Integrand size = 27, antiderivative size = 196 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx=\frac {2 (b c-a d) (b e-a f) (b g-a h) \sqrt {e+f x}}{b^4}+\frac {2 (b c-a d) (b g-a h) (e+f x)^{3/2}}{3 b^3}-\frac {2 (d (b e+a f) h-b f (d g+c h)) (e+f x)^{5/2}}{5 b^2 f^2}+\frac {2 d h (e+f x)^{7/2}}{7 b f^2}-\frac {2 (b c-a d) (b e-a f)^{3/2} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{9/2}} \] Output:
2*(-a*d+b*c)*(-a*f+b*e)*(-a*h+b*g)*(f*x+e)^(1/2)/b^4+2/3*(-a*d+b*c)*(-a*h+ b*g)*(f*x+e)^(3/2)/b^3-2/5*(d*(a*f+b*e)*h-b*f*(c*h+d*g))*(f*x+e)^(5/2)/b^2 /f^2+2/7*d*h*(f*x+e)^(7/2)/b/f^2-2*(-a*d+b*c)*(-a*f+b*e)^(3/2)*(-a*h+b*g)* arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(9/2)
Time = 0.34 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx=\frac {2 \sqrt {e+f x} \left (-105 a^3 d f^3 h+35 a^2 b f^2 (3 c f h+d (3 f g+4 e h+f h x))-7 a b^2 f \left (5 c f (3 f g+4 e h+f h x)+d \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )\right )+b^3 \left (-3 d (e+f x)^2 (-7 f g+2 e h-5 f h x)+7 c f \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )\right )\right )}{105 b^4 f^2}+\frac {2 (b c-a d) (-b e+a f)^{3/2} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{9/2}} \] Input:
Integrate[((c + d*x)*(e + f*x)^(3/2)*(g + h*x))/(a + b*x),x]
Output:
(2*Sqrt[e + f*x]*(-105*a^3*d*f^3*h + 35*a^2*b*f^2*(3*c*f*h + d*(3*f*g + 4* e*h + f*h*x)) - 7*a*b^2*f*(5*c*f*(3*f*g + 4*e*h + f*h*x) + d*(3*e^2*h + f^ 2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h*x))) + b^3*(-3*d*(e + f*x)^2*(-7*f*g + 2*e*h - 5*f*h*x) + 7*c*f*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h *x)))))/(105*b^4*f^2) + (2*(b*c - a*d)*(-(b*e) + a*f)^(3/2)*(b*g - a*h)*Ar cTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/b^(9/2)
Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {164, 60, 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.
| \(\displaystyle \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {(b c-a d) (b g-a h) \int \frac {(e+f x)^{3/2}}{a+b x}dx}{b^2}-\frac {2 (e+f x)^{5/2} (7 a d f h-7 b f (c h+d g)+2 b d e h-5 b d f h x)}{35 b^2 f^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b c-a d) (b g-a h) \left (\frac {(b e-a f) \int \frac {\sqrt {e+f x}}{a+b x}dx}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{b^2}-\frac {2 (e+f x)^{5/2} (7 a d f h-7 b f (c h+d g)+2 b d e h-5 b d f h x)}{35 b^2 f^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b c-a d) (b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{b^2}-\frac {2 (e+f x)^{5/2} (7 a d f h-7 b f (c h+d g)+2 b d e h-5 b d f h x)}{35 b^2 f^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(b c-a d) (b g-a h) \left (\frac {(b e-a f) \left (\frac {2 (b e-a f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b f}+\frac {2 \sqrt {e+f x}}{b}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{b^2}-\frac {2 (e+f x)^{5/2} (7 a d f h-7 b f (c h+d g)+2 b d e h-5 b d f h x)}{35 b^2 f^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(b c-a d) (b g-a h) \left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x}}{b}-\frac {2 \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (e+f x)^{3/2}}{3 b}\right )}{b^2}-\frac {2 (e+f x)^{5/2} (7 a d f h-7 b f (c h+d g)+2 b d e h-5 b d f h x)}{35 b^2 f^2}\) |
