Integrand size = 20, antiderivative size = 123 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {2 c^2 \left (b c^2+a d^2\right ) (c+d x)^{3/2}}{3 d^5}-\frac {4 c \left (2 b c^2+a d^2\right ) (c+d x)^{5/2}}{5 d^5}+\frac {2 \left (6 b c^2+a d^2\right ) (c+d x)^{7/2}}{7 d^5}-\frac {8 b c (c+d x)^{9/2}}{9 d^5}+\frac {2 b (c+d x)^{11/2}}{11 d^5} \] Output:
2/3*c^2*(a*d^2+b*c^2)*(d*x+c)^(3/2)/d^5-4/5*c*(a*d^2+2*b*c^2)*(d*x+c)^(5/2 )/d^5+2/7*(a*d^2+6*b*c^2)*(d*x+c)^(7/2)/d^5-8/9*b*c*(d*x+c)^(9/2)/d^5+2/11 *b*(d*x+c)^(11/2)/d^5
Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {2 (c+d x)^{3/2} \left (33 a d^2 \left (8 c^2-12 c d x+15 d^2 x^2\right )+b \left (128 c^4-192 c^3 d x+240 c^2 d^2 x^2-280 c d^3 x^3+315 d^4 x^4\right )\right )}{3465 d^5} \] Input:
Integrate[x^2*Sqrt[c + d*x]*(a + b*x^2),x]
Output:
(2*(c + d*x)^(3/2)*(33*a*d^2*(8*c^2 - 12*c*d*x + 15*d^2*x^2) + b*(128*c^4 - 192*c^3*d*x + 240*c^2*d^2*x^2 - 280*c*d^3*x^3 + 315*d^4*x^4)))/(3465*d^5 )
Time = 0.44 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
\(\displaystyle \int x^2 \left (a+b x^2\right ) \sqrt {c+d x} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (-\frac {2 (c+d x)^{3/2} \left (a c d^2+2 b c^3\right )}{d^4}+\frac {(c+d x)^{5/2} \left (a d^2+6 b c^2\right )}{d^4}+\frac {\sqrt {c+d x} \left (a c^2 d^2+b c^4\right )}{d^4}+\frac {b (c+d x)^{9/2}}{d^4}-\frac {4 b c (c+d x)^{7/2}}{d^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 (c+d x)^{7/2} \left (a d^2+6 b c^2\right )}{7 d^5}-\frac {4 c (c+d x)^{5/2} \left (a d^2+2 b c^2\right )}{5 d^5}+\frac {2 c^2 (c+d x)^{3/2} \left (a d^2+b c^2\right )}{3 d^5}+\frac {2 b (c+d x)^{11/2}}{11 d^5}-\frac {8 b c (c+d x)^{9/2}}{9 d^5}\) |
Input:
Int[x^2*Sqrt[c + d*x]*(a + b*x^2),x]
Output:
(2*c^2*(b*c^2 + a*d^2)*(c + d*x)^(3/2))/(3*d^5) - (4*c*(2*b*c^2 + a*d^2)*( c + d*x)^(5/2))/(5*d^5) + (2*(6*b*c^2 + a*d^2)*(c + d*x)^(7/2))/(7*d^5) - (8*b*c*(c + d*x)^(9/2))/(9*d^5) + (2*b*(c + d*x)^(11/2))/(11*d^5)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.60
method | result | size |
pseudoelliptic | \(\frac {16 \left (d x +c \right )^{\frac {3}{2}} \left (\frac {15 \left (\frac {7 b \,x^{2}}{11}+a \right ) x^{2} d^{4}}{8}-\frac {3 x c \left (\frac {70 b \,x^{2}}{99}+a \right ) d^{3}}{2}+c^{2} \left (\frac {10 b \,x^{2}}{11}+a \right ) d^{2}-\frac {8 b \,c^{3} d x}{11}+\frac {16 b \,c^{4}}{33}\right )}{105 d^{5}}\) | \(74\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (315 b \,x^{4} d^{4}-280 c b \,d^{3} x^{3}+495 a \,d^{4} x^{2}+240 b \,c^{2} d^{2} x^{2}-396 a c \,d^{3} x -192 b \,c^{3} d x +264 a \,c^{2} d^{2}+128 b \,c^{4}\right )}{3465 d^{5}}\) | \(85\) |
orering | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (315 b \,x^{4} d^{4}-280 c b \,d^{3} x^{3}+495 a \,d^{4} x^{2}+240 b \,c^{2} d^{2} x^{2}-396 a c \,d^{3} x -192 b \,c^{3} d x +264 a \,c^{2} d^{2}+128 b \,c^{4}\right )}{3465 d^{5}}\) | \(85\) |
