Integrand size = 21, antiderivative size = 391 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=-\frac {3 b^3 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2}}{1024 c^5}+\frac {b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) x \sqrt {b x+c x^2}}{512 c^4}+\frac {3 b (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) x^2 \sqrt {b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) x^3 \sqrt {b x+c x^2}}{64 c^2}+\frac {e \left (24 c^2 d^2-14 b c d e+3 b^2 e^2\right ) \left (b x+c x^2\right )^{5/2}}{40 c^3}+\frac {e^2 (14 c d-3 b e) x \left (b x+c x^2\right )^{5/2}}{28 c^2}+\frac {e^3 x^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}} \] Output:
-3/1024*b^3*(-b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*(c*x^2+b*x)^(1/2) /c^5+1/512*b^2*(-b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*x*(c*x^2+b*x)^ (1/2)/c^4+3/128*b*(-b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*x^2*(c*x^2+ b*x)^(1/2)/c^3+1/64*(-b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*x^3*(c*x^ 2+b*x)^(1/2)/c^2+1/40*e*(3*b^2*e^2-14*b*c*d*e+24*c^2*d^2)*(c*x^2+b*x)^(5/2 )/c^3+1/28*e^2*(-3*b*e+14*c*d)*x*(c*x^2+b*x)^(5/2)/c^2+1/7*e^3*x^2*(c*x^2+ b*x)^(5/2)/c+3/1024*b^4*(-b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*arcta nh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c^(11/2)
Time = 1.92 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.95 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c^2 e \left (90 d^2+35 d e x+6 e^2 x^2\right )+32 b^2 c^4 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )-16 b^3 c^3 \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )+256 c^6 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+128 b c^5 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )\right )+630 b^5 e \left (8 c^2 d^2+b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+420 b^4 c d \left (8 c^2 d^2+7 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{35840 c^{11/2} \sqrt {x (b+c x)}} \] Input:
Integrate[(d + e*x)^3*(b*x + c*x^2)^(3/2),x]
Output:
(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(315*b^6*e^3 - 210*b ^5*c*e^2*(7*d + e*x) + 28*b^4*c^2*e*(90*d^2 + 35*d*e*x + 6*e^2*x^2) + 32*b ^2*c^4*x*(35*d^3 + 42*d^2*e*x + 21*d*e^2*x^2 + 4*e^3*x^3) - 16*b^3*c^3*(10 5*d^3 + 105*d^2*e*x + 49*d*e^2*x^2 + 9*e^3*x^3) + 256*c^6*x^3*(35*d^3 + 84 *d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 128*b*c^5*x^2*(105*d^3 + 231*d^2*e *x + 182*d*e^2*x^2 + 50*e^3*x^3)) + 630*b^5*e*(8*c^2*d^2 + b^2*e^2)*ArcTan h[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 420*b^4*c*d*(8*c^2*d^2 + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(35840* c^(11/2)*Sqrt[x*(b + c*x)])
Time = 0.73 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.59, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1166, 27, 1225, 1087, 1087, 1091, 219}
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.