Input:
Int[((c + d*x)*(e + f*x)^(3/2)*(g + h*x))/(a + b*x),x]
Output:
(-2*(e + f*x)^(5/2)*(2*b*d*e*h + 7*a*d*f*h - 7*b*f*(d*g + c*h) - 5*b*d*f*h *x))/(35*b^2*f^2) + ((b*c - a*d)*(b*g - a*h)*((2*(e + f*x)^(3/2))/(3*b) + ((b*e - a*f)*((2*Sqrt[e + f*x])/b - (2*Sqrt[b*e - a*f]*ArcTanh[(Sqrt[b]*Sq rt[e + f*x])/Sqrt[b*e - a*f]])/b^(3/2)))/b))/b^2
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[(-(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_)^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 = 0.44 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.36
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\sqrt {\left (a f -b e \right ) b}\, \left (\left (-\frac {\left (\frac {5}{7} d f x +c f -\frac {2}{7} d e \right ) \left (f x +e \right )^{2} b^{3}}{5}+\frac {4 a \left (\frac {3 d \,f^{2} x^{2}}{20}+\left (\frac {1}{4} c \,f^{2}+\frac {3}{10} d e f \right ) x +e \left (c f +\frac {3 d e}{20}\right )\right ) f \,b^{2}}{3}-a^{2} f^{2} \left (\frac {1}{3} d f x +c f +\frac {4}{3} d e \right ) b +a^{3} d \,f^{3}\right ) h -\left (\left (\frac {d \,f^{2} x^{2}}{5}+\left (\frac {2}{5} d e f +\frac {1}{3} c \,f^{2}\right ) x +\frac {d \,e^{2}}{5}+\frac {4 c e f}{3}\right ) b^{2}-a f \left (\frac {1}{3} d f x +c f +\frac {4}{3} d e \right ) b +a^{2} d \,f^{2}\right ) b g f \right ) \sqrt {f x +e}-f^{2} \left (a f -b e \right )^{2} \left (a h -b g \right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )\right )}{\sqrt {\left (a f -b e \right ) b}\, f^{2} b^{4}}\) | \(266\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {h d \left (f x +e \right )^{\frac {7}{2}} b^{3}}{7}+\frac {a \,b^{2} d f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{3} d e h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} d f g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a^{2} b d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a \,b^{2} c \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a \,b^{2} d \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{3} c \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d \,f^{3} h \sqrt {f x +e}-a^{2} b c \,f^{3} h \sqrt {f x +e}-a^{2} b d e \,f^{2} h \sqrt {f x +e}-a^{2} b d \,f^{3} g \sqrt {f x +e}+a \,b^{2} c e \,f^{2} h \sqrt {f x +e}+a \,b^{2} c \,f^{3} g \sqrt {f x +e}+a \,b^{2} d e \,f^{2} g \sqrt {f x +e}-b^{3} c e \,f^{2} g \sqrt {f x +e}\right )}{b^{4}}+\frac {2 f^{2} \left (a^{4} d \,f^{2} h -a^{3} b c \,f^{2} h -2 a^{3} b d e f h -a^{3} b d \,f^{2} g +2 a^{2} b^{2} c e f h +a^{2} b^{2} c \,f^{2} g +a^{2} b^{2} d \,e^{2} h +2 a^{2} b^{2} d e f g -a \,b^{3} c \,e^{2} h -2 a \,b^{3} c e f g -a \,b^{3} d \,e^{2} g +c g \,e^{2} b^{4}\right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{b^{4} \sqrt {\left (a f -b e \right ) b}}}{f^{2}}\) | \(473\) |
