Integrand size = 22, antiderivative size = 279 \[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 a^6 (b c-a d) \left (a x^2+b x^3\right )^{7/2}}{7 b^8 x^7}-\frac {2 a^5 (6 b c-7 a d) \left (a x^2+b x^3\right )^{9/2}}{9 b^8 x^9}+\frac {6 a^4 (5 b c-7 a d) \left (a x^2+b x^3\right )^{11/2}}{11 b^8 x^{11}}-\frac {10 a^3 (4 b c-7 a d) \left (a x^2+b x^3\right )^{13/2}}{13 b^8 x^{13}}+\frac {2 a^2 (3 b c-7 a d) \left (a x^2+b x^3\right )^{15/2}}{3 b^8 x^{15}}-\frac {6 a (2 b c-7 a d) \left (a x^2+b x^3\right )^{17/2}}{17 b^8 x^{17}}+\frac {2 (b c-7 a d) \left (a x^2+b x^3\right )^{19/2}}{19 b^8 x^{19}}+\frac {2 d \left (a x^2+b x^3\right )^{21/2}}{21 b^8 x^{21}} \] Output:
2/7*a^6*(-a*d+b*c)*(b*x^3+a*x^2)^(7/2)/b^8/x^7-2/9*a^5*(-7*a*d+6*b*c)*(b*x ^3+a*x^2)^(9/2)/b^8/x^9+6/11*a^4*(-7*a*d+5*b*c)*(b*x^3+a*x^2)^(11/2)/b^8/x ^11-10/13*a^3*(-7*a*d+4*b*c)*(b*x^3+a*x^2)^(13/2)/b^8/x^13+2/3*a^2*(-7*a*d +3*b*c)*(b*x^3+a*x^2)^(15/2)/b^8/x^15-6/17*a*(-7*a*d+2*b*c)*(b*x^3+a*x^2)^ (17/2)/b^8/x^17+2/19*(-7*a*d+b*c)*(b*x^3+a*x^2)^(19/2)/b^8/x^19+2/21*d*(b* x^3+a*x^2)^(21/2)/b^8/x^21
Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.55 \[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 x (a+b x)^4 \left (-2048 a^7 d+72072 a^2 b^5 x^4 (c+d x)-5376 a^5 b^2 x (2 c+3 d x)+1024 a^6 b (3 c+7 d x)+2688 a^4 b^3 x^2 (9 c+11 d x)-3696 a^3 b^4 x^3 (12 c+13 d x)-6006 a b^6 x^5 (18 c+17 d x)+7293 b^7 x^6 (21 c+19 d x)\right )}{2909907 b^8 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[x*(c + d*x)*(a*x^2 + b*x^3)^(5/2),x]
Output:
(2*x*(a + b*x)^4*(-2048*a^7*d + 72072*a^2*b^5*x^4*(c + d*x) - 5376*a^5*b^2 *x*(2*c + 3*d*x) + 1024*a^6*b*(3*c + 7*d*x) + 2688*a^4*b^3*x^2*(9*c + 11*d *x) - 3696*a^3*b^4*x^3*(12*c + 13*d*x) - 6006*a*b^6*x^5*(18*c + 17*d*x) + 7293*b^7*x^6*(21*c + 19*d*x)))/(2909907*b^8*Sqrt[x^2*(a + b*x)])
Time = 0.88 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1945, 1922, 1908, 1922, 1922, 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.