derivativedivides | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {8 b c \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,d^{2}+6 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-2 b \,c^{3}-2 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c^{2} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{5}}\) | \(102\) |
default | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {8 b c \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,d^{2}+6 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-2 b \,c^{3}-2 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c^{2} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{5}}\) | \(102\) |
trager | \(\frac {2 \left (315 b \,d^{5} x^{5}+35 b c \,d^{4} x^{4}+495 x^{3} a \,d^{5}-40 b \,c^{2} d^{3} x^{3}+99 x^{2} a c \,d^{4}+48 b \,c^{3} d^{2} x^{2}-132 a \,c^{2} d^{3} x -64 b \,c^{4} d x +264 a \,c^{3} d^{2}+128 b \,c^{5}\right ) \sqrt {d x +c}}{3465 d^{5}}\) | \(109\) |
risch | \(\frac {2 \left (315 b \,d^{5} x^{5}+35 b c \,d^{4} x^{4}+495 x^{3} a \,d^{5}-40 b \,c^{2} d^{3} x^{3}+99 x^{2} a c \,d^{4}+48 b \,c^{3} d^{2} x^{2}-132 a \,c^{2} d^{3} x -64 b \,c^{4} d x +264 a \,c^{3} d^{2}+128 b \,c^{5}\right ) \sqrt {d x +c}}{3465 d^{5}}\) | \(109\) |
Input:
int(x^2*(d*x+c)^(1/2)*(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
16/105*(d*x+c)^(3/2)*(15/8*(7/11*b*x^2+a)*x^2*d^4-3/2*x*c*(70/99*b*x^2+a)* d^3+c^2*(10/11*b*x^2+a)*d^2-8/11*b*c^3*d*x+16/33*b*c^4)/d^5
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (315 \, b d^{5} x^{5} + 35 \, b c d^{4} x^{4} + 128 \, b c^{5} + 264 \, a c^{3} d^{2} - 5 \, {\left (8 \, b c^{2} d^{3} - 99 \, a d^{5}\right )} x^{3} + 3 \, {\left (16 \, b c^{3} d^{2} + 33 \, a c d^{4}\right )} x^{2} - 4 \, {\left (16 \, b c^{4} d + 33 \, a c^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{3465 \, d^{5}} \] Input:
integrate(x^2*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="fricas")
Output:
2/3465*(315*b*d^5*x^5 + 35*b*c*d^4*x^4 + 128*b*c^5 + 264*a*c^3*d^2 - 5*(8* b*c^2*d^3 - 99*a*d^5)*x^3 + 3*(16*b*c^3*d^2 + 33*a*c*d^4)*x^2 - 4*(16*b*c^ 4*d + 33*a*c^2*d^3)*x)*sqrt(d*x + c)/d^5
Time = 0.83 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.17 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\begin {cases} \frac {2 \left (- \frac {4 b c \left (c + d x\right )^{\frac {9}{2}}}{9 d^{2}} + \frac {b \left (c + d x\right )^{\frac {11}{2}}}{11 d^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (a d^{2} + 6 b c^{2}\right )}{7 d^{2}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (- 2 a c d^{2} - 4 b c^{3}\right )}{5 d^{2}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a c^{2} d^{2} + b c^{4}\right )}{3 d^{2}}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\frac {a x^{3}}{3} + \frac {b x^{5}}{5}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(d*x+c)**(1/2)*(b*x**2+a),x)
Output:
Piecewise((2*(-4*b*c*(c + d*x)**(9/2)/(9*d**2) + b*(c + d*x)**(11/2)/(11*d **2) + (c + d*x)**(7/2)*(a*d**2 + 6*b*c**2)/(7*d**2) + (c + d*x)**(5/2)*(- 2*a*c*d**2 - 4*b*c**3)/(5*d**2) + (c + d*x)**(3/2)*(a*c**2*d**2 + b*c**4)/ (3*d**2))/d**3, Ne(d, 0)), (sqrt(c)*(a*x**3/3 + b*x**5/5), True))
