Integrand size = 24, antiderivative size = 240 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=-\frac {2 a^5 (b c-a d) \left (a x^2+b x^3\right )^{5/2}}{5 b^7 x^5}+\frac {2 a^4 (5 b c-6 a d) \left (a x^2+b x^3\right )^{7/2}}{7 b^7 x^7}-\frac {10 a^3 (2 b c-3 a d) \left (a x^2+b x^3\right )^{9/2}}{9 b^7 x^9}+\frac {20 a^2 (b c-2 a d) \left (a x^2+b x^3\right )^{11/2}}{11 b^7 x^{11}}-\frac {10 a (b c-3 a d) \left (a x^2+b x^3\right )^{13/2}}{13 b^7 x^{13}}+\frac {2 (b c-6 a d) \left (a x^2+b x^3\right )^{15/2}}{15 b^7 x^{15}}+\frac {2 d \left (a x^2+b x^3\right )^{17/2}}{17 b^7 x^{17}} \] Output:
-2/5*a^5*(-a*d+b*c)*(b*x^3+a*x^2)^(5/2)/b^7/x^5+2/7*a^4*(-6*a*d+5*b*c)*(b* x^3+a*x^2)^(7/2)/b^7/x^7-10/9*a^3*(-3*a*d+2*b*c)*(b*x^3+a*x^2)^(9/2)/b^7/x ^9+20/11*a^2*(-2*a*d+b*c)*(b*x^3+a*x^2)^(11/2)/b^7/x^11-10/13*a*(-3*a*d+b* c)*(b*x^3+a*x^2)^(13/2)/b^7/x^13+2/15*(-6*a*d+b*c)*(b*x^3+a*x^2)^(15/2)/b^ 7/x^15+2/17*d*(b*x^3+a*x^2)^(17/2)/b^7/x^17
Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.57 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 x (a+b x)^3 \left (3072 a^6 d+3003 b^6 x^5 (17 c+15 d x)-1120 a^3 b^3 x^2 (17 c+18 d x)+640 a^4 b^2 x (17 c+21 d x)-256 a^5 b (17 c+30 d x)+840 a^2 b^4 x^3 (34 c+33 d x)-462 a b^5 x^4 (85 c+78 d x)\right )}{765765 b^7 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[x^2*(c + d*x)*(a*x^2 + b*x^3)^(3/2),x]
Output:
(2*x*(a + b*x)^3*(3072*a^6*d + 3003*b^6*x^5*(17*c + 15*d*x) - 1120*a^3*b^3 *x^2*(17*c + 18*d*x) + 640*a^4*b^2*x*(17*c + 21*d*x) - 256*a^5*b*(17*c + 3 0*d*x) + 840*a^2*b^4*x^3*(34*c + 33*d*x) - 462*a*b^5*x^4*(85*c + 78*d*x))) /(765765*b^7*Sqrt[x^2*(a + b*x)])
Time = 0.79 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1945, 1922, 1922, 1908, 1922, 1922, 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.
