Integrand size = 24, antiderivative size = 167 \[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=-\frac {2 a^3 (b c-a d) \left (a x^2+b x^3\right )^{3/2}}{3 b^5 x^3}+\frac {2 a^2 (3 b c-4 a d) \left (a x^2+b x^3\right )^{5/2}}{5 b^5 x^5}-\frac {6 a (b c-2 a d) \left (a x^2+b x^3\right )^{7/2}}{7 b^5 x^7}+\frac {2 (b c-4 a d) \left (a x^2+b x^3\right )^{9/2}}{9 b^5 x^9}+\frac {2 d \left (a x^2+b x^3\right )^{11/2}}{11 b^5 x^{11}} \] Output:
-2/3*a^3*(-a*d+b*c)*(b*x^3+a*x^2)^(3/2)/b^5/x^3+2/5*a^2*(-4*a*d+3*b*c)*(b* x^3+a*x^2)^(5/2)/b^5/x^5-6/7*a*(-2*a*d+b*c)*(b*x^3+a*x^2)^(7/2)/b^5/x^7+2/ 9*(-4*a*d+b*c)*(b*x^3+a*x^2)^(9/2)/b^5/x^9+2/11*d*(b*x^3+a*x^2)^(11/2)/b^5 /x^11
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.56 \[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=\frac {2 \left (x^2 (a+b x)\right )^{3/2} \left (128 a^4 d+35 b^4 x^3 (11 c+9 d x)+24 a^2 b^2 x (11 c+10 d x)-16 a^3 b (11 c+12 d x)-10 a b^3 x^2 (33 c+28 d x)\right )}{3465 b^5 x^3} \] Input:
Integrate[x^2*(c + d*x)*Sqrt[a*x^2 + b*x^3],x]
Output:
(2*(x^2*(a + b*x))^(3/2)*(128*a^4*d + 35*b^4*x^3*(11*c + 9*d*x) + 24*a^2*b ^2*x*(11*c + 10*d*x) - 16*a^3*b*(11*c + 12*d*x) - 10*a*b^3*x^2*(33*c + 28* d*x)))/(3465*b^5*x^3)
Time = 0.61 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1945, 1922, 1922, 1908, 1920}
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.
| \(\displaystyle \int x^2 \sqrt {a x^2+b x^3} (c+d x) \, dx\) |
| \(\Big \downarrow \) 1945 |
| \(\displaystyle \frac {(11 b c-8 a d) \int x^2 \sqrt {b x^3+a x^2}dx}{11 b}+\frac {2 d x \left (a x^2+b x^3\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(11 b c-8 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {2 a \int x \sqrt {b x^3+a x^2}dx}{3 b}\right )}{11 b}+\frac {2 d x \left (a x^2+b x^3\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(11 b c-8 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{7 b x}-\frac {4 a \int \sqrt {b x^3+a x^2}dx}{7 b}\right )}{3 b}\right )}{11 b}+\frac {2 d x \left (a x^2+b x^3\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {(11 b c-8 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{7 b x}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{5 b x^2}-\frac {2 a \int \frac {\sqrt {b x^3+a x^2}}{x}dx}{5 b}\right )}{7 b}\right )}{3 b}\right )}{11 b}+\frac {2 d x \left (a x^2+b x^3\right )^{3/2}}{11 b}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {\left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{9 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{7 b x}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{3/2}}{5 b x^2}-\frac {4 a \left (a x^2+b x^3\right )^{3/2}}{15 b^2 x^3}\right )}{7 b}\right )}{3 b}\right ) (11 b c-8 a d)}{11 b}+\frac {2 d x \left (a x^2+b x^3\right )^{3/2}}{11 b}\) |
Input:
Int[x^2*(c + d*x)*Sqrt[a*x^2 + b*x^3],x]
Output:
(2*d*x*(a*x^2 + b*x^3)^(3/2))/(11*b) + ((11*b*c - 8*a*d)*((2*(a*x^2 + b*x^ 3)^(3/2))/(9*b) - (2*a*((2*(a*x^2 + b*x^3)^(3/2))/(7*b*x) - (4*a*((-4*a*(a *x^2 + b*x^3)^(3/2))/(15*b^2*x^3) + (2*(a*x^2 + b*x^3)^(3/2))/(5*b*x^2)))/ (7*b)))/(3*b)))/(11*b)
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j - 1)), x] - Simp[b*((n*p + n - j + 1)/(a*( j*p + 1))) Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[ n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1) /(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)) Int[(e* x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p *(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
