Integrand size = 27, antiderivative size = 196 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=-\frac {2 (b c-a d) (d e-c f) (d g-c h) \sqrt {e+f x}}{d^4}-\frac {2 (b c-a d) (d g-c h) (e+f x)^{3/2}}{3 d^3}-\frac {2 (b (d e+c f) h-d f (b g+a h)) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b h (e+f x)^{7/2}}{7 d f^2}+\frac {2 (b c-a d) (d e-c f)^{3/2} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}} \] Output:
-2*(-a*d+b*c)*(-c*f+d*e)*(-c*h+d*g)*(f*x+e)^(1/2)/d^4-2/3*(-a*d+b*c)*(-c*h +d*g)*(f*x+e)^(3/2)/d^3-2/5*(b*(c*f+d*e)*h-d*f*(a*h+b*g))*(f*x+e)^(5/2)/d^ 2/f^2+2/7*b*h*(f*x+e)^(7/2)/d/f^2+2*(-a*d+b*c)*(-c*f+d*e)^(3/2)*(-c*h+d*g) *arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(9/2)
Time = 0.33 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (7 a d f \left (15 c^2 f^2 h-5 c d f (3 f g+4 e h+f h x)+d^2 \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )\right )+b \left (-105 c^3 f^3 h-3 d^3 (e+f x)^2 (-7 f g+2 e h-5 f h x)+35 c^2 d f^2 (3 f g+4 e h+f h x)-7 c d^2 f \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )\right )\right )}{105 d^4 f^2}+\frac {2 (-b c+a d) (-d e+c f)^{3/2} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{9/2}} \] Input:
Integrate[((a + b*x)*(e + f*x)^(3/2)*(g + h*x))/(c + d*x),x]
Output:
(2*Sqrt[e + f*x]*(7*a*d*f*(15*c^2*f^2*h - 5*c*d*f*(3*f*g + 4*e*h + f*h*x) + d^2*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h*x))) + b*(-105*c^3* f^3*h - 3*d^3*(e + f*x)^2*(-7*f*g + 2*e*h - 5*f*h*x) + 35*c^2*d*f^2*(3*f*g + 4*e*h + f*h*x) - 7*c*d^2*f*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h*x)))))/(105*d^4*f^2) + (2*(-(b*c) + a*d)*(-(d*e) + c*f)^(3/2)*(d*g - c*h)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(9/2)
Time = 0.29 (sec) , antiderivative size = 163, 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.
| \(\displaystyle \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {(b c-a d) (d g-c h) \int \frac {(e+f x)^{3/2}}{c+d x}dx}{d^2}-\frac {2 (e+f x)^{5/2} (-7 d f (a h+b g)+7 b c f h+2 b d e h-5 b d f h x)}{35 d^2 f^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {(b c-a d) (d g-c h) \left (\frac {(d e-c f) \int \frac {\sqrt {e+f x}}{c+d x}dx}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}-\frac {2 (e+f x)^{5/2} (-7 d f (a h+b g)+7 b c f h+2 b d e h-5 b d f h x)}{35 d^2 f^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {(b c-a d) (d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}-\frac {2 (e+f x)^{5/2} (-7 d f (a h+b g)+7 b c f h+2 b d e h-5 b d f h x)}{35 d^2 f^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(b c-a d) (d g-c h) \left (\frac {(d e-c f) \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}-\frac {2 (e+f x)^{5/2} (-7 d f (a h+b g)+7 b c f h+2 b d e h-5 b d f h x)}{35 d^2 f^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(b c-a d) (d g-c h) \left (\frac {(d e-c f) \left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}-\frac {2 (e+f x)^{5/2} (-7 d f (a h+b g)+7 b c f h+2 b d e h-5 b d f h x)}{35 d^2 f^2}\) |
