Integrand size = 21, antiderivative size = 391 \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=-\frac {3 a^3 (2 b c-a d) \left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \sqrt {a x+b x^2}}{1024 b^5}+\frac {a^2 (2 b c-a d) \left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x \sqrt {a x+b x^2}}{512 b^4}+\frac {3 a (2 b c-a d) \left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x^2 \sqrt {a x+b x^2}}{128 b^3}+\frac {(2 b c-a d) \left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x^3 \sqrt {a x+b x^2}}{64 b^2}+\frac {d \left (24 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \left (a x+b x^2\right )^{5/2}}{40 b^3}+\frac {d^2 (14 b c-3 a d) x \left (a x+b x^2\right )^{5/2}}{28 b^2}+\frac {d^3 x^2 \left (a x+b x^2\right )^{5/2}}{7 b}+\frac {3 a^4 (2 b c-a d) \left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{1024 b^{11/2}} \] Output:
-3/1024*a^3*(-a*d+2*b*c)*(3*a^2*d^2-8*a*b*c*d+8*b^2*c^2)*(b*x^2+a*x)^(1/2) /b^5+1/512*a^2*(-a*d+2*b*c)*(3*a^2*d^2-8*a*b*c*d+8*b^2*c^2)*x*(b*x^2+a*x)^ (1/2)/b^4+3/128*a*(-a*d+2*b*c)*(3*a^2*d^2-8*a*b*c*d+8*b^2*c^2)*x^2*(b*x^2+ a*x)^(1/2)/b^3+1/64*(-a*d+2*b*c)*(3*a^2*d^2-8*a*b*c*d+8*b^2*c^2)*x^3*(b*x^ 2+a*x)^(1/2)/b^2+1/40*d*(3*a^2*d^2-14*a*b*c*d+24*b^2*c^2)*(b*x^2+a*x)^(5/2 )/b^3+1/28*d^2*(-3*a*d+14*b*c)*x*(b*x^2+a*x)^(5/2)/b^2+1/7*d^3*x^2*(b*x^2+ a*x)^(5/2)/b+3/1024*a^4*(-a*d+2*b*c)*(3*a^2*d^2-8*a*b*c*d+8*b^2*c^2)*arcta nh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(11/2)
Time = 1.82 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.95 \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (315 a^6 d^3-210 a^5 b d^2 (7 c+d x)+28 a^4 b^2 d \left (90 c^2+35 c d x+6 d^2 x^2\right )+32 a^2 b^4 x \left (35 c^3+42 c^2 d x+21 c d^2 x^2+4 d^3 x^3\right )-16 a^3 b^3 \left (105 c^3+105 c^2 d x+49 c d^2 x^2+9 d^3 x^3\right )+256 b^6 x^3 \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+128 a b^5 x^2 \left (105 c^3+231 c^2 d x+182 c d^2 x^2+50 d^3 x^3\right )\right )+630 a^5 d \left (8 b^2 c^2+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )+420 a^4 b c \left (8 b^2 c^2+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{35840 b^{11/2} \sqrt {x (a+b x)}} \] Input:
Integrate[(c + d*x)^3*(a*x + b*x^2)^(3/2),x]
Output:
(Sqrt[x]*Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(315*a^6*d^3 - 210*a ^5*b*d^2*(7*c + d*x) + 28*a^4*b^2*d*(90*c^2 + 35*c*d*x + 6*d^2*x^2) + 32*a ^2*b^4*x*(35*c^3 + 42*c^2*d*x + 21*c*d^2*x^2 + 4*d^3*x^3) - 16*a^3*b^3*(10 5*c^3 + 105*c^2*d*x + 49*c*d^2*x^2 + 9*d^3*x^3) + 256*b^6*x^3*(35*c^3 + 84 *c^2*d*x + 70*c*d^2*x^2 + 20*d^3*x^3) + 128*a*b^5*x^2*(105*c^3 + 231*c^2*d *x + 182*c*d^2*x^2 + 50*d^3*x^3)) + 630*a^5*d*(8*b^2*c^2 + a^2*d^2)*ArcTan h[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqrt[a + b*x])] + 420*a^4*b*c*(8*b^2*c^2 + 7*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(35840* b^(11/2)*Sqrt[x*(a + b*x)])
Time = 0.71 (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.
