2*h)*b^2+5*a*(f*(c*h+d*g)-12/5* d*e*h)*f*b+a^2*d*f^2*h)*(b*x+a)^3*(f*x+e)^(1/2)*f*arctan(b*(f*x+e)^(1/2)/( (a*f-b*e)*b)^(1/2))-((35*c*f^3*g*x^3+35/3*x^2*(18/7*(-c*h-d*g)*x+c*g)*e*f^ 2-14/3*x*(-36/7*d*h*x^2+15/7*(c*h+d*g)*x+c*g)*e^2*f+8/3*e^3*(3*d*h*x^2+3/2 *(c*h+d*g)*x+c*g))*b^4+4/3*a*(5*(3/4*(-c*h-d*g)*x^3+14*c*g*x^2)*f^3+49/2*x *e*(18/49*d*h*x^2+5/2*(-c*h-d*g)*x+c*g)*f^2-19/2*(-102/19*d*h*x^2+41/19*(c *h+d*g)*x+c*g)*e^2*f+e^3*(6*d*h*x+c*h+d*g))*b^3-28/3*a^2*((3/28*d*h*x^3+10 /7*(c*h+d*g)*x^2-33/4*c*g*x)*f^3-87/28*e*(95/87*d*h*x^2+212/87*(-c*h-d*g)* x+c*g)*f^2+e^2*(-125/14*d*h*x+c*h+d*g)*f-2/7*d*e^3*h)*b^2-27*a^3*(1/27*(8/ 3*d*h*x^2+11*(c*h+d*g)*x-16*c*g)*f^2+e*(-38/81*d*h*x+c*h+d*g)*f-94/81*d*e^ 2*h)*f*b+a^4*d*f^2*h*(f*x+e))*((a*f-b*e)*b)^(1/2))/(f*x+e)^(1/2)/((a*f-b*e )*b)^(1/2)/(b*x+a)^3/(a*f-b*e)^4/b
Leaf count of result is larger than twice the leaf count of optimal. 1891 vs. \(2 (347) = 694\).
Time = 0.49 (sec) , antiderivative size = 3796, normalized size of antiderivative = 10.15 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)^4/(f*x+e)^(3/2),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)**4/(f*x+e)**(3/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)^4/(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(a*f-b*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1033 vs. \(2 (347) = 694\).
Time = 0.15 (sec) , antiderivative size = 1033, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(h*x+g)/(b*x+a)^4/(f*x+e)^(3/2),x, algorithm="giac")
Output:
1/8*(30*b^2*d*e*f^2*g - 35*b^2*c*f^3*g + 5*a*b*d*f^3*g - 24*b^2*d*e^2*f*h + 30*b^2*c*e*f^2*h - 12*a*b*d*e*f^2*h + 5*a*b*c*f^3*h + a^2*d*f^3*h)*arcta n(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^5*e^4 - 4*a*b^4*e^3*f + 6*a^2* b^3*e^2*f^2 - 4*a^3*b^2*e*f^3 + a^4*b*f^4)*sqrt(-b^2*e + a*b*f)) + 2*(d*e* f^2*g - c*f^3*g - d*e^2*f*h + c*e*f^2*h)/((b^4*e^4 - 4*a*b^3*e^3*f + 6*a^2 *b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*sqrt(f*x + e)) + 1/24*(42*(f*x + e )^(5/2)*b^4*d*e*f^2*g - 96*(f*x + e)^(3/2)*b^4*d*e^2*f^2*g + 54*sqrt(f*x + e)*b^4*d*e^3*f^2*g - 57*(f*x + e)^(5/2)*b^4*c*f^3*g + 15*(f*x + e)^(5/2)* a*b^3*d*f^3*g + 136*(f*x + e)^(3/2)*b^4*c*e*f^3*g + 56*(f*x + e)^(3/2)*a*b ^3*d*e*f^3*g - 87*sqrt(f*x + e)*b^4*c*e^2*f^3*g - 75*sqrt(f*x + e)*a*b^3*d *e^2*f^3*g - 136*(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 + 174*sqrt(f*x + e)*a*b^3*c*e*f^4*g - 12*sqrt(f*x + e)*a^2*b^2*d *e*f^4*g - 87*sqrt(f*x + e)*a^2*b^2*c*f^5*g + 33*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 + 42*(f*x + e)^(5/2)*b^4*c*e*f^2*h - 36*(f *x + e)^(5/2)*a*b^3*d*e*f^2*h - 96*(f*x + e)^(3/2)*b^4*c*e^2*f^2*h + 48*(f *x + e)^(3/2)*a*b^3*d*e^2*f^2*h + 54*sqrt(f*x + e)*b^4*c*e^3*f^2*h - 12*sq rt(f*x + e)*a*b^3*d*e^3*f^2*h + 15*(f*x + e)^(5/2)*a*b^3*c*f^3*h + 3*(f*x + e)^(5/2)*a^2*b^2*d*f^3*h + 56*(f*x + e)^(3/2)*a*b^3*c*e*f^3*h - 104*(f*x + e)^(3/2)*a^2*b^2*d*e*f^3*h - 75*sqrt(f*x + e)*a*b^3*c*e^2*f^3*h + 93...