| default | \(\frac {-\frac {2 \left (-\frac {h d \left (f x +e \right )^{\frac {7}{2}} b^{3}}{7}+\frac {a \,b^{2} d f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c f h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{3} d e h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} d f g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a^{2} b d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a \,b^{2} c \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a \,b^{2} d \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{3} c \,f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d \,f^{3} h \sqrt {f x +e}-a^{2} b c \,f^{3} h \sqrt {f x +e}-a^{2} b d e \,f^{2} h \sqrt {f x +e}-a^{2} b d \,f^{3} g \sqrt {f x +e}+a \,b^{2} c e \,f^{2} h \sqrt {f x +e}+a \,b^{2} c \,f^{3} g \sqrt {f x +e}+a \,b^{2} d e \,f^{2} g \sqrt {f x +e}-b^{3} c e \,f^{2} g \sqrt {f x +e}\right )}{b^{4}}+\frac {2 f^{2} \left (a^{4} d \,f^{2} h -a^{3} b c \,f^{2} h -2 a^{3} b d e f h -a^{3} b d \,f^{2} g +2 a^{2} b^{2} c e f h +a^{2} b^{2} c \,f^{2} g +a^{2} b^{2} d \,e^{2} h +2 a^{2} b^{2} d e f g -a \,b^{3} c \,e^{2} h -2 a \,b^{3} c e f g -a \,b^{3} d \,e^{2} g +c g \,e^{2} b^{4}\right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{b^{4} \sqrt {\left (a f -b e \right ) b}}}{f^{2}}\) | \(473\) |
| risch | \(-\frac {2 \left (-15 d \,f^{3} h \,b^{3} x^{3}+21 a \,b^{2} d \,f^{3} h \,x^{2}-21 b^{3} c \,f^{3} h \,x^{2}-24 b^{3} d e \,f^{2} h \,x^{2}-21 b^{3} d \,f^{3} g \,x^{2}-35 a^{2} b d \,f^{3} h x +35 a \,b^{2} c \,f^{3} h x +42 a \,b^{2} d e \,f^{2} h x +35 a \,b^{2} d \,f^{3} g x -42 b^{3} c e \,f^{2} h x -35 b^{3} c \,f^{3} g x -3 b^{3} d \,e^{2} f h x -42 b^{3} d e \,f^{2} g x +105 a^{3} d \,f^{3} h -105 a^{2} b c \,f^{3} h -140 a^{2} b d e \,f^{2} h -105 a^{2} b d \,f^{3} g +140 a \,b^{2} c e \,f^{2} h +105 a \,b^{2} c \,f^{3} g +21 a \,b^{2} d \,e^{2} f h +140 a \,b^{2} d e \,f^{2} g -21 b^{3} c \,e^{2} f h -140 b^{3} c e \,f^{2} g +6 b^{3} d \,e^{3} h -21 b^{3} d \,e^{2} f g \right ) \sqrt {f x +e}}{105 f^{2} b^{4}}+\frac {2 \left (a^{4} d \,f^{2} h -a^{3} b c \,f^{2} h -2 a^{3} b d e f h -a^{3} b d \,f^{2} g +2 a^{2} b^{2} c e f h +a^{2} b^{2} c \,f^{2} g +a^{2} b^{2} d \,e^{2} h +2 a^{2} b^{2} d e f g -a \,b^{3} c \,e^{2} h -2 a \,b^{3} c e f g -a \,b^{3} d \,e^{2} g +c g \,e^{2} b^{4}\right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{b^{4} \sqrt {\left (a f -b e \right ) b}}\) | \(485\) |
Input:
int((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
-2*(((a*f-b*e)*b)^(1/2)*((-1/5*(5/7*d*f*x+c*f-2/7*d*e)*(f*x+e)^2*b^3+4/3*a *(3/20*d*f^2*x^2+(1/4*c*f^2+3/10*d*e*f)*x+e*(c*f+3/20*d*e))*f*b^2-a^2*f^2* (1/3*d*f*x+c*f+4/3*d*e)*b+a^3*d*f^3)*h-((1/5*d*f^2*x^2+(2/5*d*e*f+1/3*c*f^ 2)*x+1/5*d*e^2+4/3*c*e*f)*b^2-a*f*(1/3*d*f*x+c*f+4/3*d*e)*b+a^2*d*f^2)*b*g *f)*(f*x+e)^(1/2)-f^2*(a*f-b*e)^2*(a*h-b*g)*(a*d-b*c)*arctan(b*(f*x+e)^(1/ 2)/((a*f-b*e)*b)^(1/2)))/((a*f-b*e)*b)^(1/2)/f^2/b^4
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (172) = 344\).