| \(\displaystyle \int x^2 \left (a x^2+b x^3\right )^{3/2} (c+d x) \, dx\) |
| \(\Big \downarrow \) 1945 |
| \(\displaystyle \frac {(17 b c-12 a d) \int x^2 \left (b x^3+a x^2\right )^{3/2}dx}{17 b}+\frac {2 d x \left (a x^2+b x^3\right )^{5/2}}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-12 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac {2 a \int x \left (b x^3+a x^2\right )^{3/2}dx}{3 b}\right )}{17 b}+\frac {2 d x \left (a x^2+b x^3\right )^{5/2}}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-12 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}-\frac {8 a \int \left (b x^3+a x^2\right )^{3/2}dx}{13 b}\right )}{3 b}\right )}{17 b}+\frac {2 d x \left (a x^2+b x^3\right )^{5/2}}{17 b}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {(17 b c-12 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}-\frac {6 a \int \frac {\left (b x^3+a x^2\right )^{3/2}}{x}dx}{11 b}\right )}{13 b}\right )}{3 b}\right )}{17 b}+\frac {2 d x \left (a x^2+b x^3\right )^{5/2}}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-12 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3}-\frac {4 a \int \frac {\left (b x^3+a x^2\right )^{3/2}}{x^2}dx}{9 b}\right )}{11 b}\right )}{13 b}\right )}{3 b}\right )}{17 b}+\frac {2 d x \left (a x^2+b x^3\right )^{5/2}}{17 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-12 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{7 b x^4}-\frac {2 a \int \frac {\left (b x^3+a x^2\right )^{3/2}}{x^3}dx}{7 b}\right )}{9 b}\right )}{11 b}\right )}{13 b}\right )}{3 b}\right )}{17 b}+\frac {2 d x \left (a x^2+b x^3\right )^{5/2}}{17 b}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {\left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{9 b x^3}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{5/2}}{7 b x^4}-\frac {4 a \left (a x^2+b x^3\right )^{5/2}}{35 b^2 x^5}\right )}{9 b}\right )}{11 b}\right )}{13 b}\right )}{3 b}\right ) (17 b c-12 a d)}{17 b}+\frac {2 d x \left (a x^2+b x^3\right )^{5/2}}{17 b}\) |
Input:
Int[x^2*(c + d*x)*(a*x^2 + b*x^3)^(3/2),x]
Output:
(2*d*x*(a*x^2 + b*x^3)^(5/2))/(17*b) + ((17*b*c - 12*a*d)*((2*(a*x^2 + b*x ^3)^(5/2))/(15*b) - (2*a*((2*(a*x^2 + b*x^3)^(5/2))/(13*b*x) - (8*a*((2*(a *x^2 + b*x^3)^(5/2))/(11*b*x^2) - (6*a*((2*(a*x^2 + b*x^3)^(5/2))/(9*b*x^3 ) - (4*a*((-4*a*(a*x^2 + b*x^3)^(5/2))/(35*b^2*x^5) + (2*(a*x^2 + b*x^3)^( 5/2))/(7*b*x^4)))/(9*b)))/(11*b)))/(13*b)))/(3*b)))/(17*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 = 0.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.24
| method | result | size |