Input:
Int[((a + b*x)*(e + f*x)^(3/2)*(g + h*x))/(c + d*x),x]
Output:
(-2*(e + f*x)^(5/2)*(2*b*d*e*h + 7*b*c*f*h - 7*d*f*(b*g + a*h) - 5*b*d*f*h *x))/(35*d^2*f^2) - ((b*c - a*d)*(d*g - c*h)*((2*(e + f*x)^(3/2))/(3*d) + ((d*e - c*f)*((2*Sqrt[e + f*x])/d - (2*Sqrt[d*e - c*f]*ArcTanh[(Sqrt[d]*Sq rt[e + f*x])/Sqrt[d*e - c*f]])/d^(3/2)))/d))/d^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 = 0.46 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.36
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\sqrt {\left (c f -d e \right ) d}\, \left (\left (\frac {\left (\frac {5}{7} b f x +a f -\frac {2}{7} b e \right ) \left (f x +e \right )^{2} d^{3}}{5}-\frac {4 c \left (\frac {3 b \,f^{2} x^{2}}{20}+\left (\frac {1}{4} a \,f^{2}+\frac {3}{10} b e f \right ) x +e \left (a f +\frac {3 b e}{20}\right )\right ) f \,d^{2}}{3}+c^{2} f^{2} \left (\frac {1}{3} b f x +\frac {4}{3} b e +a f \right ) d -b \,c^{3} f^{3}\right ) h -d g f \left (\left (-\frac {b \,f^{2} x^{2}}{5}+\left (-\frac {2}{5} b e f -\frac {1}{3} a \,f^{2}\right ) x -\frac {4 a e f}{3}-\frac {b \,e^{2}}{5}\right ) d^{2}+c f \left (\frac {1}{3} b f x +\frac {4}{3} b e +a f \right ) d -b \,c^{2} f^{2}\right )\right ) \sqrt {f x +e}+f^{2} \left (c f -d e \right )^{2} \left (c h -d g \right ) \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )\right )}{\sqrt {\left (c f -d e \right ) d}\, f^{2} d^{4}}\) | \(266\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {h b \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {a \,d^{3} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b c \,d^{2} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b \,d^{3} e h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b \,d^{3} f g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a c \,d^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a \,d^{3} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b \,c^{2} d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b c \,d^{2} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+a \,c^{2} d \,f^{3} h \sqrt {f x +e}-a c \,d^{2} e \,f^{2} h \sqrt {f x +e}-a c \,d^{2} f^{3} g \sqrt {f x +e}+a \,d^{3} e \,f^{2} g \sqrt {f x +e}-b \,c^{3} f^{3} h \sqrt {f x +e}+b \,c^{2} d e \,f^{2} h \sqrt {f x +e}+b \,c^{2} d \,f^{3} g \sqrt {f x +e}-b c \,d^{2} e \,f^{2} g \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{2} \left (a \,c^{3} d \,f^{2} h -2 a \,c^{2} d^{2} e f h -a \,c^{2} d^{2} f^{2} g +a c \,d^{3} e^{2} h +2 a c \,d^{3} e f g -g a \,e^{2} d^{4}-c^{4} b \,f^{2} h +2 b \,c^{3} d e f h +b \,c^{3} d \,f^{2} g -b \,c^{2} d^{2} e^{2} h -2 b \,c^{2} d^{2} e f g +b c \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(473\) |