Time = 0.72 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.10 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx=\frac {f\,\mathrm {atan}\left (\frac {f\,\sqrt {e+f\,x}\,\left (a^4\,b\,f^4-4\,a^3\,b^2\,e\,f^3+6\,a^2\,b^3\,e^2\,f^2-4\,a\,b^4\,e^3\,f+b^5\,e^4\right )\,\left (a^2\,d\,f^2\,h-35\,b^2\,c\,f^2\,g-24\,b^2\,d\,e^2\,h+5\,a\,b\,c\,f^2\,h+5\,a\,b\,d\,f^2\,g+30\,b^2\,c\,e\,f\,h+30\,b^2\,d\,e\,f\,g-12\,a\,b\,d\,e\,f\,h\right )}{\sqrt {b}\,{\left (a\,f-b\,e\right )}^{9/2}\,\left (a^2\,d\,f^3\,h-35\,b^2\,c\,f^3\,g+30\,b^2\,c\,e\,f^2\,h+30\,b^2\,d\,e\,f^2\,g-24\,b^2\,d\,e^2\,f\,h+5\,a\,b\,c\,f^3\,h+5\,a\,b\,d\,f^3\,g-12\,a\,b\,d\,e\,f^2\,h\right )}\right )\,\left (a^2\,d\,f^2\,h-35\,b^2\,c\,f^2\,g-24\,b^2\,d\,e^2\,h+5\,a\,b\,c\,f^2\,h+5\,a\,b\,d\,f^2\,g+30\,b^2\,c\,e\,f\,h+30\,b^2\,d\,e\,f\,g-12\,a\,b\,d\,e\,f\,h\right )}{8\,b^{3/2}\,{\left (a\,f-b\,e\right )}^{9/2}}-\frac {\frac {2\,\left (c\,f^3\,g-c\,e\,f^2\,h-d\,e\,f^2\,g+d\,e^2\,f\,h\right )}{a\,f-b\,e}-\frac {{\left (e+f\,x\right )}^2\,\left (a^2\,d\,f^3\,h-35\,b^2\,c\,f^3\,g+30\,b^2\,c\,e\,f^2\,h+30\,b^2\,d\,e\,f^2\,g-24\,b^2\,d\,e^2\,f\,h+5\,a\,b\,c\,f^3\,h+5\,a\,b\,d\,f^3\,g-12\,a\,b\,d\,e\,f^2\,h\right )}{3\,{\left (a\,f-b\,e\right )}^3}-\frac {b\,{\left (e+f\,x\right )}^3\,\left (a^2\,d\,f^3\,h-35\,b^2\,c\,f^3\,g+30\,b^2\,c\,e\,f^2\,h+30\,b^2\,d\,e\,f^2\,g-24\,b^2\,d\,e^2\,f\,h+5\,a\,b\,c\,f^3\,h+5\,a\,b\,d\,f^3\,g-12\,a\,b\,d\,e\,f^2\,h\right )}{8\,{\left (a\,f-b\,e\right )}^4}+\frac {\left (e+f\,x\right )\,\left (77\,b^2\,c\,f^3\,g+a^2\,d\,f^3\,h-66\,b^2\,c\,e\,f^2\,h-66\,b^2\,d\,e\,f^2\,g+56\,b^2\,d\,e^2\,f\,h-11\,a\,b\,c\,f^3\,h-11\,a\,b\,d\,f^3\,g+20\,a\,b\,d\,e\,f^2\,h\right )}{8\,b\,{\left (a\,f-b\,e\right )}^2}}{\sqrt {e+f\,x}\,\left (a^3\,f^3-3\,a^2\,b\,e\,f^2+3\,a\,b^2\,e^2\,f-b^3\,e^3\right )+b^3\,{\left (e+f\,x\right )}^{7/2}-{\left (e+f\,x\right )}^{5/2}\,\left (3\,b^3\,e-3\,a\,b^2\,f\right )+{\left (e+f\,x\right )}^{3/2}\,\left (3\,a^2\,b\,f^2-6\,a\,b^2\,e\,f+3\,b^3\,e^2\right )} \] Input:
int(((g + h*x)*(c + d*x))/((e + f*x)^(3/2)*(a + b*x)^4),x)
Output:
(f*atan((f*(e + f*x)^(1/2)*(b^5*e^4 + a^4*b*f^4 - 4*a^3*b^2*e*f^3 + 6*a^2* b^3*e^2*f^2 - 4*a*b^4*e^3*f)*(a^2*d*f^2*h - 35*b^2*c*f^2*g - 24*b^2*d*e^2* h + 5*a*b*c*f^2*h + 5*a*b*d*f^2*g + 30*b^2*c*e*f*h + 30*b^2*d*e*f*g - 12*a *b*d*e*f*h))/(b^(1/2)*(a*f - b*e)^(9/2)*(a^2*d*f^3*h - 35*b^2*c*f^3*g + 30 *b^2*c*e*f^2*h + 30*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 - 12*a*b*d*e*f^2*h)))*(a^2*d*f^2*h - 35*b^2*c*f^2*g - 24*b^2*d* e^2*h + 5*a*b*c*f^2*h + 5*a*b*d*f^2*g + 30*b^2*c*e*f*h + 30*b^2*d*e*f*g - 12*a*b*d*e*f*h))/(8*b^(3/2)*(a*f - b*e)^(9/2)) - ((2*(c*f^3*g - c*e*f^2*h - d*e*f^2*g + d*e^2*f*h))/(a*f - b*e) - ((e + f*x)^2*(a^2*d*f^3*h - 35*b^2 *c*f^3*g + 30*b^2*c*e*f^2*h + 30*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 - 12*a*b*d*e*f^2*h))/(3*(a*f - b*e)^3) - (b*(e + f *x)^3*(a^2*d*f^3*h - 35*b^2*c*f^3*g + 30*b^2*c*e*f^2*h + 30*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 - 12*a*b*d*e*f^2*h))/(8 *(a*f - b*e)^4) + ((e + f*x)*(77*b^2*c*f^3*g + a^2*d*f^3*h - 66*b^2*c*e*f^ 2*h - 66*b^2*d*e*f^2*g + 56*b^2*d*e^2*f*h - 11*a*b*c*f^3*h - 11*a*b*d*f^3* g + 20*a*b*d*e*f^2*h))/(8*b*(a*f - b*e)^2))/((e + f*x)^(1/2)*(a^3*f^3 - b^ 3*e^3 + 3*a*b^2*e^2*f - 3*a^2*b*e*f^2) + b^3*(e + f*x)^(7/2) - (e + f*x)^( 5/2)*(3*b^3*e - 3*a*b^2*f) + (e + f*x)^(3/2)*(3*b^3*e^2 + 3*a^2*b*f^2 - 6* a*b^2*e*f))
Time = 0.28 (sec) , antiderivative size = 3040, normalized size of antiderivative = 8.13 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)*(h*x+g)/(b*x+a)^4/(f*x+e)^(3/2),x)
Output:
(3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s qrt(a*f - b*e)))*a**5*d*f**3*h + 15*sqrt(b)*sqrt(e + f*x)*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 - 36*sqr t(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a* f - b*e)))*a**4*b*d*e*f**2*h + 15*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*at an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d*f**3*g + 9*sqrt(b )*sqrt(e + f*x)*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 + 90*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan( (sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*e*f**2*h - 105*sq rt(b)*sqrt(e + f*x)*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(e + f*x)*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 - 7 2*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sq rt(a*f - b*e)))*a**3*b**2*d*e**2*f*h + 90*sqrt(b)*sqrt(e + f*x)*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 - 108*sqrt(b)*sqrt(e + f*x)*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(e + f*x)*s qrt(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 + 9*sqrt(b)*sqrt(e + f*x)*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 + 270*sqrt(b)*sqr...