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b - 1540 \, {\left (d x + c\right )}^{\frac {9}{2}} b c + 495 \, {\left (6 \, b c^{2} + a d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 1386 \, {\left (2 \, b c^{3} + a c d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (b c^{4} + a c^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{3465 \, d^{5}} \] Input:
integrate(x^2*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="maxima")
Output:
2/3465*(315*(d*x + c)^(11/2)*b - 1540*(d*x + c)^(9/2)*b*c + 495*(6*b*c^2 + a*d^2)*(d*x + c)^(7/2) - 1386*(2*b*c^3 + a*c*d^2)*(d*x + c)^(5/2) + 1155* (b*c^4 + a*c^2*d^2)*(d*x + c)^(3/2))/d^5
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (103) = 206\).
Time = 0.13 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.89 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (\frac {231 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a c}{d^{2}} + \frac {99 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a}{d^{2}} + \frac {11 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b c}{d^{4}} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b}{d^{4}}\right )}}{3465 \, d} \] Input:
integrate(x^2*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="giac")
Output:
2/3465*(231*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c ^2)*a*c/d^2 + 99*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^ (3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a/d^2 + 11*(35*(d*x + c)^(9/2) - 180*(d* x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*s qrt(d*x + c)*c^4)*b*c/d^4 + 5*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/ 2)*c^4 - 693*sqrt(d*x + c)*c^5)*b/d^4)/d
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.87 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\left (2\,b\,c^4+2\,a\,c^2\,d^2\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}-\frac {\left (8\,b\,c^3+4\,a\,c\,d^2\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}+\frac {2\,b\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {\left (12\,b\,c^2+2\,a\,d^2\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}-\frac {8\,b\,c\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5} \] Input:
int(x^2*(a + b*x^2)*(c + d*x)^(1/2),x)
Output:
((2*b*c^4 + 2*a*c^2*d^2)*(c + d*x)^(3/2))/(3*d^5) - ((8*b*c^3 + 4*a*c*d^2) *(c + d*x)^(5/2))/(5*d^5) + (2*b*(c + d*x)^(11/2))/(11*d^5) + ((2*a*d^2 + 12*b*c^2)*(c + d*x)^(7/2))/(7*d^5) - (8*b*c*(c + d*x)^(9/2))/(9*d^5)
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.87 \[ \int x^2 \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {2 \sqrt {d x +c}\, \left (315 b \,d^{5} x^{5}+35 b c \,d^{4} x^{4}+495 a \,d^{5} x^{3}-40 b \,c^{2} d^{3} x^{3}+99 a c \,d^{4} x^{2}+48 b \,c^{3} d^{2} x^{2}-132 a \,c^{2} d^{3} x -64 b \,c^{4} d x +264 a \,c^{3} d^{2}+128 b \,c^{5}\right )}{3465 d^{5}} \] Input:
int(x^2*(d*x+c)^(1/2)*(b*x^2+a),x)
Output:
(2*sqrt(c + d*x)*(264*a*c**3*d**2 - 132*a*c**2*d**3*x + 99*a*c*d**4*x**2 + 495*a*d**5*x**3 + 128*b*c**5 - 64*b*c**4*d*x + 48*b*c**3*d**2*x**2 - 40*b *c**2*d**3*x**3 + 35*b*c*d**4*x**4 + 315*b*d**5*x**5))/(3465*d**5)