| \(\displaystyle \int \left (b x+c x^2\right )^{3/2} (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {\int \frac {1}{2} (d+e x) (d (14 c d-5 b e)+9 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}dx}{7 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d+e x) (d (14 c d-5 b e)+9 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}dx}{14 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {7 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \int \left (c x^2+b x\right )^{3/2}dx}{8 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{20 c^2}}{14 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {7 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{8 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{20 c^2}}{14 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {7 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{8 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{20 c^2}}{14 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\frac {7 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{8 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{20 c^2}}{14 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {7 \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right ) (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{8 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{20 c^2}}{14 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c}\) |
Input:
Int[(d + e*x)^3*(b*x + c*x^2)^(3/2),x]
Output:
(e*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c) + ((e*(128*c^2*d^2 - 98*b*c*d*e + 21*b^2*e^2 + 30*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(5/2))/(20*c^2) + (7* (2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(((b + 2*c*x)*(b*x + c*x ^2)^(3/2))/(8*c) - (3*b^2*(((b + 2*c*x)*Sqrt[b*x + c*x^2])/(4*c) - (b^2*Ar cTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(3/2))))/(16*c)))/(8*c^2))/(14* c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Time = 0.56 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(-\frac {9 \left (\left (b^{2} e^{2}-\frac {8}{3} b c d e +\frac {8}{3} c^{2} d^{2}\right ) \left (b e -2 c d \right ) b^{4} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\left (\frac {128 x^{2} b \left (\frac {10}{21} e^{3} x^{3}+\frac {26}{15} d \,e^{2} x^{2}+\frac {11}{5} d^{2} e x +d^{3}\right ) c^{\frac {11}{2}}}{3}+\frac {256 \left (\frac {4}{7} e^{3} x^{3}+2 d \,e^{2} x^{2}+\frac {12}{5} d^{2} e x +d^{3}\right ) x^{3} c^{\frac {13}{2}}}{9}+b^{2} \left (-\frac {16 \left (\frac {3}{35} e^{3} x^{3}+\frac {7}{15} d \,e^{2} x^{2}+d^{2} e x +d^{3}\right ) b \,c^{\frac {7}{2}}}{3}+\frac {32 x \left (\frac {4}{35} e^{3} x^{3}+\frac {3}{5} d \,e^{2} x^{2}+\frac {6}{5} d^{2} e x +d^{3}\right ) c^{\frac {9}{2}}}{9}+e \left (\left (\frac {8}{15} e^{2} x^{2}+\frac {28}{9} d e x +8 d^{2}\right ) c^{\frac {5}{2}}+\left (\left (-\frac {2 e x}{3}-\frac {14 d}{3}\right ) c^{\frac {3}{2}}+b e \sqrt {c}\right ) e b \right ) b^{2}\right )\right ) \sqrt {x \left (c x +b \right )}\right )}{1024 c^{\frac {11}{2}}}\) | \(266\) |
| risch | \(\frac {\left (5120 c^{6} e^{3} x^{6}+6400 b \,c^{5} e^{3} x^{5}+17920 c^{6} d \,e^{2} x^{5}+128 b^{2} c^{4} e^{3} x^{4}+23296 b \,c^{5} d \,e^{2} x^{4}+21504 c^{6} d^{2} e \,x^{4}-144 b^{3} c^{3} e^{3} x^{3}+672 b^{2} c^{4} d \,e^{2} x^{3}+29568 b \,c^{5} d^{2} e \,x^{3}+8960 c^{6} d^{3} x^{3}+168 b^{4} c^{2} e^{3} x^{2}-784 b^{3} c^{3} d \,e^{2} x^{2}+1344 b^{2} c^{4} d^{2} e \,x^{2}+13440 b \,c^{5} d^{3} x^{2}-210 b^{5} c \,e^{3} x +980 b^{4} c^{2} d \,e^{2} x -1680 b^{3} c^{3} d^{2} e x +1120 b^{2} c^{4} d^{3} x +315 b^{6} e^{3}-1470 b^{5} c d \,e^{2}+2520 b^{4} c^{2} d^{2} e -1680 b^{3} c^{3} d^{3}\right ) x \left (c x +b \right )}{35840 c^{5} \sqrt {x \left (c x +b \right )}}-\frac {3 b^{4} \left (3 b^{3} e^{3}-14 d \,e^{2} b^{2} c +24 d^{2} e b \,c^{2}-16 d^{3} c^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}\) | \(364\) |
| default | \(d^{3} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e^{3} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 c}-\frac {9 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )+3 d^{2} e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\) | \(509\) |