Time = 1.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.34
| method | result | size |
| pseudoelliptic | \(-\frac {32 \left (b x +a \right )^{\frac {3}{2}} \left (-\frac {45 x^{2} \left (c +\frac {7 d x}{9}\right ) b^{3}}{16}+\frac {9 x a \left (\frac {5 d x}{6}+c \right ) b^{2}}{4}-\frac {3 a^{2} \left (d x +c \right ) b}{2}+a^{3} d \right )}{315 b^{4}}\) | \(57\) |
| gosper | \(\frac {2 \left (b x +a \right ) \left (315 d \,x^{4} b^{4}-280 a \,b^{3} d \,x^{3}+385 b^{4} c \,x^{3}+240 a^{2} b^{2} d \,x^{2}-330 a \,b^{3} c \,x^{2}-192 a^{3} b d x +264 a^{2} b^{2} c x +128 a^{4} d -176 a^{3} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{3465 b^{5} x}\) | \(109\) |
| default | \(\frac {2 \left (b x +a \right ) \left (315 d \,x^{4} b^{4}-280 a \,b^{3} d \,x^{3}+385 b^{4} c \,x^{3}+240 a^{2} b^{2} d \,x^{2}-330 a \,b^{3} c \,x^{2}-192 a^{3} b d x +264 a^{2} b^{2} c x +128 a^{4} d -176 a^{3} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{3465 b^{5} x}\) | \(109\) |
| orering | \(\frac {2 \left (b x +a \right ) \left (315 d \,x^{4} b^{4}-280 a \,b^{3} d \,x^{3}+385 b^{4} c \,x^{3}+240 a^{2} b^{2} d \,x^{2}-330 a \,b^{3} c \,x^{2}-192 a^{3} b d x +264 a^{2} b^{2} c x +128 a^{4} d -176 a^{3} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{3465 b^{5} x}\) | \(109\) |
| risch | \(\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (315 b^{5} d \,x^{5}+35 a \,b^{4} d \,x^{4}+385 b^{5} c \,x^{4}-40 a^{2} b^{3} d \,x^{3}+55 a \,b^{4} c \,x^{3}+48 a^{3} b^{2} d \,x^{2}-66 a^{2} b^{3} c \,x^{2}-64 a^{4} b d x +88 a^{3} b^{2} c x +128 a^{5} d -176 a^{4} b c \right )}{3465 x \,b^{5}}\) | \(126\) |
| trager | \(\frac {2 \left (315 b^{5} d \,x^{5}+35 a \,b^{4} d \,x^{4}+385 b^{5} c \,x^{4}-40 a^{2} b^{3} d \,x^{3}+55 a \,b^{4} c \,x^{3}+48 a^{3} b^{2} d \,x^{2}-66 a^{2} b^{3} c \,x^{2}-64 a^{4} b d x +88 a^{3} b^{2} c x +128 a^{5} d -176 a^{4} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{3465 b^{5} x}\) | \(128\) |
Input:
int(x^2*(d*x+c)*(b*x^3+a*x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-32/315*(b*x+a)^(3/2)*(-45/16*x^2*(c+7/9*d*x)*b^3+9/4*x*a*(5/6*d*x+c)*b^2- 3/2*a^2*(d*x+c)*b+a^3*d)/b^4
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.77 \[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=\frac {2 \, {\left (315 \, b^{5} d x^{5} - 176 \, a^{4} b c + 128 \, a^{5} d + 35 \, {\left (11 \, b^{5} c + a b^{4} d\right )} x^{4} + 5 \, {\left (11 \, a b^{4} c - 8 \, a^{2} b^{3} d\right )} x^{3} - 6 \, {\left (11 \, a^{2} b^{3} c - 8 \, a^{3} b^{2} d\right )} x^{2} + 8 \, {\left (11 \, a^{3} b^{2} c - 8 \, a^{4} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{3465 \, b^{5} x} \] Input:
integrate(x^2*(d*x+c)*(b*x^3+a*x^2)^(1/2),x, algorithm="fricas")
Output:
2/3465*(315*b^5*d*x^5 - 176*a^4*b*c + 128*a^5*d + 35*(11*b^5*c + a*b^4*d)* x^4 + 5*(11*a*b^4*c - 8*a^2*b^3*d)*x^3 - 6*(11*a^2*b^3*c - 8*a^3*b^2*d)*x^ 2 + 8*(11*a^3*b^2*c - 8*a^4*b*d)*x)*sqrt(b*x^3 + a*x^2)/(b^5*x)
\[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=\int x^{2} \sqrt {x^{2} \left (a + b x\right )} \left (c + d x\right )\, dx \] Input:
integrate(x**2*(d*x+c)*(b*x**3+a*x**2)**(1/2),x)
Output:
Integral(x**2*sqrt(x**2*(a + b*x))*(c + d*x), x)
Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.72 \[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=\frac {2 \, {\left (35 \, b^{4} x^{4} + 5 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x + a} c}{315 \, b^{4}} + \frac {2 \, {\left (315 \, b^{5} x^{5} + 35 \, a b^{4} x^{4} - 40 \, a^{2} b^{3} x^{3} + 48 \, a^{3} b^{2} x^{2} - 64 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x + a} d}{3465 \, b^{5}} \] Input:
integrate(x^2*(d*x+c)*(b*x^3+a*x^2)^(1/2),x, algorithm="maxima")
Output:
2/315*(35*b^4*x^4 + 5*a*b^3*x^3 - 6*a^2*b^2*x^2 + 8*a^3*b*x - 16*a^4)*sqrt (b*x + a)*c/b^4 + 2/3465*(315*b^5*x^5 + 35*a*b^4*x^4 - 40*a^2*b^3*x^3 + 48 *a^3*b^2*x^2 - 64*a^4*b*x + 128*a^5)*sqrt(b*x + a)*d/b^5
Time = 0.11 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.71 \[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=\frac {2 \, {\left (\frac {99 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a c \mathrm {sgn}\left (x\right )}{b^{3}} + \frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} c \mathrm {sgn}\left (x\right )}{b^{3}} + \frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a d \mathrm {sgn}\left (x\right )}{b^{4}} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} d \mathrm {sgn}\left (x\right )}{b^{4}}\right )}}{3465 \, b} + \frac {32 \, {\left (11 \, a^{\frac {9}{2}} b c - 8 \, a^{\frac {11}{2}} d\right )} \mathrm {sgn}\left (x\right )}{3465 \, b^{5}} \] Input:
integrate(x^2*(d*x+c)*(b*x^3+a*x^2)^(1/2),x, algorithm="giac")
Output:
2/3465*(99*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)* a^2 - 35*sqrt(b*x + a)*a^3)*a*c*sgn(x)/b^3 + 11*(35*(b*x + a)^(9/2) - 180* (b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 31 5*sqrt(b*x + a)*a^4)*c*sgn(x)/b^3 + 11*(35*(b*x + a)^(9/2) - 180*(b*x + a) ^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b* x + a)*a^4)*a*d*sgn(x)/b^4 + 5*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)* a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3 /2)*a^4 - 693*sqrt(b*x + a)*a^5)*d*sgn(x)/b^4)/b + 32/3465*(11*a^(9/2)*b*c - 8*a^(11/2)*d)*sgn(x)/b^5
Time = 8.82 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.72 \[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {2\,d\,x^5}{11}+\frac {256\,a^5\,d-352\,a^4\,b\,c}{3465\,b^5}+\frac {x^4\,\left (770\,c\,b^5+70\,a\,d\,b^4\right )}{3465\,b^5}-\frac {2\,a\,x^3\,\left (8\,a\,d-11\,b\,c\right )}{693\,b^2}-\frac {16\,a^3\,x\,\left (8\,a\,d-11\,b\,c\right )}{3465\,b^4}+\frac {4\,a^2\,x^2\,\left (8\,a\,d-11\,b\,c\right )}{1155\,b^3}\right )}{x} \] Input:
int(x^2*(a*x^2 + b*x^3)^(1/2)*(c + d*x),x)
Output:
((a*x^2 + b*x^3)^(1/2)*((2*d*x^5)/11 + (256*a^5*d - 352*a^4*b*c)/(3465*b^5 ) + (x^4*(770*b^5*c + 70*a*b^4*d))/(3465*b^5) - (2*a*x^3*(8*a*d - 11*b*c)) /(693*b^2) - (16*a^3*x*(8*a*d - 11*b*c))/(3465*b^4) + (4*a^2*x^2*(8*a*d - 11*b*c))/(1155*b^3)))/x
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.70 \[ \int x^2 (c+d x) \sqrt {a x^2+b x^3} \, dx=\frac {2 \sqrt {b x +a}\, \left (315 b^{5} d \,x^{5}+35 a \,b^{4} d \,x^{4}+385 b^{5} c \,x^{4}-40 a^{2} b^{3} d \,x^{3}+55 a \,b^{4} c \,x^{3}+48 a^{3} b^{2} d \,x^{2}-66 a^{2} b^{3} c \,x^{2}-64 a^{4} b d x +88 a^{3} b^{2} c x +128 a^{5} d -176 a^{4} b c \right )}{3465 b^{5}} \] Input:
int(x^2*(d*x+c)*(b*x^3+a*x^2)^(1/2),x)
Output:
(2*sqrt(a + b*x)*(128*a**5*d - 176*a**4*b*c - 64*a**4*b*d*x + 88*a**3*b**2 *c*x + 48*a**3*b**2*d*x**2 - 66*a**2*b**3*c*x**2 - 40*a**2*b**3*d*x**3 + 5 5*a*b**4*c*x**3 + 35*a*b**4*d*x**4 + 385*b**5*c*x**4 + 315*b**5*d*x**5))/( 3465*b**5)