Integrand size = 21, antiderivative size = 167 \[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=-\frac {2 a^3 (b c-a d) \left (a x^2+b x^3\right )^{5/2}}{5 b^5 x^5}+\frac {2 a^2 (3 b c-4 a d) \left (a x^2+b x^3\right )^{7/2}}{7 b^5 x^7}-\frac {2 a (b c-2 a d) \left (a x^2+b x^3\right )^{9/2}}{3 b^5 x^9}+\frac {2 (b c-4 a d) \left (a x^2+b x^3\right )^{11/2}}{11 b^5 x^{11}}+\frac {2 d \left (a x^2+b x^3\right )^{13/2}}{13 b^5 x^{13}} \] Output:
-2/5*a^3*(-a*d+b*c)*(b*x^3+a*x^2)^(5/2)/b^5/x^5+2/7*a^2*(-4*a*d+3*b*c)*(b* x^3+a*x^2)^(7/2)/b^5/x^7-2/3*a*(-2*a*d+b*c)*(b*x^3+a*x^2)^(9/2)/b^5/x^9+2/ 11*(-4*a*d+b*c)*(b*x^3+a*x^2)^(11/2)/b^5/x^11+2/13*d*(b*x^3+a*x^2)^(13/2)/ b^5/x^13
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.59 \[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 x (a+b x)^3 \left (128 a^4 d+105 b^4 x^3 (13 c+11 d x)-70 a b^3 x^2 (13 c+12 d x)+40 a^2 b^2 x (13 c+14 d x)-16 a^3 b (13 c+20 d x)\right )}{15015 b^5 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[(c + d*x)*(a*x^2 + b*x^3)^(3/2),x]
Output:
(2*x*(a + b*x)^3*(128*a^4*d + 105*b^4*x^3*(13*c + 11*d*x) - 70*a*b^3*x^2*( 13*c + 12*d*x) + 40*a^2*b^2*x*(13*c + 14*d*x) - 16*a^3*b*(13*c + 20*d*x))) /(15015*b^5*Sqrt[x^2*(a + b*x)])
Time = 0.70 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.51, 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.
\(\displaystyle \int \left (a x^2+b x^3\right )^{3/2} (c+d x) \, dx\) |
\(\Big \downarrow \) 2450 |
\(\displaystyle \int \left (c \left (a x^2+b x^3\right )^{3/2}+d x \left (a x^2+b x^3\right )^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {256 a^4 d \left (a x^2+b x^3\right )^{5/2}}{15015 b^5 x^5}-\frac {32 a^3 c \left (a x^2+b x^3\right )^{5/2}}{1155 b^4 x^5}-\frac {128 a^3 d \left (a x^2+b x^3\right )^{5/2}}{3003 b^4 x^4}+\frac {16 a^2 c \left (a x^2+b x^3\right )^{5/2}}{231 b^3 x^4}+\frac {32 a^2 d \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac {4 a c \left (a x^2+b x^3\right )^{5/2}}{33 b^2 x^3}-\frac {16 a d \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac {2 c \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}+\frac {2 d \left (a x^2+b x^3\right )^{5/2}}{13 b x}\) |
Input:
Int[(c + d*x)*(a*x^2 + b*x^3)^(3/2),x]
Output:
(-32*a^3*c*(a*x^2 + b*x^3)^(5/2))/(1155*b^4*x^5) + (256*a^4*d*(a*x^2 + b*x ^3)^(5/2))/(15015*b^5*x^5) + (16*a^2*c*(a*x^2 + b*x^3)^(5/2))/(231*b^3*x^4 ) - (128*a^3*d*(a*x^2 + b*x^3)^(5/2))/(3003*b^4*x^4) - (4*a*c*(a*x^2 + b*x ^3)^(5/2))/(33*b^2*x^3) + (32*a^2*d*(a*x^2 + b*x^3)^(5/2))/(429*b^3*x^3) + (2*c*(a*x^2 + b*x^3)^(5/2))/(11*b*x^2) - (16*a*d*(a*x^2 + b*x^3)^(5/2))/( 143*b^2*x^2) + (2*d*(a*x^2 + b*x^3)^(5/2))/(13*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 = 0.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.16
method | result | size |
pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-5 b d x +2 a d -7 b c \right )}{35 b^{2}}\) | \(27\) |
gosper | \(\frac {2 \left (b x +a \right ) \left (1155 d \,x^{4} b^{4}-840 a \,b^{3} d \,x^{3}+1365 b^{4} c \,x^{3}+560 a^{2} b^{2} d \,x^{2}-910 a \,b^{3} c \,x^{2}-320 a^{3} b d x +520 a^{2} b^{2} c x +128 a^{4} d -208 a^{3} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{15015 b^{5} x^{3}}\) | \(109\) |
default | \(\frac {2 \left (b x +a \right ) \left (1155 d \,x^{4} b^{4}-840 a \,b^{3} d \,x^{3}+1365 b^{4} c \,x^{3}+560 a^{2} b^{2} d \,x^{2}-910 a \,b^{3} c \,x^{2}-320 a^{3} b d x +520 a^{2} b^{2} c x +128 a^{4} d -208 a^{3} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{15015 b^{5} x^{3}}\) | \(109\) |