| pseudoelliptic | \(-\frac {32 \left (-\frac {385 \left (\frac {9 d x}{11}+c \right ) x^{2} b^{3}}{48}+\frac {55 x \left (\frac {21 d x}{22}+c \right ) a \,b^{2}}{12}-\frac {11 \left (\frac {15 d x}{11}+c \right ) a^{2} b}{6}+a^{3} d \right ) \left (b x +a \right )^{\frac {5}{2}}}{1155 b^{4}}\) | \(58\) |
| gosper | \(\frac {2 \left (b x +a \right ) \left (45045 d \,x^{6} b^{6}-36036 a \,b^{5} d \,x^{5}+51051 b^{6} c \,x^{5}+27720 a^{2} b^{4} d \,x^{4}-39270 a \,b^{5} c \,x^{4}-20160 a^{3} b^{3} d \,x^{3}+28560 a^{2} b^{4} c \,x^{3}+13440 a^{4} b^{2} d \,x^{2}-19040 a^{3} b^{3} c \,x^{2}-7680 a^{5} b d x +10880 a^{4} b^{2} c x +3072 a^{6} d -4352 a^{5} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{765765 b^{7} x^{3}}\) | \(157\) |
| default | \(\frac {2 \left (b x +a \right ) \left (45045 d \,x^{6} b^{6}-36036 a \,b^{5} d \,x^{5}+51051 b^{6} c \,x^{5}+27720 a^{2} b^{4} d \,x^{4}-39270 a \,b^{5} c \,x^{4}-20160 a^{3} b^{3} d \,x^{3}+28560 a^{2} b^{4} c \,x^{3}+13440 a^{4} b^{2} d \,x^{2}-19040 a^{3} b^{3} c \,x^{2}-7680 a^{5} b d x +10880 a^{4} b^{2} c x +3072 a^{6} d -4352 a^{5} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{765765 b^{7} x^{3}}\) | \(157\) |
| orering | \(\frac {2 \left (b x +a \right ) \left (45045 d \,x^{6} b^{6}-36036 a \,b^{5} d \,x^{5}+51051 b^{6} c \,x^{5}+27720 a^{2} b^{4} d \,x^{4}-39270 a \,b^{5} c \,x^{4}-20160 a^{3} b^{3} d \,x^{3}+28560 a^{2} b^{4} c \,x^{3}+13440 a^{4} b^{2} d \,x^{2}-19040 a^{3} b^{3} c \,x^{2}-7680 a^{5} b d x +10880 a^{4} b^{2} c x +3072 a^{6} d -4352 a^{5} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{765765 b^{7} x^{3}}\) | \(157\) |
| risch | \(\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (45045 b^{8} d \,x^{8}+54054 a \,b^{7} d \,x^{7}+51051 b^{8} c \,x^{7}+693 a^{2} b^{6} d \,x^{6}+62832 a \,b^{7} c \,x^{6}-756 a^{3} b^{5} d \,x^{5}+1071 a^{2} b^{6} c \,x^{5}+840 a^{4} b^{4} d \,x^{4}-1190 a^{3} b^{5} c \,x^{4}-960 a^{5} b^{3} d \,x^{3}+1360 a^{4} b^{4} c \,x^{3}+1152 a^{6} b^{2} d \,x^{2}-1632 a^{5} b^{3} c \,x^{2}-1536 a^{7} b d x +2176 a^{6} b^{2} c x +3072 a^{8} d -4352 a^{7} b c \right )}{765765 x \,b^{7}}\) | \(198\) |
| trager | \(\frac {2 \left (45045 b^{8} d \,x^{8}+54054 a \,b^{7} d \,x^{7}+51051 b^{8} c \,x^{7}+693 a^{2} b^{6} d \,x^{6}+62832 a \,b^{7} c \,x^{6}-756 a^{3} b^{5} d \,x^{5}+1071 a^{2} b^{6} c \,x^{5}+840 a^{4} b^{4} d \,x^{4}-1190 a^{3} b^{5} c \,x^{4}-960 a^{5} b^{3} d \,x^{3}+1360 a^{4} b^{4} c \,x^{3}+1152 a^{6} b^{2} d \,x^{2}-1632 a^{5} b^{3} c \,x^{2}-1536 a^{7} b d x +2176 a^{6} b^{2} c x +3072 a^{8} d -4352 a^{7} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{765765 b^{7} x}\) | \(200\) |
Input:
int(x^2*(d*x+c)*(b*x^3+a*x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-32/1155*(-385/48*(9/11*d*x+c)*x^2*b^3+55/12*x*(21/22*d*x+c)*a*b^2-11/6*(1 5/11*d*x+c)*a^2*b+a^3*d)*(b*x+a)^(5/2)/b^4
Time = 0.08 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.84 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 \, {\left (45045 \, b^{8} d x^{8} - 4352 \, a^{7} b c + 3072 \, a^{8} d + 3003 \, {\left (17 \, b^{8} c + 18 \, a b^{7} d\right )} x^{7} + 231 \, {\left (272 \, a b^{7} c + 3 \, a^{2} b^{6} d\right )} x^{6} + 63 \, {\left (17 \, a^{2} b^{6} c - 12 \, a^{3} b^{5} d\right )} x^{5} - 70 \, {\left (17 \, a^{3} b^{5} c - 12 \, a^{4} b^{4} d\right )} x^{4} + 80 \, {\left (17 \, a^{4} b^{4} c - 12 \, a^{5} b^{3} d\right )} x^{3} - 96 \, {\left (17 \, a^{5} b^{3} c - 12 \, a^{6} b^{2} d\right )} x^{2} + 128 \, {\left (17 \, a^{6} b^{2} c - 12 \, a^{7} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{765765 \, b^{7} x} \] Input:
integrate(x^2*(d*x+c)*(b*x^3+a*x^2)^(3/2),x, algorithm="fricas")
Output:
2/765765*(45045*b^8*d*x^8 - 4352*a^7*b*c + 3072*a^8*d + 3003*(17*b^8*c + 1 8*a*b^7*d)*x^7 + 231*(272*a*b^7*c + 3*a^2*b^6*d)*x^6 + 63*(17*a^2*b^6*c - 12*a^3*b^5*d)*x^5 - 70*(17*a^3*b^5*c - 12*a^4*b^4*d)*x^4 + 80*(17*a^4*b^4* c - 12*a^5*b^3*d)*x^3 - 96*(17*a^5*b^3*c - 12*a^6*b^2*d)*x^2 + 128*(17*a^6 *b^2*c - 12*a^7*b*d)*x)*sqrt(b*x^3 + a*x^2)/(b^7*x)
\[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\int x^{2} \left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )\, dx \] Input:
integrate(x**2*(d*x+c)*(b*x**3+a*x**2)**(3/2),x)
Output:
Integral(x**2*(x**2*(a + b*x))**(3/2)*(c + d*x), x)
Time = 0.04 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.78 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 \, {\left (3003 \, b^{7} x^{7} + 3696 \, a b^{6} x^{6} + 63 \, a^{2} b^{5} x^{5} - 70 \, a^{3} b^{4} x^{4} + 80 \, a^{4} b^{3} x^{3} - 96 \, a^{5} b^{2} x^{2} + 128 \, a^{6} b x - 256 \, a^{7}\right )} \sqrt {b x + a} c}{45045 \, b^{6}} + \frac {2 \, {\left (15015 \, b^{8} x^{8} + 18018 \, a b^{7} x^{7} + 231 \, a^{2} b^{6} x^{6} - 252 \, a^{3} b^{5} x^{5} + 280 \, a^{4} b^{4} x^{4} - 320 \, a^{5} b^{3} x^{3} + 384 \, a^{6} b^{2} x^{2} - 512 \, a^{7} b x + 1024 \, a^{8}\right )} \sqrt {b x + a} d}{255255 \, b^{7}} \] Input:
integrate(x^2*(d*x+c)*(b*x^3+a*x^2)^(3/2),x, algorithm="maxima")
Output:
2/45045*(3003*b^7*x^7 + 3696*a*b^6*x^6 + 63*a^2*b^5*x^5 - 70*a^3*b^4*x^4 + 80*a^4*b^3*x^3 - 96*a^5*b^2*x^2 + 128*a^6*b*x - 256*a^7)*sqrt(b*x + a)*c/ b^6 + 2/255255*(15015*b^8*x^8 + 18018*a*b^7*x^7 + 231*a^2*b^6*x^6 - 252*a^ 3*b^5*x^5 + 280*a^4*b^4*x^4 - 320*a^5*b^3*x^3 + 384*a^6*b^2*x^2 - 512*a^7* b*x + 1024*a^8)*sqrt(b*x + a)*d/b^7
Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (212) = 424\).