Time = 0.10 (sec) , antiderivative size = 854, normalized size of antiderivative = 4.36 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a),x, algorithm="fricas")
Output:
[-1/105*(105*(((b^3*c - a*b^2*d)*e*f^2 - (a*b^2*c - a^2*b*d)*f^3)*g - ((a* b^2*c - a^2*b*d)*e*f^2 - (a^2*b*c - a^3*d)*f^3)*h)*sqrt((b*e - a*f)/b)*log ((b*f*x + 2*b*e - a*f + 2*sqrt(f*x + e)*b*sqrt((b*e - a*f)/b))/(b*x + a)) - 2*(15*b^3*d*f^3*h*x^3 + 3*(7*b^3*d*f^3*g + (8*b^3*d*e*f^2 + 7*(b^3*c - a *b^2*d)*f^3)*h)*x^2 + 7*(3*b^3*d*e^2*f + 20*(b^3*c - a*b^2*d)*e*f^2 - 15*( a*b^2*c - a^2*b*d)*f^3)*g - (6*b^3*d*e^3 - 21*(b^3*c - a*b^2*d)*e^2*f + 14 0*(a*b^2*c - a^2*b*d)*e*f^2 - 105*(a^2*b*c - a^3*d)*f^3)*h + (7*(6*b^3*d*e *f^2 + 5*(b^3*c - a*b^2*d)*f^3)*g + (3*b^3*d*e^2*f + 42*(b^3*c - a*b^2*d)* e*f^2 - 35*(a*b^2*c - a^2*b*d)*f^3)*h)*x)*sqrt(f*x + e))/(b^4*f^2), -2/105 *(105*(((b^3*c - a*b^2*d)*e*f^2 - (a*b^2*c - a^2*b*d)*f^3)*g - ((a*b^2*c - a^2*b*d)*e*f^2 - (a^2*b*c - a^3*d)*f^3)*h)*sqrt(-(b*e - a*f)/b)*arctan(-s qrt(f*x + e)*b*sqrt(-(b*e - a*f)/b)/(b*e - a*f)) - (15*b^3*d*f^3*h*x^3 + 3 *(7*b^3*d*f^3*g + (8*b^3*d*e*f^2 + 7*(b^3*c - a*b^2*d)*f^3)*h)*x^2 + 7*(3* b^3*d*e^2*f + 20*(b^3*c - a*b^2*d)*e*f^2 - 15*(a*b^2*c - a^2*b*d)*f^3)*g - (6*b^3*d*e^3 - 21*(b^3*c - a*b^2*d)*e^2*f + 140*(a*b^2*c - a^2*b*d)*e*f^2 - 105*(a^2*b*c - a^3*d)*f^3)*h + (7*(6*b^3*d*e*f^2 + 5*(b^3*c - a*b^2*d)* f^3)*g + (3*b^3*d*e^2*f + 42*(b^3*c - a*b^2*d)*e*f^2 - 35*(a*b^2*c - a^2*b *d)*f^3)*h)*x)*sqrt(f*x + e))/(b^4*f^2)]
Time = 14.75 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.76 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {d h \left (e + f x\right )^{\frac {7}{2}}}{7 b f} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (- a d f h + b c f h - b d e h + b d f g\right )}{5 b^{2} f} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (a^{2} d f h - a b c f h - a b d f g + b^{2} c f g\right )}{3 b^{3}} + \frac {\sqrt {e + f x} \left (- a^{3} d f^{2} h + a^{2} b c f^{2} h + a^{2} b d e f h + a^{2} b d f^{2} g - a b^{2} c e f h - a b^{2} c f^{2} g - a b^{2} d e f g + b^{3} c e f g\right )}{b^{4}} + \frac {f \left (a d - b c\right ) \left (a f - b e\right )^{2} \left (a h - b g\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {a f - b e}{b}}} \right )}}{b^{5} \sqrt {\frac {a f - b e}{b}}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (\frac {d h x^{2}}{2 b} + \frac {x \left (- a d h + b c h + b d g\right )}{b^{2}} + \frac {\left (a d - b c\right ) \left (a h - b g\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{2}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)*(f*x+e)**(3/2)*(h*x+g)/(b*x+a),x)
