Integrand size = 20, antiderivative size = 123 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 c^2 \left (b c^2+a d^2\right ) (c+d x)^{7/2}}{7 d^5}-\frac {4 c \left (2 b c^2+a d^2\right ) (c+d x)^{9/2}}{9 d^5}+\frac {2 \left (6 b c^2+a d^2\right ) (c+d x)^{11/2}}{11 d^5}-\frac {8 b c (c+d x)^{13/2}}{13 d^5}+\frac {2 b (c+d x)^{15/2}}{15 d^5} \] Output:
2/7*c^2*(a*d^2+b*c^2)*(d*x+c)^(7/2)/d^5-4/9*c*(a*d^2+2*b*c^2)*(d*x+c)^(9/2 )/d^5+2/11*(a*d^2+6*b*c^2)*(d*x+c)^(11/2)/d^5-8/13*b*c*(d*x+c)^(13/2)/d^5+ 2/15*b*(d*x+c)^(15/2)/d^5
Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 (c+d x)^{7/2} \left (65 a d^2 \left (8 c^2-28 c d x+63 d^2 x^2\right )+b \left (128 c^4-448 c^3 d x+1008 c^2 d^2 x^2-1848 c d^3 x^3+3003 d^4 x^4\right )\right )}{45045 d^5} \] Input:
Integrate[x^2*(c + d*x)^(5/2)*(a + b*x^2),x]
Output:
(2*(c + d*x)^(7/2)*(65*a*d^2*(8*c^2 - 28*c*d*x + 63*d^2*x^2) + b*(128*c^4 - 448*c^3*d*x + 1008*c^2*d^2*x^2 - 1848*c*d^3*x^3 + 3003*d^4*x^4)))/(45045 *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.
\(\displaystyle \int x^2 \left (a+b x^2\right ) (c+d x)^{5/2} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (-\frac {2 (c+d x)^{7/2} \left (a c d^2+2 b c^3\right )}{d^4}+\frac {(c+d x)^{9/2} \left (a d^2+6 b c^2\right )}{d^4}+\frac {(c+d x)^{5/2} \left (a c^2 d^2+b c^4\right )}{d^4}+\frac {b (c+d x)^{13/2}}{d^4}-\frac {4 b c (c+d x)^{11/2}}{d^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 (c+d x)^{11/2} \left (a d^2+6 b c^2\right )}{11 d^5}-\frac {4 c (c+d x)^{9/2} \left (a d^2+2 b c^2\right )}{9 d^5}+\frac {2 c^2 (c+d x)^{7/2} \left (a d^2+b c^2\right )}{7 d^5}+\frac {2 b (c+d x)^{15/2}}{15 d^5}-\frac {8 b c (c+d x)^{13/2}}{13 d^5}\) |
Input:
Int[x^2*(c + d*x)^(5/2)*(a + b*x^2),x]
Output:
(2*c^2*(b*c^2 + a*d^2)*(c + d*x)^(7/2))/(7*d^5) - (4*c*(2*b*c^2 + a*d^2)*( c + d*x)^(9/2))/(9*d^5) + (2*(6*b*c^2 + a*d^2)*(c + d*x)^(11/2))/(11*d^5) - (8*b*c*(c + d*x)^(13/2))/(13*d^5) + (2*b*(c + d*x)^(15/2))/(15*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 = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.60
method | result | size |
pseudoelliptic | \(\frac {16 \left (d x +c \right )^{\frac {7}{2}} \left (\frac {63 x^{2} \left (\frac {11 b \,x^{2}}{15}+a \right ) d^{4}}{8}-\frac {7 x \left (\frac {66 b \,x^{2}}{65}+a \right ) c \,d^{3}}{2}+c^{2} \left (\frac {126 b \,x^{2}}{65}+a \right ) d^{2}-\frac {56 b \,c^{3} d x}{65}+\frac {16 b \,c^{4}}{65}\right )}{693 d^{5}}\) | \(74\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (3003 b \,x^{4} d^{4}-1848 c b \,d^{3} x^{3}+4095 a \,d^{4} x^{2}+1008 b \,c^{2} d^{2} x^{2}-1820 a c \,d^{3} x -448 b \,c^{3} d x +520 a \,c^{2} d^{2}+128 b \,c^{4}\right )}{45045 d^{5}}\) | \(85\) |
orering | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (3003 b \,x^{4} d^{4}-1848 c b \,d^{3} x^{3}+4095 a \,d^{4} x^{2}+1008 b \,c^{2} d^{2} x^{2}-1820 a c \,d^{3} x -448 b \,c^{3} d x +520 a \,c^{2} d^{2}+128 b \,c^{4}\right )}{45045 d^{5}}\) | \(85\) |
derivativedivides | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {15}{2}}}{15}-\frac {8 b c \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (a \,d^{2}+6 b \,c^{2}\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-2 b \,c^{3}-2 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 c^{2} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{5}}\) | \(102\) |
default | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {15}{2}}}{15}-\frac {8 b c \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (a \,d^{2}+6 b \,c^{2}\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-2 b \,c^{3}-2 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 c^{2} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{5}}\) | \(102\) |