Time = 0.19 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.50 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate(x^2*(d*x+c)*(b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Output:
2/765765*(1105*(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*sq rt(b*x + a)*a^5)*a^2*c*sgn(x)/b^5 + 510*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009 *(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)* a*c*sgn(x)/b^5 + 255*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 500 5*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^ 4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*a^2*d*sgn(x)/b^6 + 119*(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/ 2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(b*x + a)^(7/2)*a^4 - 27027*(b* x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqrt(b*x + a)*a^7)*c*s gn(x)/b^5 + 238*(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b *x + a)^(11/2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(b*x + a)^(7/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqrt(b*x + a)*a^7)*a*d*sgn(x)/b^6 + 7*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2 )*a + 235620*(b*x + a)^(13/2)*a^2 - 556920*(b*x + a)^(11/2)*a^3 + 850850*( b*x + a)^(9/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a)^(5/2)*a ^6 - 291720*(b*x + a)^(3/2)*a^7 + 109395*sqrt(b*x + a)*a^8)*d*sgn(x)/b^6)/ b + 512/765765*(17*a^(15/2)*b*c - 12*a^(17/2)*d)*sgn(x)/b^7
Time = 9.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.71 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (x^7\,\left (\frac {12\,a\,d}{85}+\frac {2\,b\,c}{15}\right )+\frac {512\,a^7\,\left (12\,a\,d-17\,b\,c\right )}{765765\,b^7}+\frac {2\,b\,d\,x^8}{17}-\frac {256\,a^6\,x\,\left (12\,a\,d-17\,b\,c\right )}{765765\,b^6}+\frac {2\,a\,x^6\,\left (3\,a\,d+272\,b\,c\right )}{3315\,b}-\frac {2\,a^2\,x^5\,\left (12\,a\,d-17\,b\,c\right )}{12155\,b^2}+\frac {4\,a^3\,x^4\,\left (12\,a\,d-17\,b\,c\right )}{21879\,b^3}-\frac {32\,a^4\,x^3\,\left (12\,a\,d-17\,b\,c\right )}{153153\,b^4}+\frac {64\,a^5\,x^2\,\left (12\,a\,d-17\,b\,c\right )}{255255\,b^5}\right )}{x} \] Input:
int(x^2*(a*x^2 + b*x^3)^(3/2)*(c + d*x),x)
Output:
((a*x^2 + b*x^3)^(1/2)*(x^7*((12*a*d)/85 + (2*b*c)/15) + (512*a^7*(12*a*d - 17*b*c))/(765765*b^7) + (2*b*d*x^8)/17 - (256*a^6*x*(12*a*d - 17*b*c))/( 765765*b^6) + (2*a*x^6*(3*a*d + 272*b*c))/(3315*b) - (2*a^2*x^5*(12*a*d - 17*b*c))/(12155*b^2) + (4*a^3*x^4*(12*a*d - 17*b*c))/(21879*b^3) - (32*a^4 *x^3*(12*a*d - 17*b*c))/(153153*b^4) + (64*a^5*x^2*(12*a*d - 17*b*c))/(255 255*b^5)))/x
Time = 0.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.79 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 \sqrt {b x +a}\, \left (45045 b^{8} d \,x^{8}+54054 a \,b^{7} d \,x^{7}+51051 b^{8} c \,x^{7}+693 a^{2} b^{6} d \,x^{6}+62832 a \,b^{7} c \,x^{6}-756 a^{3} b^{5} d \,x^{5}+1071 a^{2} b^{6} c \,x^{5}+840 a^{4} b^{4} d \,x^{4}-1190 a^{3} b^{5} c \,x^{4}-960 a^{5} b^{3} d \,x^{3}+1360 a^{4} b^{4} c \,x^{3}+1152 a^{6} b^{2} d \,x^{2}-1632 a^{5} b^{3} c \,x^{2}-1536 a^{7} b d x +2176 a^{6} b^{2} c x +3072 a^{8} d -4352 a^{7} b c \right )}{765765 b^{7}} \] Input:
int(x^2*(d*x+c)*(b*x^3+a*x^2)^(3/2),x)
Output:
(2*sqrt(a + b*x)*(3072*a**8*d - 4352*a**7*b*c - 1536*a**7*b*d*x + 2176*a** 6*b**2*c*x + 1152*a**6*b**2*d*x**2 - 1632*a**5*b**3*c*x**2 - 960*a**5*b**3 *d*x**3 + 1360*a**4*b**4*c*x**3 + 840*a**4*b**4*d*x**4 - 1190*a**3*b**5*c* x**4 - 756*a**3*b**5*d*x**5 + 1071*a**2*b**6*c*x**5 + 693*a**2*b**6*d*x**6 + 62832*a*b**7*c*x**6 + 54054*a*b**7*d*x**7 + 51051*b**8*c*x**7 + 45045*b **8*d*x**8))/(765765*b**7)