| \(\displaystyle \int x \left (a x^2+b x^3\right )^{5/2} (c+d x) \, dx\) |
| \(\Big \downarrow \) 1945 |
| \(\displaystyle \frac {(3 b c-2 a d) \int x \left (b x^3+a x^2\right )^{5/2}dx}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(3 b c-2 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \int \left (b x^3+a x^2\right )^{5/2}dx}{19 b}\right )}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {(3 b c-2 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x}dx}{17 b}\right )}{19 b}\right )}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(3 b c-2 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^2}dx}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(3 b c-2 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^3}dx}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(3 b c-2 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^4}dx}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(3 b c-2 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {2 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^5}dx}{9 b}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {\left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {4 a \left (a x^2+b x^3\right )^{7/2}}{63 b^2 x^7}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right ) (3 b c-2 a d)}{3 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{21 b}\) |
Input:
Int[x*(c + d*x)*(a*x^2 + b*x^3)^(5/2),x]
Output:
(2*d*(a*x^2 + b*x^3)^(7/2))/(21*b) + ((3*b*c - 2*a*d)*((2*(a*x^2 + b*x^3)^ (7/2))/(19*b*x) - (12*a*((2*(a*x^2 + b*x^3)^(7/2))/(17*b*x^2) - (10*a*((2* (a*x^2 + b*x^3)^(7/2))/(15*b*x^3) - (8*a*((2*(a*x^2 + b*x^3)^(7/2))/(13*b* x^4) - (6*a*((2*(a*x^2 + b*x^3)^(7/2))/(11*b*x^5) - (4*a*((-4*a*(a*x^2 + b *x^3)^(7/2))/(63*b^2*x^7) + (2*(a*x^2 + b*x^3)^(7/2))/(9*b*x^6)))/(11*b))) /(13*b)))/(15*b)))/(17*b)))/(19*b)))/(3*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 = 0.60 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.15
| method | result | size |
| pseudoelliptic | \(\frac {16 \left (b x +a \right )^{\frac {7}{2}} \left (\frac {77 \left (\frac {9 d x}{11}+c \right ) x \,b^{2}}{8}-\frac {11 \left (\frac {14 d x}{11}+c \right ) a b}{4}+a^{2} d \right )}{693 b^{3}}\) | \(41\) |
| gosper | \(-\frac {2 \left (b x +a \right ) \left (-138567 d \,x^{7} b^{7}+102102 a \,b^{6} d \,x^{6}-153153 b^{7} c \,x^{6}-72072 a^{2} b^{5} d \,x^{5}+108108 a \,b^{6} c \,x^{5}+48048 a^{3} b^{4} d \,x^{4}-72072 a^{2} b^{5} c \,x^{4}-29568 a^{4} b^{3} d \,x^{3}+44352 a^{3} b^{4} c \,x^{3}+16128 a^{5} b^{2} d \,x^{2}-24192 a^{4} b^{3} c \,x^{2}-7168 a^{6} b d x +10752 a^{5} b^{2} c x +2048 a^{7} d -3072 a^{6} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{2909907 b^{8} x^{5}}\) | \(181\) |
| default | \(-\frac {2 \left (b x +a \right ) \left (-138567 d \,x^{7} b^{7}+102102 a \,b^{6} d \,x^{6}-153153 b^{7} c \,x^{6}-72072 a^{2} b^{5} d \,x^{5}+108108 a \,b^{6} c \,x^{5}+48048 a^{3} b^{4} d \,x^{4}-72072 a^{2} b^{5} c \,x^{4}-29568 a^{4} b^{3} d \,x^{3}+44352 a^{3} b^{4} c \,x^{3}+16128 a^{5} b^{2} d \,x^{2}-24192 a^{4} b^{3} c \,x^{2}-7168 a^{6} b d x +10752 a^{5} b^{2} c x +2048 a^{7} d -3072 a^{6} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{2909907 b^{8} x^{5}}\) | \(181\) |