| \(\displaystyle \int \left (a x+b x^2\right )^{3/2} (c+d x)^3 \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {\int \frac {1}{2} (c+d x) (c (14 b c-5 a d)+9 d (2 b c-a d) x) \left (b x^2+a x\right )^{3/2}dx}{7 b}+\frac {d \left (a x+b x^2\right )^{5/2} (c+d x)^2}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (c+d x) (c (14 b c-5 a d)+9 d (2 b c-a d) x) \left (b x^2+a x\right )^{3/2}dx}{14 b}+\frac {d \left (a x+b x^2\right )^{5/2} (c+d x)^2}{7 b}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {7 (2 b c-a d) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \left (b x^2+a x\right )^{3/2}dx}{8 b^2}+\frac {d \left (a x+b x^2\right )^{5/2} \left (21 a^2 d^2+30 b d x (2 b c-a d)-98 a b c d+128 b^2 c^2\right )}{20 b^2}}{14 b}+\frac {d \left (a x+b x^2\right )^{5/2} (c+d x)^2}{7 b}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {7 (2 b c-a d) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \int \sqrt {b x^2+a x}dx}{16 b}\right )}{8 b^2}+\frac {d \left (a x+b x^2\right )^{5/2} \left (21 a^2 d^2+30 b d x (2 b c-a d)-98 a b c d+128 b^2 c^2\right )}{20 b^2}}{14 b}+\frac {d \left (a x+b x^2\right )^{5/2} (c+d x)^2}{7 b}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {7 (2 b c-a d) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{16 b}\right )}{8 b^2}+\frac {d \left (a x+b x^2\right )^{5/2} \left (21 a^2 d^2+30 b d x (2 b c-a d)-98 a b c d+128 b^2 c^2\right )}{20 b^2}}{14 b}+\frac {d \left (a x+b x^2\right )^{5/2} (c+d x)^2}{7 b}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\frac {7 (2 b c-a d) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{16 b}\right )}{8 b^2}+\frac {d \left (a x+b x^2\right )^{5/2} \left (21 a^2 d^2+30 b d x (2 b c-a d)-98 a b c d+128 b^2 c^2\right )}{20 b^2}}{14 b}+\frac {d \left (a x+b x^2\right )^{5/2} (c+d x)^2}{7 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {7 \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right )}{16 b}\right ) (2 b c-a d) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{8 b^2}+\frac {d \left (a x+b x^2\right )^{5/2} \left (21 a^2 d^2+30 b d x (2 b c-a d)-98 a b c d+128 b^2 c^2\right )}{20 b^2}}{14 b}+\frac {d \left (a x+b x^2\right )^{5/2} (c+d x)^2}{7 b}\) |
Input:
Int[(c + d*x)^3*(a*x + b*x^2)^(3/2),x]
Output:
(d*(c + d*x)^2*(a*x + b*x^2)^(5/2))/(7*b) + ((d*(128*b^2*c^2 - 98*a*b*c*d + 21*a^2*d^2 + 30*b*d*(2*b*c - a*d)*x)*(a*x + b*x^2)^(5/2))/(20*b^2) + (7* (2*b*c - a*d)*(8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*(((a + 2*b*x)*(a*x + b*x ^2)^(3/2))/(8*b) - (3*a^2*(((a + 2*b*x)*Sqrt[a*x + b*x^2])/(4*b) - (a^2*Ar cTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(4*b^(3/2))))/(16*b)))/(8*b^2))/(14* b)