Output:
Piecewise((2*(d*h*(e + f*x)**(7/2)/(7*b*f) + (e + f*x)**(5/2)*(-a*d*f*h + b*c*f*h - b*d*e*h + b*d*f*g)/(5*b**2*f) + (e + f*x)**(3/2)*(a**2*d*f*h - a *b*c*f*h - a*b*d*f*g + b**2*c*f*g)/(3*b**3) + sqrt(e + f*x)*(-a**3*d*f**2* h + a**2*b*c*f**2*h + a**2*b*d*e*f*h + a**2*b*d*f**2*g - a*b**2*c*e*f*h - a*b**2*c*f**2*g - a*b**2*d*e*f*g + b**3*c*e*f*g)/b**4 + f*(a*d - b*c)*(a*f - b*e)**2*(a*h - b*g)*atan(sqrt(e + f*x)/sqrt((a*f - b*e)/b))/(b**5*sqrt( (a*f - b*e)/b)))/f, Ne(f, 0)), (e**(3/2)*(d*h*x**2/(2*b) + x*(-a*d*h + b*c *h + b*d*g)/b**2 + (a*d - b*c)*(a*h - b*g)*Piecewise((x/a, Eq(b, 0)), (log (a + b*x)/b, True))/b**2), True))
Exception generated. \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a),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. 497 vs. \(2 (172) = 344\).
Time = 0.14 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.54 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx=\frac {2 \, {\left (b^{4} c e^{2} g - a b^{3} d e^{2} g - 2 \, a b^{3} c e f g + 2 \, a^{2} b^{2} d e f g + a^{2} b^{2} c f^{2} g - a^{3} b d f^{2} g - a b^{3} c e^{2} h + a^{2} b^{2} d e^{2} h + 2 \, a^{2} b^{2} c e f h - 2 \, a^{3} b d e f h - a^{3} b c f^{2} h + a^{4} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{\sqrt {-b^{2} e + a b f} b^{4}} + \frac {2 \, {\left (21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{6} d f^{13} g + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{6} c f^{14} g - 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{5} d f^{14} g + 105 \, \sqrt {f x + e} b^{6} c e f^{14} g - 105 \, \sqrt {f x + e} a b^{5} d e f^{14} g - 105 \, \sqrt {f x + e} a b^{5} c f^{15} g + 105 \, \sqrt {f x + e} a^{2} b^{4} d f^{15} g + 15 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{6} d f^{12} h - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{6} d e f^{12} h + 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{6} c f^{13} h - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{5} d f^{13} h - 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{5} c f^{14} h + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b^{4} d f^{14} h - 105 \, \sqrt {f x + e} a b^{5} c e f^{14} h + 105 \, \sqrt {f x + e} a^{2} b^{4} d e f^{14} h + 105 \, \sqrt {f x + e} a^{2} b^{4} c f^{15} h - 105 \, \sqrt {f x + e} a^{3} b^{3} d f^{15} h\right )}}{105 \, b^{7} f^{14}} \] Input:
integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a),x, algorithm="giac")
