Integrand size = 21, antiderivative size = 240 \[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=-\frac {2 a^5 (b c-a d) \left (a x^2+b x^3\right )^{7/2}}{7 b^7 x^7}+\frac {2 a^4 (5 b c-6 a d) \left (a x^2+b x^3\right )^{9/2}}{9 b^7 x^9}-\frac {10 a^3 (2 b c-3 a d) \left (a x^2+b x^3\right )^{11/2}}{11 b^7 x^{11}}+\frac {20 a^2 (b c-2 a d) \left (a x^2+b x^3\right )^{13/2}}{13 b^7 x^{13}}-\frac {2 a (b c-3 a d) \left (a x^2+b x^3\right )^{15/2}}{3 b^7 x^{15}}+\frac {2 (b c-6 a d) \left (a x^2+b x^3\right )^{17/2}}{17 b^7 x^{17}}+\frac {2 d \left (a x^2+b x^3\right )^{19/2}}{19 b^7 x^{19}} \] Output:
-2/7*a^5*(-a*d+b*c)*(b*x^3+a*x^2)^(7/2)/b^7/x^7+2/9*a^4*(-6*a*d+5*b*c)*(b* x^3+a*x^2)^(9/2)/b^7/x^9-10/11*a^3*(-3*a*d+2*b*c)*(b*x^3+a*x^2)^(11/2)/b^7 /x^11+20/13*a^2*(-2*a*d+b*c)*(b*x^3+a*x^2)^(13/2)/b^7/x^13-2/3*a*(-3*a*d+b *c)*(b*x^3+a*x^2)^(15/2)/b^7/x^15+2/17*(-6*a*d+b*c)*(b*x^3+a*x^2)^(17/2)/b ^7/x^17+2/19*d*(b*x^3+a*x^2)^(19/2)/b^7/x^19
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.57 \[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 x (a+b x)^4 \left (3072 a^6 d+9009 b^6 x^5 (19 c+17 d x)-6006 a b^5 x^4 (19 c+18 d x)-2016 a^3 b^3 x^2 (19 c+22 d x)+896 a^4 b^2 x (19 c+27 d x)+1848 a^2 b^4 x^3 (38 c+39 d x)-256 a^5 b (19 c+42 d x)\right )}{2909907 b^7 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[(c + d*x)*(a*x^2 + b*x^3)^(5/2),x]
Output:
(2*x*(a + b*x)^4*(3072*a^6*d + 9009*b^6*x^5*(19*c + 17*d*x) - 6006*a*b^5*x ^4*(19*c + 18*d*x) - 2016*a^3*b^3*x^2*(19*c + 22*d*x) + 896*a^4*b^2*x*(19* c + 27*d*x) + 1848*a^2*b^4*x^3*(38*c + 39*d*x) - 256*a^5*b*(19*c + 42*d*x) ))/(2909907*b^7*Sqrt[x^2*(a + b*x)])
Time = 0.95 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2450, 2009}
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.
| \(\displaystyle \int \left (a x^2+b x^3\right )^{5/2} (c+d x) \, dx\) |
| \(\Big \downarrow \) 2450 |
| \(\displaystyle \int \left (c \left (a x^2+b x^3\right )^{5/2}+d x \left (a x^2+b x^3\right )^{5/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2048 a^6 d \left (a x^2+b x^3\right )^{7/2}}{969969 b^7 x^7}-\frac {512 a^5 c \left (a x^2+b x^3\right )^{7/2}}{153153 b^6 x^7}-\frac {1024 a^5 d \left (a x^2+b x^3\right )^{7/2}}{138567 b^6 x^6}+\frac {256 a^4 c \left (a x^2+b x^3\right )^{7/2}}{21879 b^5 x^6}+\frac {768 a^4 d \left (a x^2+b x^3\right )^{7/2}}{46189 b^5 x^5}-\frac {64 a^3 c \left (a x^2+b x^3\right )^{7/2}}{2431 b^4 x^5}-\frac {128 a^3 d \left (a x^2+b x^3\right )^{7/2}}{4199 b^4 x^4}+\frac {32 a^2 c \left (a x^2+b x^3\right )^{7/2}}{663 b^3 x^4}+\frac {16 a^2 d \left (a x^2+b x^3\right )^{7/2}}{323 b^3 x^3}-\frac {4 a c \left (a x^2+b x^3\right )^{7/2}}{51 b^2 x^3}-\frac {24 a d \left (a x^2+b x^3\right )^{7/2}}{323 b^2 x^2}+\frac {2 c \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{19 b x}\) |
Input:
Int[(c + d*x)*(a*x^2 + b*x^3)^(5/2),x]
Output:
(-512*a^5*c*(a*x^2 + b*x^3)^(7/2))/(153153*b^6*x^7) + (2048*a^6*d*(a*x^2 + b*x^3)^(7/2))/(969969*b^7*x^7) + (256*a^4*c*(a*x^2 + b*x^3)^(7/2))/(21879 *b^5*x^6) - (1024*a^5*d*(a*x^2 + b*x^3)^(7/2))/(138567*b^6*x^6) - (64*a^3* c*(a*x^2 + b*x^3)^(7/2))/(2431*b^4*x^5) + (768*a^4*d*(a*x^2 + b*x^3)^(7/2) )/(46189*b^5*x^5) + (32*a^2*c*(a*x^2 + b*x^3)^(7/2))/(663*b^3*x^4) - (128* a^3*d*(a*x^2 + b*x^3)^(7/2))/(4199*b^4*x^4) - (4*a*c*(a*x^2 + b*x^3)^(7/2) )/(51*b^2*x^3) + (16*a^2*d*(a*x^2 + b*x^3)^(7/2))/(323*b^3*x^3) + (2*c*(a* x^2 + b*x^3)^(7/2))/(17*b*x^2) - (24*a*d*(a*x^2 + b*x^3)^(7/2))/(323*b^2*x ^2) + (2*d*(a*x^2 + b*x^3)^(7/2))/(19*b*x)
Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[Expan dIntegrand[Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && (Po lyQ[Pq, x] || PolyQ[Pq, x^n]) && !IntegerQ[p] && NeQ[n, j]
Time = 0.60 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.11
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-7 b d x +2 a d -9 b c \right )}{63 b^{2}}\) | \(27\) |
| gosper | \(\frac {2 \left (b x +a \right ) \left (153153 d \,x^{6} b^{6}-108108 a \,b^{5} d \,x^{5}+171171 b^{6} c \,x^{5}+72072 a^{2} b^{4} d \,x^{4}-114114 a \,b^{5} c \,x^{4}-44352 a^{3} b^{3} d \,x^{3}+70224 a^{2} b^{4} c \,x^{3}+24192 a^{4} b^{2} d \,x^{2}-38304 a^{3} b^{3} c \,x^{2}-10752 a^{5} b d x +17024 a^{4} b^{2} c x +3072 a^{6} d -4864 a^{5} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{2909907 b^{7} x^{5}}\) | \(157\) |
| default | \(\frac {2 \left (b x +a \right ) \left (153153 d \,x^{6} b^{6}-108108 a \,b^{5} d \,x^{5}+171171 b^{6} c \,x^{5}+72072 a^{2} b^{4} d \,x^{4}-114114 a \,b^{5} c \,x^{4}-44352 a^{3} b^{3} d \,x^{3}+70224 a^{2} b^{4} c \,x^{3}+24192 a^{4} b^{2} d \,x^{2}-38304 a^{3} b^{3} c \,x^{2}-10752 a^{5} b d x +17024 a^{4} b^{2} c x +3072 a^{6} d -4864 a^{5} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{2909907 b^{7} x^{5}}\) | \(157\) |
| orering | \(\frac {2 \left (b x +a \right ) \left (153153 d \,x^{6} b^{6}-108108 a \,b^{5} d \,x^{5}+171171 b^{6} c \,x^{5}+72072 a^{2} b^{4} d \,x^{4}-114114 a \,b^{5} c \,x^{4}-44352 a^{3} b^{3} d \,x^{3}+70224 a^{2} b^{4} c \,x^{3}+24192 a^{4} b^{2} d \,x^{2}-38304 a^{3} b^{3} c \,x^{2}-10752 a^{5} b d x +17024 a^{4} b^{2} c x +3072 a^{6} d -4864 a^{5} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{2909907 b^{7} x^{5}}\) | \(157\) |
| risch | \(\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (153153 b^{9} d \,x^{9}+351351 a \,b^{8} d \,x^{8}+171171 b^{9} c \,x^{8}+207207 a^{2} b^{7} d \,x^{7}+399399 a \,b^{8} c \,x^{7}+693 a^{3} b^{6} d \,x^{6}+241395 a^{2} b^{7} c \,x^{6}-756 a^{4} b^{5} d \,x^{5}+1197 a^{3} b^{6} c \,x^{5}+840 a^{5} b^{4} d \,x^{4}-1330 a^{4} b^{5} c \,x^{4}-960 a^{6} b^{3} d \,x^{3}+1520 a^{5} b^{4} c \,x^{3}+1152 a^{7} b^{2} d \,x^{2}-1824 a^{6} b^{3} c \,x^{2}-1536 a^{8} b d x +2432 a^{7} b^{2} c x +3072 a^{9} d -4864 a^{8} b c \right )}{2909907 x \,b^{7}}\) | \(222\) |
| trager | \(\frac {2 \left (153153 b^{9} d \,x^{9}+351351 a \,b^{8} d \,x^{8}+171171 b^{9} c \,x^{8}+207207 a^{2} b^{7} d \,x^{7}+399399 a \,b^{8} c \,x^{7}+693 a^{3} b^{6} d \,x^{6}+241395 a^{2} b^{7} c \,x^{6}-756 a^{4} b^{5} d \,x^{5}+1197 a^{3} b^{6} c \,x^{5}+840 a^{5} b^{4} d \,x^{4}-1330 a^{4} b^{5} c \,x^{4}-960 a^{6} b^{3} d \,x^{3}+1520 a^{5} b^{4} c \,x^{3}+1152 a^{7} b^{2} d \,x^{2}-1824 a^{6} b^{3} c \,x^{2}-1536 a^{8} b d x +2432 a^{7} b^{2} c x +3072 a^{9} d -4864 a^{8} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{2909907 b^{7} x}\) | \(224\) |
Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/63*(b*x+a)^(7/2)*(-7*b*d*x+2*a*d-9*b*c)/b^2
Time = 0.15 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.94 \[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (153153 \, b^{9} d x^{9} - 4864 \, a^{8} b c + 3072 \, a^{9} d + 9009 \, {\left (19 \, b^{9} c + 39 \, a b^{8} d\right )} x^{8} + 3003 \, {\left (133 \, a b^{8} c + 69 \, a^{2} b^{7} d\right )} x^{7} + 231 \, {\left (1045 \, a^{2} b^{7} c + 3 \, a^{3} b^{6} d\right )} x^{6} + 63 \, {\left (19 \, a^{3} b^{6} c - 12 \, a^{4} b^{5} d\right )} x^{5} - 70 \, {\left (19 \, a^{4} b^{5} c - 12 \, a^{5} b^{4} d\right )} x^{4} + 80 \, {\left (19 \, a^{5} b^{4} c - 12 \, a^{6} b^{3} d\right )} x^{3} - 96 \, {\left (19 \, a^{6} b^{3} c - 12 \, a^{7} b^{2} d\right )} x^{2} + 128 \, {\left (19 \, a^{7} b^{2} c - 12 \, a^{8} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{2909907 \, b^{7} x} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2),x, algorithm="fricas")
Output:
2/2909907*(153153*b^9*d*x^9 - 4864*a^8*b*c + 3072*a^9*d + 9009*(19*b^9*c + 39*a*b^8*d)*x^8 + 3003*(133*a*b^8*c + 69*a^2*b^7*d)*x^7 + 231*(1045*a^2*b ^7*c + 3*a^3*b^6*d)*x^6 + 63*(19*a^3*b^6*c - 12*a^4*b^5*d)*x^5 - 70*(19*a^ 4*b^5*c - 12*a^5*b^4*d)*x^4 + 80*(19*a^5*b^4*c - 12*a^6*b^3*d)*x^3 - 96*(1 9*a^6*b^3*c - 12*a^7*b^2*d)*x^2 + 128*(19*a^7*b^2*c - 12*a^8*b*d)*x)*sqrt( b*x^3 + a*x^2)/(b^7*x)
\[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\int \left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}} \left (c + d x\right )\, dx \] Input:
integrate((d*x+c)*(b*x**3+a*x**2)**(5/2),x)
Output:
Integral((x**2*(a + b*x))**(5/2)*(c + d*x), x)
Time = 0.05 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.87 \[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{8} x^{8} + 21021 \, a b^{7} x^{7} + 12705 \, a^{2} b^{6} x^{6} + 63 \, a^{3} b^{5} x^{5} - 70 \, a^{4} b^{4} x^{4} + 80 \, a^{5} b^{3} x^{3} - 96 \, a^{6} b^{2} x^{2} + 128 \, a^{7} b x - 256 \, a^{8}\right )} \sqrt {b x + a} c}{153153 \, b^{6}} + \frac {2 \, {\left (51051 \, b^{9} x^{9} + 117117 \, a b^{8} x^{8} + 69069 \, a^{2} b^{7} x^{7} + 231 \, a^{3} b^{6} x^{6} - 252 \, a^{4} b^{5} x^{5} + 280 \, a^{5} b^{4} x^{4} - 320 \, a^{6} b^{3} x^{3} + 384 \, a^{7} b^{2} x^{2} - 512 \, a^{8} b x + 1024 \, a^{9}\right )} \sqrt {b x + a} d}{969969 \, b^{7}} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2),x, algorithm="maxima")
Output:
2/153153*(9009*b^8*x^8 + 21021*a*b^7*x^7 + 12705*a^2*b^6*x^6 + 63*a^3*b^5* x^5 - 70*a^4*b^4*x^4 + 80*a^5*b^3*x^3 - 96*a^6*b^2*x^2 + 128*a^7*b*x - 256 *a^8)*sqrt(b*x + a)*c/b^6 + 2/969969*(51051*b^9*x^9 + 117117*a*b^8*x^8 + 6 9069*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 - 3 20*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)*d /b^7
Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (212) = 424\).
