Integrand size = 22, antiderivative size = 129 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2 a c^3 (e x)^{3/2}}{3 e}+\frac {6 a c^2 d (e x)^{5/2}}{5 e^2}+\frac {2 c \left (b c^2+3 a d^2\right ) (e x)^{7/2}}{7 e^3}+\frac {2 d \left (3 b c^2+a d^2\right ) (e x)^{9/2}}{9 e^4}+\frac {6 b c d^2 (e x)^{11/2}}{11 e^5}+\frac {2 b d^3 (e x)^{13/2}}{13 e^6} \] Output:
2/3*a*c^3*(e*x)^(3/2)/e+6/5*a*c^2*d*(e*x)^(5/2)/e^2+2/7*c*(3*a*d^2+b*c^2)* (e*x)^(7/2)/e^3+2/9*d*(a*d^2+3*b*c^2)*(e*x)^(9/2)/e^4+6/11*b*c*d^2*(e*x)^( 11/2)/e^5+2/13*b*d^3*(e*x)^(13/2)/e^6
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2 x \sqrt {e x} \left (143 a \left (105 c^3+189 c^2 d x+135 c d^2 x^2+35 d^3 x^3\right )+15 b x^2 \left (429 c^3+1001 c^2 d x+819 c d^2 x^2+231 d^3 x^3\right )\right )}{45045} \] Input:
Integrate[Sqrt[e*x]*(c + d*x)^3*(a + b*x^2),x]
Output:
(2*x*Sqrt[e*x]*(143*a*(105*c^3 + 189*c^2*d*x + 135*c*d^2*x^2 + 35*d^3*x^3) + 15*b*x^2*(429*c^3 + 1001*c^2*d*x + 819*c*d^2*x^2 + 231*d^3*x^3)))/45045
Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int \sqrt {e x} \left (a+b x^2\right ) (c+d x)^3 \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {d (e x)^{7/2} \left (a d^2+3 b c^2\right )}{e^3}+\frac {c (e x)^{5/2} \left (3 a d^2+b c^2\right )}{e^2}+a c^3 \sqrt {e x}+\frac {3 a c^2 d (e x)^{3/2}}{e}+\frac {3 b c d^2 (e x)^{9/2}}{e^4}+\frac {b d^3 (e x)^{11/2}}{e^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 d (e x)^{9/2} \left (a d^2+3 b c^2\right )}{9 e^4}+\frac {2 c (e x)^{7/2} \left (3 a d^2+b c^2\right )}{7 e^3}+\frac {2 a c^3 (e x)^{3/2}}{3 e}+\frac {6 a c^2 d (e x)^{5/2}}{5 e^2}+\frac {6 b c d^2 (e x)^{11/2}}{11 e^5}+\frac {2 b d^3 (e x)^{13/2}}{13 e^6}\) |
Input:
Int[Sqrt[e*x]*(c + d*x)^3*(a + b*x^2),x]
Output:
(2*a*c^3*(e*x)^(3/2))/(3*e) + (6*a*c^2*d*(e*x)^(5/2))/(5*e^2) + (2*c*(b*c^ 2 + 3*a*d^2)*(e*x)^(7/2))/(7*e^3) + (2*d*(3*b*c^2 + a*d^2)*(e*x)^(9/2))/(9 *e^4) + (6*b*c*d^2*(e*x)^(11/2))/(11*e^5) + (2*b*d^3*(e*x)^(13/2))/(13*e^6 )
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 = 77, normalized size of antiderivative = 0.60
method | result | size |
pseudoelliptic | \(\frac {2 x \sqrt {e x}\, \left (\frac {3 b \,d^{3} x^{5}}{13}+\frac {9 b c \,d^{2} x^{4}}{11}+\left (\frac {1}{3} a \,d^{3}+b \,c^{2} d \right ) x^{3}+\frac {3 \left (3 a \,d^{2} c +b \,c^{3}\right ) x^{2}}{7}+\frac {9 a d x \,c^{2}}{5}+c^{3} a \right )}{3}\) | \(77\) |
gosper | \(\frac {2 x \left (3465 b \,d^{3} x^{5}+12285 b c \,d^{2} x^{4}+5005 a \,x^{3} d^{3}+15015 b \,c^{2} d \,x^{3}+19305 a \,d^{2} x^{2} c +6435 b \,c^{3} x^{2}+27027 a d x \,c^{2}+15015 c^{3} a \right ) \sqrt {e x}}{45045}\) | \(81\) |
trager | \(\frac {2 x \left (3465 b \,d^{3} x^{5}+12285 b c \,d^{2} x^{4}+5005 a \,x^{3} d^{3}+15015 b \,c^{2} d \,x^{3}+19305 a \,d^{2} x^{2} c +6435 b \,c^{3} x^{2}+27027 a d x \,c^{2}+15015 c^{3} a \right ) \sqrt {e x}}{45045}\) | \(81\) |