orering | \(\frac {2 \left (b x +a \right ) \left (1155 d \,x^{4} b^{4}-840 a \,b^{3} d \,x^{3}+1365 b^{4} c \,x^{3}+560 a^{2} b^{2} d \,x^{2}-910 a \,b^{3} c \,x^{2}-320 a^{3} b d x +520 a^{2} b^{2} c x +128 a^{4} d -208 a^{3} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{15015 b^{5} x^{3}}\) | \(109\) |
risch | \(\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (1155 d \,x^{6} b^{6}+1470 a \,b^{5} d \,x^{5}+1365 b^{6} c \,x^{5}+35 a^{2} b^{4} d \,x^{4}+1820 a \,b^{5} c \,x^{4}-40 a^{3} b^{3} d \,x^{3}+65 a^{2} b^{4} c \,x^{3}+48 a^{4} b^{2} d \,x^{2}-78 a^{3} b^{3} c \,x^{2}-64 a^{5} b d x +104 a^{4} b^{2} c x +128 a^{6} d -208 a^{5} b c \right )}{15015 x \,b^{5}}\) | \(150\) |
trager | \(\frac {2 \left (1155 d \,x^{6} b^{6}+1470 a \,b^{5} d \,x^{5}+1365 b^{6} c \,x^{5}+35 a^{2} b^{4} d \,x^{4}+1820 a \,b^{5} c \,x^{4}-40 a^{3} b^{3} d \,x^{3}+65 a^{2} b^{4} c \,x^{3}+48 a^{4} b^{2} d \,x^{2}-78 a^{3} b^{3} c \,x^{2}-64 a^{5} b d x +104 a^{4} b^{2} c x +128 a^{6} d -208 a^{5} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{15015 b^{5} x}\) | \(152\) |
Input:
int((d*x+c)*(b*x^3+a*x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/35*(b*x+a)^(5/2)*(-5*b*d*x+2*a*d-7*b*c)/b^2
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.91 \[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, b^{6} d x^{6} - 208 \, a^{5} b c + 128 \, a^{6} d + 105 \, {\left (13 \, b^{6} c + 14 \, a b^{5} d\right )} x^{5} + 35 \, {\left (52 \, a b^{5} c + a^{2} b^{4} d\right )} x^{4} + 5 \, {\left (13 \, a^{2} b^{4} c - 8 \, a^{3} b^{3} d\right )} x^{3} - 6 \, {\left (13 \, a^{3} b^{3} c - 8 \, a^{4} b^{2} d\right )} x^{2} + 8 \, {\left (13 \, a^{4} b^{2} c - 8 \, a^{5} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{15015 \, b^{5} x} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(3/2),x, algorithm="fricas")
Output:
2/15015*(1155*b^6*d*x^6 - 208*a^5*b*c + 128*a^6*d + 105*(13*b^6*c + 14*a*b ^5*d)*x^5 + 35*(52*a*b^5*c + a^2*b^4*d)*x^4 + 5*(13*a^2*b^4*c - 8*a^3*b^3* d)*x^3 - 6*(13*a^3*b^3*c - 8*a^4*b^2*d)*x^2 + 8*(13*a^4*b^2*c - 8*a^5*b*d) *x)*sqrt(b*x^3 + a*x^2)/(b^5*x)
\[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\int \left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )\, dx \] Input:
integrate((d*x+c)*(b*x**3+a*x**2)**(3/2),x)
Output:
Integral((x**2*(a + b*x))**(3/2)*(c + d*x), x)
Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.85 \[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 \, {\left (105 \, b^{5} x^{5} + 140 \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{3} - 6 \, a^{3} b^{2} x^{2} + 8 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x + a} c}{1155 \, b^{4}} + \frac {2 \, {\left (1155 \, b^{6} x^{6} + 1470 \, a b^{5} x^{5} + 35 \, a^{2} b^{4} x^{4} - 40 \, a^{3} b^{3} x^{3} + 48 \, a^{4} b^{2} x^{2} - 64 \, a^{5} b x + 128 \, a^{6}\right )} \sqrt {b x + a} d}{15015 \, b^{5}} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(3/2),x, algorithm="maxima")
Output:
2/1155*(105*b^5*x^5 + 140*a*b^4*x^4 + 5*a^2*b^3*x^3 - 6*a^3*b^2*x^2 + 8*a^ 4*b*x - 16*a^5)*sqrt(b*x + a)*c/b^4 + 2/15015*(1155*b^6*x^6 + 1470*a*b^5*x ^5 + 35*a^2*b^4*x^4 - 40*a^3*b^3*x^3 + 48*a^4*b^2*x^2 - 64*a^5*b*x + 128*a ^6)*sqrt(b*x + a)*d/b^5
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (147) = 294\).