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 = 0.64 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(-\frac {9 \left (\left (a^{2} d^{2}-\frac {8}{3} a b c d +\frac {8}{3} b^{2} c^{2}\right ) \left (a d -2 b c \right ) a^{4} \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\left (\frac {128 \left (\frac {10}{21} d^{3} x^{3}+\frac {26}{15} c \,d^{2} x^{2}+\frac {11}{5} c^{2} d x +c^{3}\right ) x^{2} a \,b^{\frac {11}{2}}}{3}+\frac {256 x^{3} \left (\frac {4}{7} d^{3} x^{3}+2 c \,d^{2} x^{2}+\frac {12}{5} c^{2} d x +c^{3}\right ) b^{\frac {13}{2}}}{9}+\left (-\frac {16 a \left (\frac {3}{35} d^{3} x^{3}+\frac {7}{15} c \,d^{2} x^{2}+c^{2} d x +c^{3}\right ) b^{\frac {7}{2}}}{3}+\frac {32 x \left (\frac {4}{35} d^{3} x^{3}+\frac {3}{5} c \,d^{2} x^{2}+\frac {6}{5} c^{2} d x +c^{3}\right ) b^{\frac {9}{2}}}{9}+d \left (\left (\frac {8}{15} d^{2} x^{2}+\frac {28}{9} c d x +8 c^{2}\right ) b^{\frac {5}{2}}+d \left (\left (-\frac {2 d x}{3}-\frac {14 c}{3}\right ) b^{\frac {3}{2}}+\sqrt {b}\, a d \right ) a \right ) a^{2}\right ) a^{2}\right ) \sqrt {x \left (b x +a \right )}\right )}{1024 b^{\frac {11}{2}}}\) | \(266\) |
| risch | \(\frac {\left (5120 b^{6} d^{3} x^{6}+6400 a \,b^{5} d^{3} x^{5}+17920 b^{6} c \,d^{2} x^{5}+128 a^{2} b^{4} d^{3} x^{4}+23296 a \,b^{5} c \,d^{2} x^{4}+21504 b^{6} c^{2} d \,x^{4}-144 a^{3} b^{3} d^{3} x^{3}+672 a^{2} b^{4} c \,d^{2} x^{3}+29568 a \,b^{5} c^{2} d \,x^{3}+8960 b^{6} c^{3} x^{3}+168 a^{4} b^{2} d^{3} x^{2}-784 a^{3} b^{3} c \,d^{2} x^{2}+1344 a^{2} b^{4} c^{2} d \,x^{2}+13440 a \,b^{5} c^{3} x^{2}-210 a^{5} b \,d^{3} x +980 a^{4} b^{2} c \,d^{2} x -1680 a^{3} b^{3} c^{2} d x +1120 a^{2} b^{4} c^{3} x +315 a^{6} d^{3}-1470 a^{5} b c \,d^{2}+2520 a^{4} b^{2} c^{2} d -1680 a^{3} b^{3} c^{3}\right ) x \left (b x +a \right )}{35840 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {3 a^{4} \left (3 a^{3} d^{3}-14 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2048 b^{\frac {11}{2}}}\) | \(364\) |
| default | \(c^{3} \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )+d^{3} \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 b}-\frac {9 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}\right )+3 c \,d^{2} \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )+3 c^{2} d \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )\) | \(509\) |
Input:
int((d*x+c)^3*(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-9/1024/b^(11/2)*((a^2*d^2-8/3*a*b*c*d+8/3*b^2*c^2)*(a*d-2*b*c)*a^4*arctan h((x*(b*x+a))^(1/2)/x/b^(1/2))-(128/3*(10/21*d^3*x^3+26/15*c*d^2*x^2+11/5* c^2*d*x+c^3)*x^2*a*b^(11/2)+256/9*x^3*(4/7*d^3*x^3+2*c*d^2*x^2+12/5*c^2*d* x+c^3)*b^(13/2)+(-16/3*a*(3/35*d^3*x^3+7/15*c*d^2*x^2+c^2*d*x+c^3)*b^(7/2) +32/9*x*(4/35*d^3*x^3+3/5*c*d^2*x^2+6/5*c^2*d*x+c^3)*b^(9/2)+d*((8/15*d^2* x^2+28/9*c*d*x+8*c^2)*b^(5/2)+d*((-2/3*d*x-14/3*c)*b^(3/2)+b^(1/2)*a*d)*a) *a^2)*a^2)*(x*(b*x+a))^(1/2))