Time = 0.13 (sec) , antiderivative size = 842, normalized size of antiderivative = 3.51 \[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2),x, algorithm="giac")
Output:
2/14549535*(20995*(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)*a^3*c*sgn(x)/b^5 + 14535*(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^2*c*sgn(x)/b^5 + 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*d*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*c*sgn(x)/b^5 + 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 - 643 5*sqrt(b*x + a)*a^7)*a^2*d*sgn(x)/b^6 + 133*(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 )*c*sgn(x)/b^5 + 399*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 2 35620*(b*x + a)^(13/2)*a^2 - 556920*(b*x + a)^(11/2)*a^3 + 850850*(b*x ...
Time = 9.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.80 \[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {2\,a\,x^7\,\left (69\,a\,d+133\,b\,c\right )}{969}+\frac {2\,b\,x^8\,\left (39\,a\,d+19\,b\,c\right )}{323}+\frac {2\,b^2\,d\,x^9}{19}+\frac {512\,a^8\,\left (12\,a\,d-19\,b\,c\right )}{2909907\,b^7}-\frac {256\,a^7\,x\,\left (12\,a\,d-19\,b\,c\right )}{2909907\,b^6}-\frac {2\,a^3\,x^5\,\left (12\,a\,d-19\,b\,c\right )}{46189\,b^2}+\frac {20\,a^4\,x^4\,\left (12\,a\,d-19\,b\,c\right )}{415701\,b^3}-\frac {160\,a^5\,x^3\,\left (12\,a\,d-19\,b\,c\right )}{2909907\,b^4}+\frac {64\,a^6\,x^2\,\left (12\,a\,d-19\,b\,c\right )}{969969\,b^5}+\frac {2\,a^2\,x^6\,\left (3\,a\,d+1045\,b\,c\right )}{12597\,b}\right )}{x} \] Input:
int((a*x^2 + b*x^3)^(5/2)*(c + d*x),x)
Output:
((a*x^2 + b*x^3)^(1/2)*((2*a*x^7*(69*a*d + 133*b*c))/969 + (2*b*x^8*(39*a* d + 19*b*c))/323 + (2*b^2*d*x^9)/19 + (512*a^8*(12*a*d - 19*b*c))/(2909907 *b^7) - (256*a^7*x*(12*a*d - 19*b*c))/(2909907*b^6) - (2*a^3*x^5*(12*a*d - 19*b*c))/(46189*b^2) + (20*a^4*x^4*(12*a*d - 19*b*c))/(415701*b^3) - (160 *a^5*x^3*(12*a*d - 19*b*c))/(2909907*b^4) + (64*a^6*x^2*(12*a*d - 19*b*c)) /(969969*b^5) + (2*a^2*x^6*(3*a*d + 1045*b*c))/(12597*b)))/x
Time = 0.20 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.89 \[ \int (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \sqrt {b x +a}\, \left (153153 b^{9} d \,x^{9}+351351 a \,b^{8} d \,x^{8}+171171 b^{9} c \,x^{8}+207207 a^{2} b^{7} d \,x^{7}+399399 a \,b^{8} c \,x^{7}+693 a^{3} b^{6} d \,x^{6}+241395 a^{2} b^{7} c \,x^{6}-756 a^{4} b^{5} d \,x^{5}+1197 a^{3} b^{6} c \,x^{5}+840 a^{5} b^{4} d \,x^{4}-1330 a^{4} b^{5} c \,x^{4}-960 a^{6} b^{3} d \,x^{3}+1520 a^{5} b^{4} c \,x^{3}+1152 a^{7} b^{2} d \,x^{2}-1824 a^{6} b^{3} c \,x^{2}-1536 a^{8} b d x +2432 a^{7} b^{2} c x +3072 a^{9} d -4864 a^{8} b c \right )}{2909907 b^{7}} \] Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2),x)
Output:
(2*sqrt(a + b*x)*(3072*a**9*d - 4864*a**8*b*c - 1536*a**8*b*d*x + 2432*a** 7*b**2*c*x + 1152*a**7*b**2*d*x**2 - 1824*a**6*b**3*c*x**2 - 960*a**6*b**3 *d*x**3 + 1520*a**5*b**4*c*x**3 + 840*a**5*b**4*d*x**4 - 1330*a**4*b**5*c* x**4 - 756*a**4*b**5*d*x**5 + 1197*a**3*b**6*c*x**5 + 693*a**3*b**6*d*x**6 + 241395*a**2*b**7*c*x**6 + 207207*a**2*b**7*d*x**7 + 399399*a*b**8*c*x** 7 + 351351*a*b**8*d*x**8 + 171171*b**9*c*x**8 + 153153*b**9*d*x**9))/(2909 907*b**7)