| orering | \(-\frac {2 \left (b x +a \right ) \left (-138567 d \,x^{7} b^{7}+102102 a \,b^{6} d \,x^{6}-153153 b^{7} c \,x^{6}-72072 a^{2} b^{5} d \,x^{5}+108108 a \,b^{6} c \,x^{5}+48048 a^{3} b^{4} d \,x^{4}-72072 a^{2} b^{5} c \,x^{4}-29568 a^{4} b^{3} d \,x^{3}+44352 a^{3} b^{4} c \,x^{3}+16128 a^{5} b^{2} d \,x^{2}-24192 a^{4} b^{3} c \,x^{2}-7168 a^{6} b d x +10752 a^{5} b^{2} c x +2048 a^{7} d -3072 a^{6} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{2909907 b^{8} x^{5}}\) | \(181\) |
| risch | \(-\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (-138567 b^{10} d \,x^{10}-313599 a \,b^{9} d \,x^{9}-153153 b^{10} c \,x^{9}-181467 a^{2} b^{8} d \,x^{8}-351351 a \,b^{9} c \,x^{8}-429 a^{3} b^{7} d \,x^{7}-207207 a^{2} b^{8} c \,x^{7}+462 a^{4} b^{6} d \,x^{6}-693 a^{3} b^{7} c \,x^{6}-504 a^{5} b^{5} d \,x^{5}+756 a^{4} b^{6} c \,x^{5}+560 a^{6} b^{4} d \,x^{4}-840 a^{5} b^{5} c \,x^{4}-640 a^{7} b^{3} d \,x^{3}+960 a^{6} b^{4} c \,x^{3}+768 a^{8} b^{2} d \,x^{2}-1152 a^{7} b^{3} c \,x^{2}-1024 a^{9} b d x +1536 a^{8} b^{2} c x +2048 a^{10} d -3072 a^{9} b c \right )}{2909907 x \,b^{8}}\) | \(246\) |
| trager | \(-\frac {2 \left (-138567 b^{10} d \,x^{10}-313599 a \,b^{9} d \,x^{9}-153153 b^{10} c \,x^{9}-181467 a^{2} b^{8} d \,x^{8}-351351 a \,b^{9} c \,x^{8}-429 a^{3} b^{7} d \,x^{7}-207207 a^{2} b^{8} c \,x^{7}+462 a^{4} b^{6} d \,x^{6}-693 a^{3} b^{7} c \,x^{6}-504 a^{5} b^{5} d \,x^{5}+756 a^{4} b^{6} c \,x^{5}+560 a^{6} b^{4} d \,x^{4}-840 a^{5} b^{5} c \,x^{4}-640 a^{7} b^{3} d \,x^{3}+960 a^{6} b^{4} c \,x^{3}+768 a^{8} b^{2} d \,x^{2}-1152 a^{7} b^{3} c \,x^{2}-1024 a^{9} b d x +1536 a^{8} b^{2} c x +2048 a^{10} d -3072 a^{9} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{2909907 b^{8} x}\) | \(248\) |
Input:
int(x*(d*x+c)*(b*x^3+a*x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
16/693*(b*x+a)^(7/2)*(77/8*(9/11*d*x+c)*x*b^2-11/4*(14/11*d*x+c)*a*b+a^2*d )/b^3
Time = 0.07 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.89 \[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (138567 \, b^{10} d x^{10} + 3072 \, a^{9} b c - 2048 \, a^{10} d + 7293 \, {\left (21 \, b^{10} c + 43 \, a b^{9} d\right )} x^{9} + 3861 \, {\left (91 \, a b^{9} c + 47 \, a^{2} b^{8} d\right )} x^{8} + 429 \, {\left (483 \, a^{2} b^{8} c + a^{3} b^{7} d\right )} x^{7} + 231 \, {\left (3 \, a^{3} b^{7} c - 2 \, a^{4} b^{6} d\right )} x^{6} - 252 \, {\left (3 \, a^{4} b^{6} c - 2 \, a^{5} b^{5} d\right )} x^{5} + 280 \, {\left (3 \, a^{5} b^{5} c - 2 \, a^{6} b^{4} d\right )} x^{4} - 320 \, {\left (3 \, a^{6} b^{4} c - 2 \, a^{7} b^{3} d\right )} x^{3} + 384 \, {\left (3 \, a^{7} b^{3} c - 2 \, a^{8} b^{2} d\right )} x^{2} - 512 \, {\left (3 \, a^{8} b^{2} c - 2 \, a^{9} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{2909907 \, b^{8} x} \] Input:
integrate(x*(d*x+c)*(b*x^3+a*x^2)^(5/2),x, algorithm="fricas")
Output:
2/2909907*(138567*b^10*d*x^10 + 3072*a^9*b*c - 2048*a^10*d + 7293*(21*b^10 *c + 43*a*b^9*d)*x^9 + 3861*(91*a*b^9*c + 47*a^2*b^8*d)*x^8 + 429*(483*a^2 *b^8*c + a^3*b^7*d)*x^7 + 231*(3*a^3*b^7*c - 2*a^4*b^6*d)*x^6 - 252*(3*a^4 *b^6*c - 2*a^5*b^5*d)*x^5 + 280*(3*a^5*b^5*c - 2*a^6*b^4*d)*x^4 - 320*(3*a ^6*b^4*c - 2*a^7*b^3*d)*x^3 + 384*(3*a^7*b^3*c - 2*a^8*b^2*d)*x^2 - 512*(3 *a^8*b^2*c - 2*a^9*b*d)*x)*sqrt(b*x^3 + a*x^2)/(b^8*x)
\[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\int x \left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}} \left (c + d x\right )\, dx \] Input:
integrate(x*(d*x+c)*(b*x**3+a*x**2)**(5/2),x)
Output:
Integral(x*(x**2*(a + b*x))**(5/2)*(c + d*x), x)
Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.82 \[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (51051 \, b^{9} x^{9} + 117117 \, a b^{8} x^{8} + 69069 \, a^{2} b^{7} x^{7} + 231 \, a^{3} b^{6} x^{6} - 252 \, a^{4} b^{5} x^{5} + 280 \, a^{5} b^{4} x^{4} - 320 \, a^{6} b^{3} x^{3} + 384 \, a^{7} b^{2} x^{2} - 512 \, a^{8} b x + 1024 \, a^{9}\right )} \sqrt {b x + a} c}{969969 \, b^{7}} + \frac {2 \, {\left (138567 \, b^{10} x^{10} + 313599 \, a b^{9} x^{9} + 181467 \, a^{2} b^{8} x^{8} + 429 \, a^{3} b^{7} x^{7} - 462 \, a^{4} b^{6} x^{6} + 504 \, a^{5} b^{5} x^{5} - 560 \, a^{6} b^{4} x^{4} + 640 \, a^{7} b^{3} x^{3} - 768 \, a^{8} b^{2} x^{2} + 1024 \, a^{9} b x - 2048 \, a^{10}\right )} \sqrt {b x + a} d}{2909907 \, b^{8}} \] Input:
integrate(x*(d*x+c)*(b*x^3+a*x^2)^(5/2),x, algorithm="maxima")
Output:
2/969969*(51051*b^9*x^9 + 117117*a*b^8*x^8 + 69069*a^2*b^7*x^7 + 231*a^3*b ^6*x^6 - 252*a^4*b^5*x^5 + 280*a^5*b^4*x^4 - 320*a^6*b^3*x^3 + 384*a^7*b^2 *x^2 - 512*a^8*b*x + 1024*a^9)*sqrt(b*x + a)*c/b^7 + 2/2909907*(138567*b^1 0*x^10 + 313599*a*b^9*x^9 + 181467*a^2*b^8*x^8 + 429*a^3*b^7*x^7 - 462*a^4 *b^6*x^6 + 504*a^5*b^5*x^5 - 560*a^6*b^4*x^4 + 640*a^7*b^3*x^3 - 768*a^8*b ^2*x^2 + 1024*a^9*b*x - 2048*a^10)*sqrt(b*x + a)*d/b^8
Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (247) = 494\).
Time = 0.14 (sec) , antiderivative size = 938, normalized size of antiderivative = 3.36 \[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate(x*(d*x+c)*(b*x^3+a*x^2)^(5/2),x, algorithm="giac")
Output:
2/14549535*(4845*(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^3*c*sgn(x)/b^6 + 6783 *(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^2*c* sgn(x)/b^6 + 2261*(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^3*d*sgn(x)/b^7 + 399*(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 + 85 0850*(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)*a*c*sgn( x)/b^6 + 399*(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)*a^2*d*sgn(x)/b^7 + 63*(12155* (b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 554268*(b*x + a)^(15/2)*a^2 - 1492260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a)^(11/2)*a^4 - 3233230*( b*x + a)^(9/2)*a^5 + 2771340*(b*x + a)^(7/2)*a^6 - 1662804*(b*x + a)^(5...