Time = 0.11 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.81 \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (16 \, a^{4} b^{3} c^{3} - 24 \, a^{5} b^{2} c^{2} d + 14 \, a^{6} b c d^{2} - 3 \, a^{7} d^{3}\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (5120 \, b^{7} d^{3} x^{6} - 1680 \, a^{3} b^{4} c^{3} + 2520 \, a^{4} b^{3} c^{2} d - 1470 \, a^{5} b^{2} c d^{2} + 315 \, a^{6} b d^{3} + 1280 \, {\left (14 \, b^{7} c d^{2} + 5 \, a b^{6} d^{3}\right )} x^{5} + 128 \, {\left (168 \, b^{7} c^{2} d + 182 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{4} + 16 \, {\left (560 \, b^{7} c^{3} + 1848 \, a b^{6} c^{2} d + 42 \, a^{2} b^{5} c d^{2} - 9 \, a^{3} b^{4} d^{3}\right )} x^{3} + 56 \, {\left (240 \, a b^{6} c^{3} + 24 \, a^{2} b^{5} c^{2} d - 14 \, a^{3} b^{4} c d^{2} + 3 \, a^{4} b^{3} d^{3}\right )} x^{2} + 70 \, {\left (16 \, a^{2} b^{5} c^{3} - 24 \, a^{3} b^{4} c^{2} d + 14 \, a^{4} b^{3} c d^{2} - 3 \, a^{5} b^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{71680 \, b^{6}}, -\frac {105 \, {\left (16 \, a^{4} b^{3} c^{3} - 24 \, a^{5} b^{2} c^{2} d + 14 \, a^{6} b c d^{2} - 3 \, a^{7} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (5120 \, b^{7} d^{3} x^{6} - 1680 \, a^{3} b^{4} c^{3} + 2520 \, a^{4} b^{3} c^{2} d - 1470 \, a^{5} b^{2} c d^{2} + 315 \, a^{6} b d^{3} + 1280 \, {\left (14 \, b^{7} c d^{2} + 5 \, a b^{6} d^{3}\right )} x^{5} + 128 \, {\left (168 \, b^{7} c^{2} d + 182 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{4} + 16 \, {\left (560 \, b^{7} c^{3} + 1848 \, a b^{6} c^{2} d + 42 \, a^{2} b^{5} c d^{2} - 9 \, a^{3} b^{4} d^{3}\right )} x^{3} + 56 \, {\left (240 \, a b^{6} c^{3} + 24 \, a^{2} b^{5} c^{2} d - 14 \, a^{3} b^{4} c d^{2} + 3 \, a^{4} b^{3} d^{3}\right )} x^{2} + 70 \, {\left (16 \, a^{2} b^{5} c^{3} - 24 \, a^{3} b^{4} c^{2} d + 14 \, a^{4} b^{3} c d^{2} - 3 \, a^{5} b^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{35840 \, b^{6}}\right ] \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
[-1/71680*(105*(16*a^4*b^3*c^3 - 24*a^5*b^2*c^2*d + 14*a^6*b*c*d^2 - 3*a^7 *d^3)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(5120*b^7*d ^3*x^6 - 1680*a^3*b^4*c^3 + 2520*a^4*b^3*c^2*d - 1470*a^5*b^2*c*d^2 + 315* a^6*b*d^3 + 1280*(14*b^7*c*d^2 + 5*a*b^6*d^3)*x^5 + 128*(168*b^7*c^2*d + 1 82*a*b^6*c*d^2 + a^2*b^5*d^3)*x^4 + 16*(560*b^7*c^3 + 1848*a*b^6*c^2*d + 4 2*a^2*b^5*c*d^2 - 9*a^3*b^4*d^3)*x^3 + 56*(240*a*b^6*c^3 + 24*a^2*b^5*c^2* d - 14*a^3*b^4*c*d^2 + 3*a^4*b^3*d^3)*x^2 + 70*(16*a^2*b^5*c^3 - 24*a^3*b^ 4*c^2*d + 14*a^4*b^3*c*d^2 - 3*a^5*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/b^6, -1/ 35840*(105*(16*a^4*b^3*c^3 - 24*a^5*b^2*c^2*d + 14*a^6*b*c*d^2 - 3*a^7*d^3 )*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (5120*b^7*d^3*x^ 6 - 1680*a^3*b^4*c^3 + 2520*a^4*b^3*c^2*d - 1470*a^5*b^2*c*d^2 + 315*a^6*b *d^3 + 1280*(14*b^7*c*d^2 + 5*a*b^6*d^3)*x^5 + 128*(168*b^7*c^2*d + 182*a* b^6*c*d^2 + a^2*b^5*d^3)*x^4 + 16*(560*b^7*c^3 + 1848*a*b^6*c^2*d + 42*a^2 *b^5*c*d^2 - 9*a^3*b^4*d^3)*x^3 + 56*(240*a*b^6*c^3 + 24*a^2*b^5*c^2*d - 1 4*a^3*b^4*c*d^2 + 3*a^4*b^3*d^3)*x^2 + 70*(16*a^2*b^5*c^3 - 24*a^3*b^4*c^2 *d + 14*a^4*b^3*c*d^2 - 3*a^5*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/b^6]
Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (388) = 776\).