trager | \(\frac {2 \left (3003 b \,d^{7} x^{7}+7161 d^{6} b c \,x^{6}+4095 a \,d^{7} x^{5}+4473 b \,c^{2} d^{5} x^{5}+10465 a c \,d^{6} x^{4}+35 b \,c^{3} d^{4} x^{4}+7345 d^{5} a \,c^{2} x^{3}-40 b \,c^{4} d^{3} x^{3}+195 a \,c^{3} d^{4} x^{2}+48 b \,c^{5} d^{2} x^{2}-260 a \,c^{4} d^{3} x -64 b \,c^{6} d x +520 a \,c^{5} d^{2}+128 b \,c^{7}\right ) \sqrt {d x +c}}{45045 d^{5}}\) | \(157\) |
risch | \(\frac {2 \left (3003 b \,d^{7} x^{7}+7161 d^{6} b c \,x^{6}+4095 a \,d^{7} x^{5}+4473 b \,c^{2} d^{5} x^{5}+10465 a c \,d^{6} x^{4}+35 b \,c^{3} d^{4} x^{4}+7345 d^{5} a \,c^{2} x^{3}-40 b \,c^{4} d^{3} x^{3}+195 a \,c^{3} d^{4} x^{2}+48 b \,c^{5} d^{2} x^{2}-260 a \,c^{4} d^{3} x -64 b \,c^{6} d x +520 a \,c^{5} d^{2}+128 b \,c^{7}\right ) \sqrt {d x +c}}{45045 d^{5}}\) | \(157\) |
Input:
int(x^2*(d*x+c)^(5/2)*(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
16/693*(d*x+c)^(7/2)*(63/8*x^2*(11/15*b*x^2+a)*d^4-7/2*x*(66/65*b*x^2+a)*c *d^3+c^2*(126/65*b*x^2+a)*d^2-56/65*b*c^3*d*x+16/65*b*c^4)/d^5
Time = 0.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.28 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (3003 \, b d^{7} x^{7} + 7161 \, b c d^{6} x^{6} + 128 \, b c^{7} + 520 \, a c^{5} d^{2} + 63 \, {\left (71 \, b c^{2} d^{5} + 65 \, a d^{7}\right )} x^{5} + 35 \, {\left (b c^{3} d^{4} + 299 \, a c d^{6}\right )} x^{4} - 5 \, {\left (8 \, b c^{4} d^{3} - 1469 \, a c^{2} d^{5}\right )} x^{3} + 3 \, {\left (16 \, b c^{5} d^{2} + 65 \, a c^{3} d^{4}\right )} x^{2} - 4 \, {\left (16 \, b c^{6} d + 65 \, a c^{4} d^{3}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{5}} \] Input:
integrate(x^2*(d*x+c)^(5/2)*(b*x^2+a),x, algorithm="fricas")
Output:
2/45045*(3003*b*d^7*x^7 + 7161*b*c*d^6*x^6 + 128*b*c^7 + 520*a*c^5*d^2 + 6 3*(71*b*c^2*d^5 + 65*a*d^7)*x^5 + 35*(b*c^3*d^4 + 299*a*c*d^6)*x^4 - 5*(8* b*c^4*d^3 - 1469*a*c^2*d^5)*x^3 + 3*(16*b*c^5*d^2 + 65*a*c^3*d^4)*x^2 - 4* (16*b*c^6*d + 65*a*c^4*d^3)*x)*sqrt(d*x + c)/d^5
Time = 0.75 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.17 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\begin {cases} \frac {2 \left (- \frac {4 b c \left (c + d x\right )^{\frac {13}{2}}}{13 d^{2}} + \frac {b \left (c + d x\right )^{\frac {15}{2}}}{15 d^{2}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \left (a d^{2} + 6 b c^{2}\right )}{11 d^{2}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (- 2 a c d^{2} - 4 b c^{3}\right )}{9 d^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (a c^{2} d^{2} + b c^{4}\right )}{7 d^{2}}\right )}{d^{3}} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {a x^{3}}{3} + \frac {b x^{5}}{5}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(d*x+c)**(5/2)*(b*x**2+a),x)
Output:
Piecewise((2*(-4*b*c*(c + d*x)**(13/2)/(13*d**2) + b*(c + d*x)**(15/2)/(15 *d**2) + (c + d*x)**(11/2)*(a*d**2 + 6*b*c**2)/(11*d**2) + (c + d*x)**(9/2 )*(-2*a*c*d**2 - 4*b*c**3)/(9*d**2) + (c + d*x)**(7/2)*(a*c**2*d**2 + b*c* *4)/(7*d**2))/d**3, Ne(d, 0)), (c**(5/2)*(a*x**3/3 + b*x**5/5), True))
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b - 13860 \, {\left (d x + c\right )}^{\frac {13}{2}} b c + 4095 \, {\left (6 \, b c^{2} + a d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 10010 \, {\left (2 \, b c^{3} + a c d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 6435 \, {\left (b c^{4} + a c^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{45045 \, d^{5}} \] Input:
integrate(x^2*(d*x+c)^(5/2)*(b*x^2+a),x, algorithm="maxima")
Output:
2/45045*(3003*(d*x + c)^(15/2)*b - 13860*(d*x + c)^(13/2)*b*c + 4095*(6*b* c^2 + a*d^2)*(d*x + c)^(11/2) - 10010*(2*b*c^3 + a*c*d^2)*(d*x + c)^(9/2) + 6435*(b*c^4 + a*c^2*d^2)*(d*x + c)^(7/2))/d^5
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (103) = 206\).