Output:
2*(b^4*c*e^2*g - a*b^3*d*e^2*g - 2*a*b^3*c*e*f*g + 2*a^2*b^2*d*e*f*g + a^2 *b^2*c*f^2*g - a^3*b*d*f^2*g - a*b^3*c*e^2*h + a^2*b^2*d*e^2*h + 2*a^2*b^2 *c*e*f*h - 2*a^3*b*d*e*f*h - a^3*b*c*f^2*h + a^4*d*f^2*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/(sqrt(-b^2*e + a*b*f)*b^4) + 2/105*(21*(f*x + e)^(5/2)*b^6*d*f^13*g + 35*(f*x + e)^(3/2)*b^6*c*f^14*g - 35*(f*x + e)^(3 /2)*a*b^5*d*f^14*g + 105*sqrt(f*x + e)*b^6*c*e*f^14*g - 105*sqrt(f*x + e)* a*b^5*d*e*f^14*g - 105*sqrt(f*x + e)*a*b^5*c*f^15*g + 105*sqrt(f*x + e)*a^ 2*b^4*d*f^15*g + 15*(f*x + e)^(7/2)*b^6*d*f^12*h - 21*(f*x + e)^(5/2)*b^6* d*e*f^12*h + 21*(f*x + e)^(5/2)*b^6*c*f^13*h - 21*(f*x + e)^(5/2)*a*b^5*d* f^13*h - 35*(f*x + e)^(3/2)*a*b^5*c*f^14*h + 35*(f*x + e)^(3/2)*a^2*b^4*d* f^14*h - 105*sqrt(f*x + e)*a*b^5*c*e*f^14*h + 105*sqrt(f*x + e)*a^2*b^4*d* e*f^14*h + 105*sqrt(f*x + e)*a^2*b^4*c*f^15*h - 105*sqrt(f*x + e)*a^3*b^3* d*f^15*h)/(b^7*f^14)
Time = 0.09 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.54 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx={\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,c\,f\,h-4\,d\,e\,h+2\,d\,f\,g}{5\,b\,f^2}-\frac {2\,d\,h\,\left (a\,f^3-b\,e\,f^2\right )}{5\,b^2\,f^4}\right )-{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,\left (c\,f-d\,e\right )\,\left (e\,h-f\,g\right )}{3\,b\,f^2}+\frac {\left (a\,f^3-b\,e\,f^2\right )\,\left (\frac {2\,c\,f\,h-4\,d\,e\,h+2\,d\,f\,g}{b\,f^2}-\frac {2\,d\,h\,\left (a\,f^3-b\,e\,f^2\right )}{b^2\,f^4}\right )}{3\,b\,f^2}\right )+\frac {2\,d\,h\,{\left (e+f\,x\right )}^{7/2}}{7\,b\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e+f\,x}\,\left (a\,d-b\,c\right )\,{\left (a\,f-b\,e\right )}^{3/2}\,\left (a\,h-b\,g\right )}{b^4\,c\,e^2\,g+a^4\,d\,f^2\,h-a\,b^3\,c\,e^2\,h-a\,b^3\,d\,e^2\,g-a^3\,b\,c\,f^2\,h-a^3\,b\,d\,f^2\,g+a^2\,b^2\,c\,f^2\,g+a^2\,b^2\,d\,e^2\,h-2\,a\,b^3\,c\,e\,f\,g-2\,a^3\,b\,d\,e\,f\,h+2\,a^2\,b^2\,c\,e\,f\,h+2\,a^2\,b^2\,d\,e\,f\,g}\right )\,\left (a\,d-b\,c\right )\,{\left (a\,f-b\,e\right )}^{3/2}\,\left (a\,h-b\,g\right )}{b^{9/2}}+\frac {\sqrt {e+f\,x}\,\left (\frac {2\,\left (c\,f-d\,e\right )\,\left (e\,h-f\,g\right )}{b\,f^2}+\frac {\left (a\,f^3-b\,e\,f^2\right )\,\left (\frac {2\,c\,f\,h-4\,d\,e\,h+2\,d\,f\,g}{b\,f^2}-\frac {2\,d\,h\,\left (a\,f^3-b\,e\,f^2\right )}{b^2\,f^4}\right )}{b\,f^2}\right )\,\left (a\,f^3-b\,e\,f^2\right )}{b\,f^2} \] Input:
int(((e + f*x)^(3/2)*(g + h*x)*(c + d*x))/(a + b*x),x)
Output:
(e + f*x)^(5/2)*((2*c*f*h - 4*d*e*h + 2*d*f*g)/(5*b*f^2) - (2*d*h*(a*f^3 - b*e*f^2))/(5*b^2*f^4)) - (e + f*x)^(3/2)*((2*(c*f - d*e)*(e*h - f*g))/(3* b*f^2) + ((a*f^3 - b*e*f^2)*((2*c*f*h - 4*d*e*h + 2*d*f*g)/(b*f^2) - (2*d* h*(a*f^3 - b*e*f^2))/(b^2*f^4)))/(3*b*f^2)) + (2*d*h*(e + f*x)^(7/2))/(7*b *f^2) + (2*atan((b^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)*(a*f - b*e)^(3/2)*(a* h - b*g))/(b^4*c*e^2*g + a^4*d*f^2*h - a*b^3*c*e^2*h - a*b^3*d*e^2*g - a^3 *b*c*f^2*h - a^3*b*d*f^2*g + a^2*b^2*c*f^2*g + a^2*b^2*d*e^2*h - 2*a*b^3*c *e*f*g - 2*a^3*b*d*e*f*h + 2*a^2*b^2*c*e*f*h + 2*a^2*b^2*d*e*f*g))*(a*d - b*c)*(a*f - b*e)^(3/2)*(a*h - b*g))/b^(9/2) + ((e + f*x)^(1/2)*((2*(c*f - d*e)*(e*h - f*g))/(b*f^2) + ((a*f^3 - b*e*f^2)*((2*c*f*h - 4*d*e*h + 2*d*f *g)/(b*f^2) - (2*d*h*(a*f^3 - b*e*f^2))/(b^2*f^4)))/(b*f^2))*(a*f^3 - b*e* f^2))/(b*f^2)
Time = 0.17 (sec) , antiderivative size = 835, normalized size of antiderivative = 4.26 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{a+b x} \, dx =\text {Too large to display} \] Input:
int((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a),x)
Output:
(2*(105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*d*f**3*h - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) /(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c*f**3*h - 105*sqrt(b)*sqrt(a*f - b*e)* atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*d*e*f**2*h - 105* sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* a**2*b*d*f**3*g + 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt (b)*sqrt(a*f - b*e)))*a*b**2*c*e*f**2*h + 105*sqrt(b)*sqrt(a*f - b*e)*atan ((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c*f**3*g + 105*sqrt(b )*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2 *d*e*f**2*g - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* sqrt(a*f - b*e)))*b**3*c*e*f**2*g - 105*sqrt(e + f*x)*a**3*b*d*f**3*h + 10 5*sqrt(e + f*x)*a**2*b**2*c*f**3*h + 140*sqrt(e + f*x)*a**2*b**2*d*e*f**2* h + 105*sqrt(e + f*x)*a**2*b**2*d*f**3*g + 35*sqrt(e + f*x)*a**2*b**2*d*f* *3*h*x - 140*sqrt(e + f*x)*a*b**3*c*e*f**2*h - 105*sqrt(e + f*x)*a*b**3*c* f**3*g - 35*sqrt(e + f*x)*a*b**3*c*f**3*h*x - 21*sqrt(e + f*x)*a*b**3*d*e* *2*f*h - 140*sqrt(e + f*x)*a*b**3*d*e*f**2*g - 42*sqrt(e + f*x)*a*b**3*d*e *f**2*h*x - 35*sqrt(e + f*x)*a*b**3*d*f**3*g*x - 21*sqrt(e + f*x)*a*b**3*d *f**3*h*x**2 + 21*sqrt(e + f*x)*b**4*c*e**2*f*h + 140*sqrt(e + f*x)*b**4*c *e*f**2*g + 42*sqrt(e + f*x)*b**4*c*e*f**2*h*x + 35*sqrt(e + f*x)*b**4*c*f **3*g*x + 21*sqrt(e + f*x)*b**4*c*f**3*h*x**2 - 6*sqrt(e + f*x)*b**4*d*...