| default | \(\frac {\frac {2 \left (\frac {h b \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {a \,d^{3} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b c \,d^{2} f h \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b \,d^{3} e h \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b \,d^{3} f g \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a c \,d^{2} f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a \,d^{3} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b \,c^{2} d \,f^{2} h \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b c \,d^{2} f^{2} g \left (f x +e \right )^{\frac {3}{2}}}{3}+a \,c^{2} d \,f^{3} h \sqrt {f x +e}-a c \,d^{2} e \,f^{2} h \sqrt {f x +e}-a c \,d^{2} f^{3} g \sqrt {f x +e}+a \,d^{3} e \,f^{2} g \sqrt {f x +e}-b \,c^{3} f^{3} h \sqrt {f x +e}+b \,c^{2} d e \,f^{2} h \sqrt {f x +e}+b \,c^{2} d \,f^{3} g \sqrt {f x +e}-b c \,d^{2} e \,f^{2} g \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{2} \left (a \,c^{3} d \,f^{2} h -2 a \,c^{2} d^{2} e f h -a \,c^{2} d^{2} f^{2} g +a c \,d^{3} e^{2} h +2 a c \,d^{3} e f g -g a \,e^{2} d^{4}-c^{4} b \,f^{2} h +2 b \,c^{3} d e f h +b \,c^{3} d \,f^{2} g -b \,c^{2} d^{2} e^{2} h -2 b \,c^{2} d^{2} e f g +b c \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(473\) |
| risch | \(\frac {2 \left (15 b \,f^{3} h \,d^{3} x^{3}+21 a \,d^{3} f^{3} h \,x^{2}-21 b c \,d^{2} f^{3} h \,x^{2}+24 b \,d^{3} e \,f^{2} h \,x^{2}+21 b \,d^{3} f^{3} g \,x^{2}-35 a c \,d^{2} f^{3} h x +42 a \,d^{3} e \,f^{2} h x +35 a \,d^{3} f^{3} g x +35 b \,c^{2} d \,f^{3} h x -42 b c \,d^{2} e \,f^{2} h x -35 b c \,d^{2} f^{3} g x +3 b \,d^{3} e^{2} f h x +42 b \,d^{3} e \,f^{2} g x +105 a \,c^{2} d \,f^{3} h -140 a c \,d^{2} e \,f^{2} h -105 a c \,d^{2} f^{3} g +21 a \,d^{3} e^{2} f h +140 a \,d^{3} e \,f^{2} g -105 b \,c^{3} f^{3} h +140 b \,c^{2} d e \,f^{2} h +105 b \,c^{2} d \,f^{3} g -21 b c \,d^{2} e^{2} f h -140 b c \,d^{2} e \,f^{2} g -6 b \,d^{3} e^{3} h +21 b \,d^{3} e^{2} f g \right ) \sqrt {f x +e}}{105 f^{2} d^{4}}-\frac {2 \left (a \,c^{3} d \,f^{2} h -2 a \,c^{2} d^{2} e f h -a \,c^{2} d^{2} f^{2} g +a c \,d^{3} e^{2} h +2 a c \,d^{3} e f g -g a \,e^{2} d^{4}-c^{4} b \,f^{2} h +2 b \,c^{3} d e f h +b \,c^{3} d \,f^{2} g -b \,c^{2} d^{2} e^{2} h -2 b \,c^{2} d^{2} e f g +b c \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}\) | \(485\) |
Input:
int((b*x+a)*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2/((c*f-d*e)*d)^(1/2)*(-((c*f-d*e)*d)^(1/2)*((1/5*(5/7*b*f*x+a*f-2/7*b*e) *(f*x+e)^2*d^3-4/3*c*(3/20*b*f^2*x^2+(1/4*a*f^2+3/10*b*e*f)*x+e*(a*f+3/20* b*e))*f*d^2+c^2*f^2*(1/3*b*f*x+4/3*b*e+a*f)*d-b*c^3*f^3)*h-d*g*f*((-1/5*b* f^2*x^2+(-2/5*b*e*f-1/3*a*f^2)*x-4/3*a*e*f-1/5*b*e^2)*d^2+c*f*(1/3*b*f*x+4 /3*b*e+a*f)*d-b*c^2*f^2))*(f*x+e)^(1/2)+f^2*(c*f-d*e)^2*(c*h-d*g)*(a*d-b*c )*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))/f^2/d^4
Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (172) = 344\).
Time = 0.10 (sec) , antiderivative size = 853, normalized size of antiderivative = 4.35 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x, algorithm="fricas")
Output:
[-1/105*(105*(((b*c*d^2 - a*d^3)*e*f^2 - (b*c^2*d - a*c*d^2)*f^3)*g - ((b* c^2*d - a*c*d^2)*e*f^2 - (b*c^3 - a*c^2*d)*f^3)*h)*sqrt((d*e - c*f)/d)*log ((d*f*x + 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(15*b*d^3*f^3*h*x^3 + 3*(7*b*d^3*f^3*g + (8*b*d^3*e*f^2 - 7*(b*c*d^2 - a*d^3)*f^3)*h)*x^2 + 7*(3*b*d^3*e^2*f - 20*(b*c*d^2 - a*d^3)*e*f^2 + 15*( b*c^2*d - a*c*d^2)*f^3)*g - (6*b*d^3*e^3 + 21*(b*c*d^2 - a*d^3)*e^2*f - 14 0*(b*c^2*d - a*c*d^2)*e*f^2 + 105*(b*c^3 - a*c^2*d)*f^3)*h + (7*(6*b*d^3*e *f^2 - 5*(b*c*d^2 - a*d^3)*f^3)*g + (3*b*d^3*e^2*f - 42*(b*c*d^2 - a*d^3)* e*f^2 + 35*(b*c^2*d - a*c*d^2)*f^3)*h)*x)*sqrt(f*x + e))/(d^4*f^2), 2/105* (105*(((b*c*d^2 - a*d^3)*e*f^2 - (b*c^2*d - a*c*d^2)*f^3)*g - ((b*c^2*d - a*c*d^2)*e*f^2 - (b*c^3 - a*c^2*d)*f^3)*h)*sqrt(-(d*e - c*f)/d)*arctan(-sq rt(f*x + e)*d*sqrt(-(d*e - c*f)/d)/(d*e - c*f)) + (15*b*d^3*f^3*h*x^3 + 3* (7*b*d^3*f^3*g + (8*b*d^3*e*f^2 - 7*(b*c*d^2 - a*d^3)*f^3)*h)*x^2 + 7*(3*b *d^3*e^2*f - 20*(b*c*d^2 - a*d^3)*e*f^2 + 15*(b*c^2*d - a*c*d^2)*f^3)*g - (6*b*d^3*e^3 + 21*(b*c*d^2 - a*d^3)*e^2*f - 140*(b*c^2*d - a*c*d^2)*e*f^2 + 105*(b*c^3 - a*c^2*d)*f^3)*h + (7*(6*b*d^3*e*f^2 - 5*(b*c*d^2 - a*d^3)*f ^3)*g + (3*b*d^3*e^2*f - 42*(b*c*d^2 - a*d^3)*e*f^2 + 35*(b*c^2*d - a*c*d^ 2)*f^3)*h)*x)*sqrt(f*x + e))/(d^4*f^2)]
Time = 14.01 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b h \left (e + f x\right )^{\frac {7}{2}}}{7 d f} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a d f h - b c f h - b d e h + b d f g\right )}{5 d^{2} f} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- a c d f h + a d^{2} f g + b c^{2} f h - b c d f g\right )}{3 d^{3}} + \frac {\sqrt {e + f x} \left (a c^{2} d f^{2} h - a c d^{2} e f h - a c d^{2} f^{2} g + a d^{3} e f g - b c^{3} f^{2} h + b c^{2} d e f h + b c^{2} d f^{2} g - b c d^{2} e f g\right )}{d^{4}} - \frac {f \left (a d - b c\right ) \left (c f - d e\right )^{2} \left (c h - d g\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{5} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (\frac {b h x^{2}}{2 d} + \frac {x \left (a d h - b c h + b d g\right )}{d^{2}} - \frac {\left (a d - b c\right ) \left (c h - d g\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(f*x+e)**(3/2)*(h*x+g)/(d*x+c),x)
Output:
Piecewise((2*(b*h*(e + f*x)**(7/2)/(7*d*f) + (e + f*x)**(5/2)*(a*d*f*h - b *c*f*h - b*d*e*h + b*d*f*g)/(5*d**2*f) + (e + f*x)**(3/2)*(-a*c*d*f*h + a* d**2*f*g + b*c**2*f*h - b*c*d*f*g)/(3*d**3) + sqrt(e + f*x)*(a*c**2*d*f**2 *h - a*c*d**2*e*f*h - a*c*d**2*f**2*g + a*d**3*e*f*g - b*c**3*f**2*h + b*c **2*d*e*f*h + b*c**2*d*f**2*g - b*c*d**2*e*f*g)/d**4 - f*(a*d - b*c)*(c*f - d*e)**2*(c*h - d*g)*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**5*sqrt(( c*f - d*e)/d)))/f, Ne(f, 0)), (e**(3/2)*(b*h*x**2/(2*d) + x*(a*d*h - b*c*h + b*d*g)/d**2 - (a*d - b*c)*(c*h - d*g)*Piecewise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d**2), True))
Exception generated. \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (172) = 344\).