Time = 0.16 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.73 \[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Output:
2/45045*(1287*(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^2*c*sgn(x)/b^3 + 286*(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*c*sgn(x)/b^3 + 143*(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)*a^2*d*sgn(x)/b^4 + 65*(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)*c*sgn(x)/b^3 + 130*(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*d*sgn( x)/b^4 + 15*(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)*d*sgn(x)/b^4)/b + 32/15015*( 13*a^(11/2)*b*c - 8*a^(13/2)*d)*sgn(x)/b^5
Time = 9.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (x^5\,\left (\frac {28\,a\,d}{143}+\frac {2\,b\,c}{11}\right )+\frac {256\,a^6\,d-416\,a^5\,b\,c}{15015\,b^5}+\frac {2\,b\,d\,x^6}{13}-\frac {16\,a^4\,x\,\left (8\,a\,d-13\,b\,c\right )}{15015\,b^4}+\frac {2\,a\,x^4\,\left (a\,d+52\,b\,c\right )}{429\,b}-\frac {2\,a^2\,x^3\,\left (8\,a\,d-13\,b\,c\right )}{3003\,b^2}+\frac {4\,a^3\,x^2\,\left (8\,a\,d-13\,b\,c\right )}{5005\,b^3}\right )}{x} \] Input:
int((a*x^2 + b*x^3)^(3/2)*(c + d*x),x)
Output:
((a*x^2 + b*x^3)^(1/2)*(x^5*((28*a*d)/143 + (2*b*c)/11) + (256*a^6*d - 416 *a^5*b*c)/(15015*b^5) + (2*b*d*x^6)/13 - (16*a^4*x*(8*a*d - 13*b*c))/(1501 5*b^4) + (2*a*x^4*(a*d + 52*b*c))/(429*b) - (2*a^2*x^3*(8*a*d - 13*b*c))/( 3003*b^2) + (4*a^3*x^2*(8*a*d - 13*b*c))/(5005*b^3)))/x
Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.84 \[ \int (c+d x) \left (a x^2+b x^3\right )^{3/2} \, dx=\frac {2 \sqrt {b x +a}\, \left (1155 b^{6} d \,x^{6}+1470 a \,b^{5} d \,x^{5}+1365 b^{6} c \,x^{5}+35 a^{2} b^{4} d \,x^{4}+1820 a \,b^{5} c \,x^{4}-40 a^{3} b^{3} d \,x^{3}+65 a^{2} b^{4} c \,x^{3}+48 a^{4} b^{2} d \,x^{2}-78 a^{3} b^{3} c \,x^{2}-64 a^{5} b d x +104 a^{4} b^{2} c x +128 a^{6} d -208 a^{5} b c \right )}{15015 b^{5}} \] Input:
int((d*x+c)*(b*x^3+a*x^2)^(3/2),x)
Output:
(2*sqrt(a + b*x)*(128*a**6*d - 208*a**5*b*c - 64*a**5*b*d*x + 104*a**4*b** 2*c*x + 48*a**4*b**2*d*x**2 - 78*a**3*b**3*c*x**2 - 40*a**3*b**3*d*x**3 + 65*a**2*b**4*c*x**3 + 35*a**2*b**4*d*x**4 + 1820*a*b**5*c*x**4 + 1470*a*b* *5*d*x**5 + 1365*b**6*c*x**5 + 1155*b**6*d*x**6))/(15015*b**5)