orering | \(\frac {2 x \left (3465 b \,d^{3} x^{5}+12285 b c \,d^{2} x^{4}+5005 a \,x^{3} d^{3}+15015 b \,c^{2} d \,x^{3}+19305 a \,d^{2} x^{2} c +6435 b \,c^{3} x^{2}+27027 a d x \,c^{2}+15015 c^{3} a \right ) \sqrt {e x}}{45045}\) | \(81\) |
risch | \(\frac {2 e \,x^{2} \left (3465 b \,d^{3} x^{5}+12285 b c \,d^{2} x^{4}+5005 a \,x^{3} d^{3}+15015 b \,c^{2} d \,x^{3}+19305 a \,d^{2} x^{2} c +6435 b \,c^{3} x^{2}+27027 a d x \,c^{2}+15015 c^{3} a \right )}{45045 \sqrt {e x}}\) | \(84\) |
derivativedivides | \(\frac {\frac {2 b \,d^{3} \left (e x \right )^{\frac {13}{2}}}{13}+\frac {6 c e \,d^{2} b \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,d^{3} e^{2}+3 b \,c^{2} d \,e^{2}\right ) \left (e x \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 a c \,d^{2} e^{3}+b \,c^{3} e^{3}\right ) \left (e x \right )^{\frac {7}{2}}}{7}+\frac {6 c^{2} e^{4} d a \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 c^{3} e^{5} a \left (e x \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(112\) |
default | \(\frac {\frac {2 b \,d^{3} \left (e x \right )^{\frac {13}{2}}}{13}+\frac {6 c e \,d^{2} b \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,d^{3} e^{2}+3 b \,c^{2} d \,e^{2}\right ) \left (e x \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 a c \,d^{2} e^{3}+b \,c^{3} e^{3}\right ) \left (e x \right )^{\frac {7}{2}}}{7}+\frac {6 c^{2} e^{4} d a \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 c^{3} e^{5} a \left (e x \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(112\) |
Input:
int((e*x)^(1/2)*(d*x+c)^3*(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
2/3*x*(e*x)^(1/2)*(3/13*b*d^3*x^5+9/11*b*c*d^2*x^4+(1/3*a*d^3+b*c^2*d)*x^3 +3/7*(3*a*c*d^2+b*c^3)*x^2+9/5*a*d*x*c^2+c^3*a)
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.62 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2}{45045} \, {\left (3465 \, b d^{3} x^{6} + 12285 \, b c d^{2} x^{5} + 27027 \, a c^{2} d x^{2} + 15015 \, a c^{3} x + 5005 \, {\left (3 \, b c^{2} d + a d^{3}\right )} x^{4} + 6435 \, {\left (b c^{3} + 3 \, a c d^{2}\right )} x^{3}\right )} \sqrt {e x} \] Input:
integrate((e*x)^(1/2)*(d*x+c)^3*(b*x^2+a),x, algorithm="fricas")
Output:
2/45045*(3465*b*d^3*x^6 + 12285*b*c*d^2*x^5 + 27027*a*c^2*d*x^2 + 15015*a* c^3*x + 5005*(3*b*c^2*d + a*d^3)*x^4 + 6435*(b*c^3 + 3*a*c*d^2)*x^3)*sqrt( e*x)
Time = 0.16 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.19 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2 a c^{3} x \sqrt {e x}}{3} + \frac {6 a c^{2} d x^{2} \sqrt {e x}}{5} + \frac {6 a c d^{2} x^{3} \sqrt {e x}}{7} + \frac {2 a d^{3} x^{4} \sqrt {e x}}{9} + \frac {2 b c^{3} x^{3} \sqrt {e x}}{7} + \frac {2 b c^{2} d x^{4} \sqrt {e x}}{3} + \frac {6 b c d^{2} x^{5} \sqrt {e x}}{11} + \frac {2 b d^{3} x^{6} \sqrt {e x}}{13} \] Input:
integrate((e*x)**(1/2)*(d*x+c)**3*(b*x**2+a),x)
Output:
2*a*c**3*x*sqrt(e*x)/3 + 6*a*c**2*d*x**2*sqrt(e*x)/5 + 6*a*c*d**2*x**3*sqr t(e*x)/7 + 2*a*d**3*x**4*sqrt(e*x)/9 + 2*b*c**3*x**3*sqrt(e*x)/7 + 2*b*c** 2*d*x**4*sqrt(e*x)/3 + 6*b*c*d**2*x**5*sqrt(e*x)/11 + 2*b*d**3*x**6*sqrt(e *x)/13