Time = 0.13 (sec) , antiderivative size = 564, normalized size of antiderivative = 4.59 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate(x^2*(d*x+c)^(5/2)*(b*x^2+a),x, algorithm="giac")
Output:
2/45045*(3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c) *c^2)*a*c^3/d^2 + 3861*(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*c^2/d^2 + 143*(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*sqrt(d*x + c)*c^4)*b*c^3/d^4 + 429*(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)*a*c/d^2 + 195*(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*c^2/d^4 + 65*(63*(d*x + c)^(11/2) - 38 5*(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)*a/d^2 + 45*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 30 03*sqrt(d*x + c)*c^6)*b*c/d^4 + 7*(429*(d*x + c)^(15/2) - 3465*(d*x + c)^( 13/2)*c + 12285*(d*x + c)^(11/2)*c^2 - 25025*(d*x + c)^(9/2)*c^3 + 32175*( d*x + c)^(7/2)*c^4 - 27027*(d*x + c)^(5/2)*c^5 + 15015*(d*x + c)^(3/2)*c^6 - 6435*sqrt(d*x + c)*c^7)*b/d^4)/d
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.87 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {\left (2\,b\,c^4+2\,a\,c^2\,d^2\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}-\frac {\left (8\,b\,c^3+4\,a\,c\,d^2\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,b\,{\left (c+d\,x\right )}^{15/2}}{15\,d^5}+\frac {\left (12\,b\,c^2+2\,a\,d^2\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}-\frac {8\,b\,c\,{\left (c+d\,x\right )}^{13/2}}{13\,d^5} \] Input:
int(x^2*(a + b*x^2)*(c + d*x)^(5/2),x)
Output:
((2*b*c^4 + 2*a*c^2*d^2)*(c + d*x)^(7/2))/(7*d^5) - ((8*b*c^3 + 4*a*c*d^2) *(c + d*x)^(9/2))/(9*d^5) + (2*b*(c + d*x)^(15/2))/(15*d^5) + ((2*a*d^2 + 12*b*c^2)*(c + d*x)^(11/2))/(11*d^5) - (8*b*c*(c + d*x)^(13/2))/(13*d^5)
Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.26 \[ \int x^2 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 \sqrt {d x +c}\, \left (3003 b \,d^{7} x^{7}+7161 b c \,d^{6} x^{6}+4095 a \,d^{7} x^{5}+4473 b \,c^{2} d^{5} x^{5}+10465 a c \,d^{6} x^{4}+35 b \,c^{3} d^{4} x^{4}+7345 a \,c^{2} d^{5} x^{3}-40 b \,c^{4} d^{3} x^{3}+195 a \,c^{3} d^{4} x^{2}+48 b \,c^{5} d^{2} x^{2}-260 a \,c^{4} d^{3} x -64 b \,c^{6} d x +520 a \,c^{5} d^{2}+128 b \,c^{7}\right )}{45045 d^{5}} \] Input:
int(x^2*(d*x+c)^(5/2)*(b*x^2+a),x)
Output:
(2*sqrt(c + d*x)*(520*a*c**5*d**2 - 260*a*c**4*d**3*x + 195*a*c**3*d**4*x* *2 + 7345*a*c**2*d**5*x**3 + 10465*a*c*d**6*x**4 + 4095*a*d**7*x**5 + 128* b*c**7 - 64*b*c**6*d*x + 48*b*c**5*d**2*x**2 - 40*b*c**4*d**3*x**3 + 35*b* c**3*d**4*x**4 + 4473*b*c**2*d**5*x**5 + 7161*b*c*d**6*x**6 + 3003*b*d**7* x**7))/(45045*d**5)