Input:
int((e*x+d)^3*(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-9/1024/c^(11/2)*((b^2*e^2-8/3*b*c*d*e+8/3*c^2*d^2)*(b*e-2*c*d)*b^4*arctan h((x*(c*x+b))^(1/2)/x/c^(1/2))-(128/3*x^2*b*(10/21*e^3*x^3+26/15*d*e^2*x^2 +11/5*d^2*e*x+d^3)*c^(11/2)+256/9*(4/7*e^3*x^3+2*d*e^2*x^2+12/5*d^2*e*x+d^ 3)*x^3*c^(13/2)+b^2*(-16/3*(3/35*e^3*x^3+7/15*d*e^2*x^2+d^2*e*x+d^3)*b*c^( 7/2)+32/9*x*(4/35*e^3*x^3+3/5*d*e^2*x^2+6/5*d^2*e*x+d^3)*c^(9/2)+e*((8/15* e^2*x^2+28/9*d*e*x+8*d^2)*c^(5/2)+((-2/3*e*x-14/3*d)*c^(3/2)+b*e*c^(1/2))* e*b)*b^2))*(x*(c*x+b))^(1/2))
Time = 0.10 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.81 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \, {\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \, {\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \, {\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{71680 \, c^{6}}, -\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) - {\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \, {\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \, {\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \, {\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{35840 \, c^{6}}\right ] \] Input:
integrate((e*x+d)^3*(c*x^2+b*x)^(3/2),x, algorithm="fricas")
Output:
[-1/71680*(105*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7 *e^3)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(5120*c^7*e ^3*x^6 - 1680*b^3*c^4*d^3 + 2520*b^4*c^3*d^2*e - 1470*b^5*c^2*d*e^2 + 315* b^6*c*e^3 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7*d^2*e + 1 82*b*c^6*d*e^2 + b^2*c^5*e^3)*x^4 + 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 4 2*b^2*c^5*d*e^2 - 9*b^3*c^4*e^3)*x^3 + 56*(240*b*c^6*d^3 + 24*b^2*c^5*d^2* e - 14*b^3*c^4*d*e^2 + 3*b^4*c^3*e^3)*x^2 + 70*(16*b^2*c^5*d^3 - 24*b^3*c^ 4*d^2*e + 14*b^4*c^3*d*e^2 - 3*b^5*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/c^6, -1/ 35840*(105*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3 )*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) - (5120*c^7*e^3*x^ 6 - 1680*b^3*c^4*d^3 + 2520*b^4*c^3*d^2*e - 1470*b^5*c^2*d*e^2 + 315*b^6*c *e^3 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7*d^2*e + 182*b* c^6*d*e^2 + b^2*c^5*e^3)*x^4 + 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2 *c^5*d*e^2 - 9*b^3*c^4*e^3)*x^3 + 56*(240*b*c^6*d^3 + 24*b^2*c^5*d^2*e - 1 4*b^3*c^4*d*e^2 + 3*b^4*c^3*e^3)*x^2 + 70*(16*b^2*c^5*d^3 - 24*b^3*c^4*d^2 *e + 14*b^4*c^3*d*e^2 - 3*b^5*c^2*e^3)*x)*sqrt(c*x^2 + b*x))/c^6]
Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (388) = 776\).
Time = 0.49 (sec) , antiderivative size = 864, normalized size of antiderivative = 2.21 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)**3*(c*x**2+b*x)**(3/2),x)
Output:
Piecewise((3*b**2*(b**2*d**3 - 5*b*(3*b**2*d**2*e + 2*b*c*d**3 - 7*b*(3*b* *2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e* *3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c))/ (6*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c), N e(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), T rue))/(8*c**2) + sqrt(b*x + c*x**2)*(-3*b*(b**2*d**3 - 5*b*(3*b**2*d**2*e + 2*b*c*d**3 - 7*b*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c* d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10 *c) + c**2*d**3)/(8*c))/(6*c))/(4*c**2) + c*e**3*x**6/7 + x**5*(15*b*c*e** 3/14 + 3*c**2*d*e**2)/(6*c) + x**4*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b* c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(5*c) + x**3*(3*b**2*d* e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(4*c) + x**2 *(3*b**2*d**2*e + 2*b*c*d**3 - 7*b*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b* *2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3* c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c))/(3*c) + x*(b**2*d**3 - 5*b*(3*b**2 *d**2*e + 2*b*c*d**3 - 7*b*(3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d** 2*e)/(10*c) + c**2*d**3)/(8*c))/(6*c))/(2*c)), Ne(c, 0)), (2*(d**3*(b*x)** (5/2)/5 + 3*d**2*e*(b*x)**(7/2)/(7*b) + d*e**2*(b*x)**(9/2)/(3*b**2) + ...