Time = 9.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.76 \[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {6\,a\,x^8\,\left (47\,a\,d+91\,b\,c\right )}{2261}+\frac {2\,b\,x^9\,\left (43\,a\,d+21\,b\,c\right )}{399}+\frac {2\,b^2\,d\,x^{10}}{21}-\frac {2048\,a^9\,\left (2\,a\,d-3\,b\,c\right )}{2909907\,b^8}+\frac {1024\,a^8\,x\,\left (2\,a\,d-3\,b\,c\right )}{2909907\,b^7}-\frac {2\,a^3\,x^6\,\left (2\,a\,d-3\,b\,c\right )}{12597\,b^2}+\frac {8\,a^4\,x^5\,\left (2\,a\,d-3\,b\,c\right )}{46189\,b^3}-\frac {80\,a^5\,x^4\,\left (2\,a\,d-3\,b\,c\right )}{415701\,b^4}+\frac {640\,a^6\,x^3\,\left (2\,a\,d-3\,b\,c\right )}{2909907\,b^5}-\frac {256\,a^7\,x^2\,\left (2\,a\,d-3\,b\,c\right )}{969969\,b^6}+\frac {2\,a^2\,x^7\,\left (a\,d+483\,b\,c\right )}{6783\,b}\right )}{x} \] Input:
int(x*(a*x^2 + b*x^3)^(5/2)*(c + d*x),x)
Output:
((a*x^2 + b*x^3)^(1/2)*((6*a*x^8*(47*a*d + 91*b*c))/2261 + (2*b*x^9*(43*a* d + 21*b*c))/399 + (2*b^2*d*x^10)/21 - (2048*a^9*(2*a*d - 3*b*c))/(2909907 *b^8) + (1024*a^8*x*(2*a*d - 3*b*c))/(2909907*b^7) - (2*a^3*x^6*(2*a*d - 3 *b*c))/(12597*b^2) + (8*a^4*x^5*(2*a*d - 3*b*c))/(46189*b^3) - (80*a^5*x^4 *(2*a*d - 3*b*c))/(415701*b^4) + (640*a^6*x^3*(2*a*d - 3*b*c))/(2909907*b^ 5) - (256*a^7*x^2*(2*a*d - 3*b*c))/(969969*b^6) + (2*a^2*x^7*(a*d + 483*b* c))/(6783*b)))/x
Time = 0.21 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.85 \[ \int x (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \sqrt {b x +a}\, \left (138567 b^{10} d \,x^{10}+313599 a \,b^{9} d \,x^{9}+153153 b^{10} c \,x^{9}+181467 a^{2} b^{8} d \,x^{8}+351351 a \,b^{9} c \,x^{8}+429 a^{3} b^{7} d \,x^{7}+207207 a^{2} b^{8} c \,x^{7}-462 a^{4} b^{6} d \,x^{6}+693 a^{3} b^{7} c \,x^{6}+504 a^{5} b^{5} d \,x^{5}-756 a^{4} b^{6} c \,x^{5}-560 a^{6} b^{4} d \,x^{4}+840 a^{5} b^{5} c \,x^{4}+640 a^{7} b^{3} d \,x^{3}-960 a^{6} b^{4} c \,x^{3}-768 a^{8} b^{2} d \,x^{2}+1152 a^{7} b^{3} c \,x^{2}+1024 a^{9} b d x -1536 a^{8} b^{2} c x -2048 a^{10} d +3072 a^{9} b c \right )}{2909907 b^{8}} \] Input:
int(x*(d*x+c)*(b*x^3+a*x^2)^(5/2),x)
Output:
(2*sqrt(a + b*x)*( - 2048*a**10*d + 3072*a**9*b*c + 1024*a**9*b*d*x - 1536 *a**8*b**2*c*x - 768*a**8*b**2*d*x**2 + 1152*a**7*b**3*c*x**2 + 640*a**7*b **3*d*x**3 - 960*a**6*b**4*c*x**3 - 560*a**6*b**4*d*x**4 + 840*a**5*b**5*c *x**4 + 504*a**5*b**5*d*x**5 - 756*a**4*b**6*c*x**5 - 462*a**4*b**6*d*x**6 + 693*a**3*b**7*c*x**6 + 429*a**3*b**7*d*x**7 + 207207*a**2*b**8*c*x**7 + 181467*a**2*b**8*d*x**8 + 351351*a*b**9*c*x**8 + 313599*a*b**9*d*x**9 + 1 53153*b**10*c*x**9 + 138567*b**10*d*x**10))/(2909907*b**8)