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (3465 \, \left (e x\right )^{\frac {13}{2}} b d^{3} + 12285 \, \left (e x\right )^{\frac {11}{2}} b c d^{2} e + 27027 \, \left (e x\right )^{\frac {5}{2}} a c^{2} d e^{4} + 15015 \, \left (e x\right )^{\frac {3}{2}} a c^{3} e^{5} + 5005 \, {\left (3 \, b c^{2} d + a d^{3}\right )} \left (e x\right )^{\frac {9}{2}} e^{2} + 6435 \, {\left (b c^{3} + 3 \, a c d^{2}\right )} \left (e x\right )^{\frac {7}{2}} e^{3}\right )}}{45045 \, e^{6}} \] Input:
integrate((e*x)^(1/2)*(d*x+c)^3*(b*x^2+a),x, algorithm="maxima")
Output:
2/45045*(3465*(e*x)^(13/2)*b*d^3 + 12285*(e*x)^(11/2)*b*c*d^2*e + 27027*(e *x)^(5/2)*a*c^2*d*e^4 + 15015*(e*x)^(3/2)*a*c^3*e^5 + 5005*(3*b*c^2*d + a* d^3)*(e*x)^(9/2)*e^2 + 6435*(b*c^3 + 3*a*c*d^2)*(e*x)^(7/2)*e^3)/e^6
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.89 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2}{13} \, \sqrt {e x} b d^{3} x^{6} + \frac {6}{11} \, \sqrt {e x} b c d^{2} x^{5} + \frac {2}{3} \, \sqrt {e x} b c^{2} d x^{4} + \frac {2}{9} \, \sqrt {e x} a d^{3} x^{4} + \frac {2}{7} \, \sqrt {e x} b c^{3} x^{3} + \frac {6}{7} \, \sqrt {e x} a c d^{2} x^{3} + \frac {6}{5} \, \sqrt {e x} a c^{2} d x^{2} + \frac {2}{3} \, \sqrt {e x} a c^{3} x \] Input:
integrate((e*x)^(1/2)*(d*x+c)^3*(b*x^2+a),x, algorithm="giac")
Output:
2/13*sqrt(e*x)*b*d^3*x^6 + 6/11*sqrt(e*x)*b*c*d^2*x^5 + 2/3*sqrt(e*x)*b*c^ 2*d*x^4 + 2/9*sqrt(e*x)*a*d^3*x^4 + 2/7*sqrt(e*x)*b*c^3*x^3 + 6/7*sqrt(e*x )*a*c*d^2*x^3 + 6/5*sqrt(e*x)*a*c^2*d*x^2 + 2/3*sqrt(e*x)*a*c^3*x
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2\,c\,{\left (e\,x\right )}^{7/2}\,\left (b\,c^2+3\,a\,d^2\right )}{7\,e^3}+\frac {2\,d\,{\left (e\,x\right )}^{9/2}\,\left (3\,b\,c^2+a\,d^2\right )}{9\,e^4}+\frac {2\,a\,c^3\,{\left (e\,x\right )}^{3/2}}{3\,e}+\frac {2\,b\,d^3\,{\left (e\,x\right )}^{13/2}}{13\,e^6}+\frac {6\,a\,c^2\,d\,{\left (e\,x\right )}^{5/2}}{5\,e^2}+\frac {6\,b\,c\,d^2\,{\left (e\,x\right )}^{11/2}}{11\,e^5} \] Input:
int((e*x)^(1/2)*(a + b*x^2)*(c + d*x)^3,x)
Output:
(2*c*(e*x)^(7/2)*(3*a*d^2 + b*c^2))/(7*e^3) + (2*d*(e*x)^(9/2)*(a*d^2 + 3* b*c^2))/(9*e^4) + (2*a*c^3*(e*x)^(3/2))/(3*e) + (2*b*d^3*(e*x)^(13/2))/(13 *e^6) + (6*a*c^2*d*(e*x)^(5/2))/(5*e^2) + (6*b*c*d^2*(e*x)^(11/2))/(11*e^5 )
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.61 \[ \int \sqrt {e x} (c+d x)^3 \left (a+b x^2\right ) \, dx=\frac {2 \sqrt {x}\, \sqrt {e}\, x \left (3465 b \,d^{3} x^{5}+12285 b c \,d^{2} x^{4}+5005 a \,d^{3} x^{3}+15015 b \,c^{2} d \,x^{3}+19305 a c \,d^{2} x^{2}+6435 b \,c^{3} x^{2}+27027 a \,c^{2} d x +15015 a \,c^{3}\right )}{45045} \] Input:
int((e*x)^(1/2)*(d*x+c)^3*(b*x^2+a),x)
Output:
(2*sqrt(x)*sqrt(e)*x*(15015*a*c**3 + 27027*a*c**2*d*x + 19305*a*c*d**2*x** 2 + 5005*a*d**3*x**3 + 6435*b*c**3*x**2 + 15015*b*c**2*d*x**3 + 12285*b*c* d**2*x**4 + 3465*b*d**3*x**5))/45045