Time = 0.04 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.60 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x)^(3/2),x, algorithm="maxima")
Output:
1/7*(c*x^2 + b*x)^(5/2)*e^3*x^2/c + 1/4*(c*x^2 + b*x)^(3/2)*d^3*x - 3/32*s qrt(c*x^2 + b*x)*b^2*d^3*x/c + 9/64*sqrt(c*x^2 + b*x)*b^3*d^2*e*x/c^2 - 3/ 8*(c*x^2 + b*x)^(3/2)*b*d^2*e*x/c - 21/256*sqrt(c*x^2 + b*x)*b^4*d*e^2*x/c ^3 + 7/32*(c*x^2 + b*x)^(3/2)*b^2*d*e^2*x/c^2 + 1/2*(c*x^2 + b*x)^(5/2)*d* e^2*x/c + 9/512*sqrt(c*x^2 + b*x)*b^5*e^3*x/c^4 - 3/64*(c*x^2 + b*x)^(3/2) *b^3*e^3*x/c^3 - 3/28*(c*x^2 + b*x)^(5/2)*b*e^3*x/c^2 + 3/128*b^4*d^3*log( 2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 9/256*b^5*d^2*e*log(2*c *x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 21/1024*b^6*d*e^2*log(2*c* x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 9/2048*b^7*e^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) - 3/64*sqrt(c*x^2 + b*x)*b^3*d^3 /c^2 + 1/8*(c*x^2 + b*x)^(3/2)*b*d^3/c + 9/128*sqrt(c*x^2 + b*x)*b^4*d^2*e /c^3 - 3/16*(c*x^2 + b*x)^(3/2)*b^2*d^2*e/c^2 + 3/5*(c*x^2 + b*x)^(5/2)*d^ 2*e/c - 21/512*sqrt(c*x^2 + b*x)*b^5*d*e^2/c^4 + 7/64*(c*x^2 + b*x)^(3/2)* b^3*d*e^2/c^3 - 7/20*(c*x^2 + b*x)^(5/2)*b*d*e^2/c^2 + 9/1024*sqrt(c*x^2 + b*x)*b^6*e^3/c^5 - 3/128*(c*x^2 + b*x)^(3/2)*b^4*e^3/c^4 + 3/40*(c*x^2 + b*x)^(5/2)*b^2*e^3/c^3
Time = 0.13 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.95 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{35840} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c e^{3} x + \frac {14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac {168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac {560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}}{c^{6}}\right )} x + \frac {7 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )}}{c^{6}}\right )} x - \frac {105 \, {\left (16 \, b^{3} c^{4} d^{3} - 24 \, b^{4} c^{3} d^{2} e + 14 \, b^{5} c^{2} d e^{2} - 3 \, b^{6} c e^{3}\right )}}{c^{6}}\right )} - \frac {3 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x)^(3/2),x, algorithm="giac")
Output:
1/35840*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(4*c*e^3*x + (14*c^7*d*e^2 + 5*b *c^6*e^3)/c^6)*x + (168*c^7*d^2*e + 182*b*c^6*d*e^2 + b^2*c^5*e^3)/c^6)*x + (560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2 - 9*b^3*c^4*e^3)/c^6) *x + 7*(240*b*c^6*d^3 + 24*b^2*c^5*d^2*e - 14*b^3*c^4*d*e^2 + 3*b^4*c^3*e^ 3)/c^6)*x + 35*(16*b^2*c^5*d^3 - 24*b^3*c^4*d^2*e + 14*b^4*c^3*d*e^2 - 3*b ^5*c^2*e^3)/c^6)*x - 105*(16*b^3*c^4*d^3 - 24*b^4*c^3*d^2*e + 14*b^5*c^2*d *e^2 - 3*b^6*c*e^3)/c^6) - 3/2048*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14* b^6*c*d*e^2 - 3*b^7*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(11/2)
Timed out. \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^3 \,d x \] Input:
int((b*x + c*x^2)^(3/2)*(d + e*x)^3,x)
Output:
int((b*x + c*x^2)^(3/2)*(d + e*x)^3, x)