Time = 0.13 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.54 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx=-\frac {2 \, {\left (b c d^{3} e^{2} g - a d^{4} e^{2} g - 2 \, b c^{2} d^{2} e f g + 2 \, a c d^{3} e f g + b c^{3} d f^{2} g - a c^{2} d^{2} f^{2} g - b c^{2} d^{2} e^{2} h + a c d^{3} e^{2} h + 2 \, b c^{3} d e f h - 2 \, a c^{2} d^{2} e f h - b c^{4} f^{2} h + a c^{3} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{4}} + \frac {2 \, {\left (21 \, {\left (f x + e\right )}^{\frac {5}{2}} b d^{6} f^{13} g - 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b c d^{5} f^{14} g + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{6} f^{14} g - 105 \, \sqrt {f x + e} b c d^{5} e f^{14} g + 105 \, \sqrt {f x + e} a d^{6} e f^{14} g + 105 \, \sqrt {f x + e} b c^{2} d^{4} f^{15} g - 105 \, \sqrt {f x + e} a c d^{5} f^{15} g + 15 \, {\left (f x + e\right )}^{\frac {7}{2}} b d^{6} f^{12} h - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b d^{6} e f^{12} h - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b c d^{5} f^{13} h + 21 \, {\left (f x + e\right )}^{\frac {5}{2}} a d^{6} f^{13} h + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b c^{2} d^{4} f^{14} h - 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a c d^{5} f^{14} h + 105 \, \sqrt {f x + e} b c^{2} d^{4} e f^{14} h - 105 \, \sqrt {f x + e} a c d^{5} e f^{14} h - 105 \, \sqrt {f x + e} b c^{3} d^{3} f^{15} h + 105 \, \sqrt {f x + e} a c^{2} d^{4} f^{15} h\right )}}{105 \, d^{7} f^{14}} \] Input:
integrate((b*x+a)*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x, algorithm="giac")
Output:
-2*(b*c*d^3*e^2*g - a*d^4*e^2*g - 2*b*c^2*d^2*e*f*g + 2*a*c*d^3*e*f*g + b* c^3*d*f^2*g - a*c^2*d^2*f^2*g - b*c^2*d^2*e^2*h + a*c*d^3*e^2*h + 2*b*c^3* d*e*f*h - 2*a*c^2*d^2*e*f*h - b*c^4*f^2*h + a*c^3*d*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^4) + 2/105*(21*(f*x + e)^(5/2)*b*d^6*f^13*g - 35*(f*x + e)^(3/2)*b*c*d^5*f^14*g + 35*(f*x + e) ^(3/2)*a*d^6*f^14*g - 105*sqrt(f*x + e)*b*c*d^5*e*f^14*g + 105*sqrt(f*x + e)*a*d^6*e*f^14*g + 105*sqrt(f*x + e)*b*c^2*d^4*f^15*g - 105*sqrt(f*x + e) *a*c*d^5*f^15*g + 15*(f*x + e)^(7/2)*b*d^6*f^12*h - 21*(f*x + e)^(5/2)*b*d ^6*e*f^12*h - 21*(f*x + e)^(5/2)*b*c*d^5*f^13*h + 21*(f*x + e)^(5/2)*a*d^6 *f^13*h + 35*(f*x + e)^(3/2)*b*c^2*d^4*f^14*h - 35*(f*x + e)^(3/2)*a*c*d^5 *f^14*h + 105*sqrt(f*x + e)*b*c^2*d^4*e*f^14*h - 105*sqrt(f*x + e)*a*c*d^5 *e*f^14*h - 105*sqrt(f*x + e)*b*c^3*d^3*f^15*h + 105*sqrt(f*x + e)*a*c^2*d ^4*f^15*h)/(d^7*f^14)
Time = 0.10 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.54 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx={\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,a\,f\,h-4\,b\,e\,h+2\,b\,f\,g}{5\,d\,f^2}-\frac {2\,b\,h\,\left (c\,f^3-d\,e\,f^2\right )}{5\,d^2\,f^4}\right )-{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,\left (a\,f-b\,e\right )\,\left (e\,h-f\,g\right )}{3\,d\,f^2}+\frac {\left (c\,f^3-d\,e\,f^2\right )\,\left (\frac {2\,a\,f\,h-4\,b\,e\,h+2\,b\,f\,g}{d\,f^2}-\frac {2\,b\,h\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )}{3\,d\,f^2}\right )+\frac {2\,b\,h\,{\left (e+f\,x\right )}^{7/2}}{7\,d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,\left (a\,d-b\,c\right )\,{\left (c\,f-d\,e\right )}^{3/2}\,\left (c\,h-d\,g\right )}{a\,d^4\,e^2\,g+b\,c^4\,f^2\,h-a\,c\,d^3\,e^2\,h-b\,c\,d^3\,e^2\,g-a\,c^3\,d\,f^2\,h-b\,c^3\,d\,f^2\,g+a\,c^2\,d^2\,f^2\,g+b\,c^2\,d^2\,e^2\,h-2\,a\,c\,d^3\,e\,f\,g-2\,b\,c^3\,d\,e\,f\,h+2\,a\,c^2\,d^2\,e\,f\,h+2\,b\,c^2\,d^2\,e\,f\,g}\right )\,\left (a\,d-b\,c\right )\,{\left (c\,f-d\,e\right )}^{3/2}\,\left (c\,h-d\,g\right )}{d^{9/2}}+\frac {\sqrt {e+f\,x}\,\left (\frac {2\,\left (a\,f-b\,e\right )\,\left (e\,h-f\,g\right )}{d\,f^2}+\frac {\left (c\,f^3-d\,e\,f^2\right )\,\left (\frac {2\,a\,f\,h-4\,b\,e\,h+2\,b\,f\,g}{d\,f^2}-\frac {2\,b\,h\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )}{d\,f^2}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2} \] Input:
int(((e + f*x)^(3/2)*(g + h*x)*(a + b*x))/(c + d*x),x)
Output:
(e + f*x)^(5/2)*((2*a*f*h - 4*b*e*h + 2*b*f*g)/(5*d*f^2) - (2*b*h*(c*f^3 - d*e*f^2))/(5*d^2*f^4)) - (e + f*x)^(3/2)*((2*(a*f - b*e)*(e*h - f*g))/(3* d*f^2) + ((c*f^3 - d*e*f^2)*((2*a*f*h - 4*b*e*h + 2*b*f*g)/(d*f^2) - (2*b* h*(c*f^3 - d*e*f^2))/(d^2*f^4)))/(3*d*f^2)) + (2*b*h*(e + f*x)^(7/2))/(7*d *f^2) + (2*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)*(c*f - d*e)^(3/2)*(c* h - d*g))/(a*d^4*e^2*g + b*c^4*f^2*h - a*c*d^3*e^2*h - b*c*d^3*e^2*g - a*c ^3*d*f^2*h - b*c^3*d*f^2*g + a*c^2*d^2*f^2*g + b*c^2*d^2*e^2*h - 2*a*c*d^3 *e*f*g - 2*b*c^3*d*e*f*h + 2*a*c^2*d^2*e*f*h + 2*b*c^2*d^2*e*f*g))*(a*d - b*c)*(c*f - d*e)^(3/2)*(c*h - d*g))/d^(9/2) + ((e + f*x)^(1/2)*((2*(a*f - b*e)*(e*h - f*g))/(d*f^2) + ((c*f^3 - d*e*f^2)*((2*a*f*h - 4*b*e*h + 2*b*f *g)/(d*f^2) - (2*b*h*(c*f^3 - d*e*f^2))/(d^2*f^4)))/(d*f^2))*(c*f^3 - d*e* f^2))/(d*f^2)
Time = 0.17 (sec) , antiderivative size = 835, normalized size of antiderivative = 4.26 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{c+d x} \, dx =\text {Too large to display} \] Input:
int((b*x+a)*(f*x+e)^(3/2)*(h*x+g)/(d*x+c),x)
Output:
(2*( - 105*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c* f - d*e)))*a*c**2*d*f**3*h + 105*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f* x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c*d**2*e*f**2*h + 105*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c*d**2*f**3*g - 105*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d* e)))*a*d**3*e*f**2*g + 105*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/ (sqrt(d)*sqrt(c*f - d*e)))*b*c**3*f**3*h - 105*sqrt(d)*sqrt(c*f - d*e)*ata n((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**2*d*e*f**2*h - 105*sqr t(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c **2*d*f**3*g + 105*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d) *sqrt(c*f - d*e)))*b*c*d**2*e*f**2*g + 105*sqrt(e + f*x)*a*c**2*d**2*f**3* h - 140*sqrt(e + f*x)*a*c*d**3*e*f**2*h - 105*sqrt(e + f*x)*a*c*d**3*f**3* g - 35*sqrt(e + f*x)*a*c*d**3*f**3*h*x + 21*sqrt(e + f*x)*a*d**4*e**2*f*h + 140*sqrt(e + f*x)*a*d**4*e*f**2*g + 42*sqrt(e + f*x)*a*d**4*e*f**2*h*x + 35*sqrt(e + f*x)*a*d**4*f**3*g*x + 21*sqrt(e + f*x)*a*d**4*f**3*h*x**2 - 105*sqrt(e + f*x)*b*c**3*d*f**3*h + 140*sqrt(e + f*x)*b*c**2*d**2*e*f**2*h + 105*sqrt(e + f*x)*b*c**2*d**2*f**3*g + 35*sqrt(e + f*x)*b*c**2*d**2*f** 3*h*x - 21*sqrt(e + f*x)*b*c*d**3*e**2*f*h - 140*sqrt(e + f*x)*b*c*d**3*e* f**2*g - 42*sqrt(e + f*x)*b*c*d**3*e*f**2*h*x - 35*sqrt(e + f*x)*b*c*d**3* f**3*g*x - 21*sqrt(e + f*x)*b*c*d**3*f**3*h*x**2 - 6*sqrt(e + f*x)*b*d*...