Time = 0.51 (sec) , antiderivative size = 864, normalized size of antiderivative = 2.21 \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)**3*(b*x**2+a*x)**(3/2),x)
Output:
Piecewise((3*a**2*(a**2*c**3 - 5*a*(3*a**2*c**2*d + 2*a*b*c**3 - 7*a*(3*a* *2*c*d**2 + 6*a*b*c**2*d - 9*a*(a**2*d**3 + 6*a*b*c*d**2 - 11*a*(15*a*b*d* *3/14 + 3*b**2*c*d**2)/(12*b) + 3*b**2*c**2*d)/(10*b) + b**2*c**3)/(8*b))/ (6*b))*Piecewise((log(a + 2*sqrt(b)*sqrt(a*x + b*x**2) + 2*b*x)/sqrt(b), N e(a**2/b, 0)), ((a/(2*b) + x)*log(a/(2*b) + x)/sqrt(b*(a/(2*b) + x)**2), T rue))/(8*b**2) + sqrt(a*x + b*x**2)*(-3*a*(a**2*c**3 - 5*a*(3*a**2*c**2*d + 2*a*b*c**3 - 7*a*(3*a**2*c*d**2 + 6*a*b*c**2*d - 9*a*(a**2*d**3 + 6*a*b* c*d**2 - 11*a*(15*a*b*d**3/14 + 3*b**2*c*d**2)/(12*b) + 3*b**2*c**2*d)/(10 *b) + b**2*c**3)/(8*b))/(6*b))/(4*b**2) + b*d**3*x**6/7 + x**5*(15*a*b*d** 3/14 + 3*b**2*c*d**2)/(6*b) + x**4*(a**2*d**3 + 6*a*b*c*d**2 - 11*a*(15*a* b*d**3/14 + 3*b**2*c*d**2)/(12*b) + 3*b**2*c**2*d)/(5*b) + x**3*(3*a**2*c* d**2 + 6*a*b*c**2*d - 9*a*(a**2*d**3 + 6*a*b*c*d**2 - 11*a*(15*a*b*d**3/14 + 3*b**2*c*d**2)/(12*b) + 3*b**2*c**2*d)/(10*b) + b**2*c**3)/(4*b) + x**2 *(3*a**2*c**2*d + 2*a*b*c**3 - 7*a*(3*a**2*c*d**2 + 6*a*b*c**2*d - 9*a*(a* *2*d**3 + 6*a*b*c*d**2 - 11*a*(15*a*b*d**3/14 + 3*b**2*c*d**2)/(12*b) + 3* b**2*c**2*d)/(10*b) + b**2*c**3)/(8*b))/(3*b) + x*(a**2*c**3 - 5*a*(3*a**2 *c**2*d + 2*a*b*c**3 - 7*a*(3*a**2*c*d**2 + 6*a*b*c**2*d - 9*a*(a**2*d**3 + 6*a*b*c*d**2 - 11*a*(15*a*b*d**3/14 + 3*b**2*c*d**2)/(12*b) + 3*b**2*c** 2*d)/(10*b) + b**2*c**3)/(8*b))/(6*b))/(2*b)), Ne(b, 0)), (2*(c**3*(a*x)** (5/2)/5 + 3*c**2*d*(a*x)**(7/2)/(7*a) + c*d**2*(a*x)**(9/2)/(3*a**2) + ...