Time = 26.31 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.48 \[ \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx=\frac {-315 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{7} e^{3}+315 \sqrt {x}\, \sqrt {c x +b}\, b^{6} c \,e^{3}-1680 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{4} d^{3}+8960 \sqrt {x}\, \sqrt {c x +b}\, c^{7} d^{3} x^{3}+5120 \sqrt {x}\, \sqrt {c x +b}\, c^{7} e^{3} x^{6}+1680 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{4} c^{3} d^{3}-1470 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{2} d \,e^{2}-210 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{2} e^{3} x +2520 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{3} d^{2} e +168 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{3} e^{3} x^{2}-144 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{4} e^{3} x^{3}+1120 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{5} d^{3} x +128 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{5} e^{3} x^{4}+13440 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{6} d^{3} x^{2}+6400 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{6} e^{3} x^{5}+21504 \sqrt {x}\, \sqrt {c x +b}\, c^{7} d^{2} e \,x^{4}+17920 \sqrt {x}\, \sqrt {c x +b}\, c^{7} d \,e^{2} x^{5}+1470 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{6} c d \,e^{2}-2520 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{5} c^{2} d^{2} e +980 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{3} d \,e^{2} x -1680 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{4} d^{2} e x -784 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{4} d \,e^{2} x^{2}+1344 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{5} d^{2} e \,x^{2}+672 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{5} d \,e^{2} x^{3}+29568 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{6} d^{2} e \,x^{3}+23296 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{6} d \,e^{2} x^{4}}{35840 c^{6}} \] Input:
int((e*x+d)^3*(c*x^2+b*x)^(3/2),x)
Output:
(315*sqrt(x)*sqrt(b + c*x)*b**6*c*e**3 - 1470*sqrt(x)*sqrt(b + c*x)*b**5*c **2*d*e**2 - 210*sqrt(x)*sqrt(b + c*x)*b**5*c**2*e**3*x + 2520*sqrt(x)*sqr t(b + c*x)*b**4*c**3*d**2*e + 980*sqrt(x)*sqrt(b + c*x)*b**4*c**3*d*e**2*x + 168*sqrt(x)*sqrt(b + c*x)*b**4*c**3*e**3*x**2 - 1680*sqrt(x)*sqrt(b + c *x)*b**3*c**4*d**3 - 1680*sqrt(x)*sqrt(b + c*x)*b**3*c**4*d**2*e*x - 784*s qrt(x)*sqrt(b + c*x)*b**3*c**4*d*e**2*x**2 - 144*sqrt(x)*sqrt(b + c*x)*b** 3*c**4*e**3*x**3 + 1120*sqrt(x)*sqrt(b + c*x)*b**2*c**5*d**3*x + 1344*sqrt (x)*sqrt(b + c*x)*b**2*c**5*d**2*e*x**2 + 672*sqrt(x)*sqrt(b + c*x)*b**2*c **5*d*e**2*x**3 + 128*sqrt(x)*sqrt(b + c*x)*b**2*c**5*e**3*x**4 + 13440*sq rt(x)*sqrt(b + c*x)*b*c**6*d**3*x**2 + 29568*sqrt(x)*sqrt(b + c*x)*b*c**6* d**2*e*x**3 + 23296*sqrt(x)*sqrt(b + c*x)*b*c**6*d*e**2*x**4 + 6400*sqrt(x )*sqrt(b + c*x)*b*c**6*e**3*x**5 + 8960*sqrt(x)*sqrt(b + c*x)*c**7*d**3*x* *3 + 21504*sqrt(x)*sqrt(b + c*x)*c**7*d**2*e*x**4 + 17920*sqrt(x)*sqrt(b + c*x)*c**7*d*e**2*x**5 + 5120*sqrt(x)*sqrt(b + c*x)*c**7*e**3*x**6 - 315*s qrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**7*e**3 + 1470*sqr t(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**6*c*d*e**2 - 2520*s qrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**5*c**2*d**2*e + 1 680*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))*b**4*c**3*d**3) /(35840*c**6)