Time = 0.04 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.60 \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
1/7*(b*x^2 + a*x)^(5/2)*d^3*x^2/b + 1/4*(b*x^2 + a*x)^(3/2)*c^3*x - 3/32*s qrt(b*x^2 + a*x)*a^2*c^3*x/b + 9/64*sqrt(b*x^2 + a*x)*a^3*c^2*d*x/b^2 - 3/ 8*(b*x^2 + a*x)^(3/2)*a*c^2*d*x/b - 21/256*sqrt(b*x^2 + a*x)*a^4*c*d^2*x/b ^3 + 7/32*(b*x^2 + a*x)^(3/2)*a^2*c*d^2*x/b^2 + 1/2*(b*x^2 + a*x)^(5/2)*c* d^2*x/b + 9/512*sqrt(b*x^2 + a*x)*a^5*d^3*x/b^4 - 3/64*(b*x^2 + a*x)^(3/2) *a^3*d^3*x/b^3 - 3/28*(b*x^2 + a*x)^(5/2)*a*d^3*x/b^2 + 3/128*a^4*c^3*log( 2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(5/2) - 9/256*a^5*c^2*d*log(2*b *x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 21/1024*a^6*c*d^2*log(2*b* x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2) - 9/2048*a^7*d^3*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) - 3/64*sqrt(b*x^2 + a*x)*a^3*c^3 /b^2 + 1/8*(b*x^2 + a*x)^(3/2)*a*c^3/b + 9/128*sqrt(b*x^2 + a*x)*a^4*c^2*d /b^3 - 3/16*(b*x^2 + a*x)^(3/2)*a^2*c^2*d/b^2 + 3/5*(b*x^2 + a*x)^(5/2)*c^ 2*d/b - 21/512*sqrt(b*x^2 + a*x)*a^5*c*d^2/b^4 + 7/64*(b*x^2 + a*x)^(3/2)* a^3*c*d^2/b^3 - 7/20*(b*x^2 + a*x)^(5/2)*a*c*d^2/b^2 + 9/1024*sqrt(b*x^2 + a*x)*a^6*d^3/b^5 - 3/128*(b*x^2 + a*x)^(3/2)*a^4*d^3/b^4 + 3/40*(b*x^2 + a*x)^(5/2)*a^2*d^3/b^3
Time = 0.14 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.95 \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\frac {1}{35840} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, b d^{3} x + \frac {14 \, b^{7} c d^{2} + 5 \, a b^{6} d^{3}}{b^{6}}\right )} x + \frac {168 \, b^{7} c^{2} d + 182 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}}{b^{6}}\right )} x + \frac {560 \, b^{7} c^{3} + 1848 \, a b^{6} c^{2} d + 42 \, a^{2} b^{5} c d^{2} - 9 \, a^{3} b^{4} d^{3}}{b^{6}}\right )} x + \frac {7 \, {\left (240 \, a b^{6} c^{3} + 24 \, a^{2} b^{5} c^{2} d - 14 \, a^{3} b^{4} c d^{2} + 3 \, a^{4} b^{3} d^{3}\right )}}{b^{6}}\right )} x + \frac {35 \, {\left (16 \, a^{2} b^{5} c^{3} - 24 \, a^{3} b^{4} c^{2} d + 14 \, a^{4} b^{3} c d^{2} - 3 \, a^{5} b^{2} d^{3}\right )}}{b^{6}}\right )} x - \frac {105 \, {\left (16 \, a^{3} b^{4} c^{3} - 24 \, a^{4} b^{3} c^{2} d + 14 \, a^{5} b^{2} c d^{2} - 3 \, a^{6} b d^{3}\right )}}{b^{6}}\right )} - \frac {3 \, {\left (16 \, a^{4} b^{3} c^{3} - 24 \, a^{5} b^{2} c^{2} d + 14 \, a^{6} b c d^{2} - 3 \, a^{7} d^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{2048 \, b^{\frac {11}{2}}} \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
1/35840*sqrt(b*x^2 + a*x)*(2*(4*(2*(8*(10*(4*b*d^3*x + (14*b^7*c*d^2 + 5*a *b^6*d^3)/b^6)*x + (168*b^7*c^2*d + 182*a*b^6*c*d^2 + a^2*b^5*d^3)/b^6)*x + (560*b^7*c^3 + 1848*a*b^6*c^2*d + 42*a^2*b^5*c*d^2 - 9*a^3*b^4*d^3)/b^6) *x + 7*(240*a*b^6*c^3 + 24*a^2*b^5*c^2*d - 14*a^3*b^4*c*d^2 + 3*a^4*b^3*d^ 3)/b^6)*x + 35*(16*a^2*b^5*c^3 - 24*a^3*b^4*c^2*d + 14*a^4*b^3*c*d^2 - 3*a ^5*b^2*d^3)/b^6)*x - 105*(16*a^3*b^4*c^3 - 24*a^4*b^3*c^2*d + 14*a^5*b^2*c *d^2 - 3*a^6*b*d^3)/b^6) - 3/2048*(16*a^4*b^3*c^3 - 24*a^5*b^2*c^2*d + 14* a^6*b*c*d^2 - 3*a^7*d^3)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(11/2)
Timed out. \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\int {\left (b\,x^2+a\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^3 \,d x \] Input:
int((a*x + b*x^2)^(3/2)*(c + d*x)^3,x)
Output:
int((a*x + b*x^2)^(3/2)*(c + d*x)^3, x)
Time = 26.38 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.48 \[ \int (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\frac {-1470 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c \,d^{2}-210 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} d^{3} x +2520 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c^{2} d +168 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} d^{3} x^{2}-144 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} d^{3} x^{3}+1120 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c^{3} x +128 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} d^{3} x^{4}+13440 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c^{3} x^{2}+6400 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} d^{3} x^{5}+21504 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c^{2} d \,x^{4}+17920 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c \,d^{2} x^{5}+1470 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6} b c \,d^{2}-2520 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5} b^{2} c^{2} d -315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{7} d^{3}+980 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c \,d^{2} x -1680 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c^{2} d x -784 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c \,d^{2} x^{2}+1344 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c^{2} d \,x^{2}+672 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c \,d^{2} x^{3}+29568 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c^{2} d \,x^{3}+23296 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c \,d^{2} x^{4}+315 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b \,d^{3}-1680 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c^{3}+8960 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c^{3} x^{3}+5120 \sqrt {x}\, \sqrt {b x +a}\, b^{7} d^{3} x^{6}+1680 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{4} b^{3} c^{3}}{35840 b^{6}} \] Input:
int((d*x+c)^3*(b*x^2+a*x)^(3/2),x)
Output:
(315*sqrt(x)*sqrt(a + b*x)*a**6*b*d**3 - 1470*sqrt(x)*sqrt(a + b*x)*a**5*b **2*c*d**2 - 210*sqrt(x)*sqrt(a + b*x)*a**5*b**2*d**3*x + 2520*sqrt(x)*sqr t(a + b*x)*a**4*b**3*c**2*d + 980*sqrt(x)*sqrt(a + b*x)*a**4*b**3*c*d**2*x + 168*sqrt(x)*sqrt(a + b*x)*a**4*b**3*d**3*x**2 - 1680*sqrt(x)*sqrt(a + b *x)*a**3*b**4*c**3 - 1680*sqrt(x)*sqrt(a + b*x)*a**3*b**4*c**2*d*x - 784*s qrt(x)*sqrt(a + b*x)*a**3*b**4*c*d**2*x**2 - 144*sqrt(x)*sqrt(a + b*x)*a** 3*b**4*d**3*x**3 + 1120*sqrt(x)*sqrt(a + b*x)*a**2*b**5*c**3*x + 1344*sqrt (x)*sqrt(a + b*x)*a**2*b**5*c**2*d*x**2 + 672*sqrt(x)*sqrt(a + b*x)*a**2*b **5*c*d**2*x**3 + 128*sqrt(x)*sqrt(a + b*x)*a**2*b**5*d**3*x**4 + 13440*sq rt(x)*sqrt(a + b*x)*a*b**6*c**3*x**2 + 29568*sqrt(x)*sqrt(a + b*x)*a*b**6* c**2*d*x**3 + 23296*sqrt(x)*sqrt(a + b*x)*a*b**6*c*d**2*x**4 + 6400*sqrt(x )*sqrt(a + b*x)*a*b**6*d**3*x**5 + 8960*sqrt(x)*sqrt(a + b*x)*b**7*c**3*x* *3 + 21504*sqrt(x)*sqrt(a + b*x)*b**7*c**2*d*x**4 + 17920*sqrt(x)*sqrt(a + b*x)*b**7*c*d**2*x**5 + 5120*sqrt(x)*sqrt(a + b*x)*b**7*d**3*x**6 - 315*s qrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**7*d**3 + 1470*sqr t(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**6*b*c*d**2 - 2520*s qrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**5*b**2*c**2*d + 1 680*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**4*b**3*c**3) /(35840*b**6)