Integrand size = 22, antiderivative size = 125 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {2 a c^3 \sqrt {e x}}{e}+\frac {2 a c^2 d (e x)^{3/2}}{e^2}+\frac {2 c \left (b c^2+3 a d^2\right ) (e x)^{5/2}}{5 e^3}+\frac {2 d \left (3 b c^2+a d^2\right ) (e x)^{7/2}}{7 e^4}+\frac {2 b c d^2 (e x)^{9/2}}{3 e^5}+\frac {2 b d^3 (e x)^{11/2}}{11 e^6} \] Output:
2*a*c^3*(e*x)^(1/2)/e+2*a*c^2*d*(e*x)^(3/2)/e^2+2/5*c*(3*a*d^2+b*c^2)*(e*x )^(5/2)/e^3+2/7*d*(a*d^2+3*b*c^2)*(e*x)^(7/2)/e^4+2/3*b*c*d^2*(e*x)^(9/2)/ e^5+2/11*b*d^3*(e*x)^(11/2)/e^6
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {66 a x \left (35 c^3+35 c^2 d x+21 c d^2 x^2+5 d^3 x^3\right )+2 b x^3 \left (231 c^3+495 c^2 d x+385 c d^2 x^2+105 d^3 x^3\right )}{1155 \sqrt {e x}} \] Input:
Integrate[((c + d*x)^3*(a + b*x^2))/Sqrt[e*x],x]
Output:
(66*a*x*(35*c^3 + 35*c^2*d*x + 21*c*d^2*x^2 + 5*d^3*x^3) + 2*b*x^3*(231*c^ 3 + 495*c^2*d*x + 385*c*d^2*x^2 + 105*d^3*x^3))/(1155*Sqrt[e*x])
Time = 0.43 (sec) , antiderivative size = 125, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right ) (c+d x)^3}{\sqrt {e x}} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {d (e x)^{5/2} \left (a d^2+3 b c^2\right )}{e^3}+\frac {c (e x)^{3/2} \left (3 a d^2+b c^2\right )}{e^2}+\frac {a c^3}{\sqrt {e x}}+\frac {3 a c^2 d \sqrt {e x}}{e}+\frac {3 b c d^2 (e x)^{7/2}}{e^4}+\frac {b d^3 (e x)^{9/2}}{e^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 d (e x)^{7/2} \left (a d^2+3 b c^2\right )}{7 e^4}+\frac {2 c (e x)^{5/2} \left (3 a d^2+b c^2\right )}{5 e^3}+\frac {2 a c^3 \sqrt {e x}}{e}+\frac {2 a c^2 d (e x)^{3/2}}{e^2}+\frac {2 b c d^2 (e x)^{9/2}}{3 e^5}+\frac {2 b d^3 (e x)^{11/2}}{11 e^6}\) |
Input:
Int[((c + d*x)^3*(a + b*x^2))/Sqrt[e*x],x]
Output:
(2*a*c^3*Sqrt[e*x])/e + (2*a*c^2*d*(e*x)^(3/2))/e^2 + (2*c*(b*c^2 + 3*a*d^ 2)*(e*x)^(5/2))/(5*e^3) + (2*d*(3*b*c^2 + a*d^2)*(e*x)^(7/2))/(7*e^4) + (2 *b*c*d^2*(e*x)^(9/2))/(3*e^5) + (2*b*d^3*(e*x)^(11/2))/(11*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 = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.57
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {b \,x^{2}}{5}+a \right ) c^{3}+d x \left (\frac {3 b \,x^{2}}{7}+a \right ) c^{2}+\frac {3 d^{2} x^{2} \left (\frac {5 b \,x^{2}}{9}+a \right ) c}{5}+\frac {d^{3} x^{3} \left (\frac {7 b \,x^{2}}{11}+a \right )}{7}\right ) \sqrt {e x}}{e}\) | \(71\) |
gosper | \(\frac {2 x \left (105 b \,d^{3} x^{5}+385 b c \,d^{2} x^{4}+165 a \,x^{3} d^{3}+495 b \,c^{2} d \,x^{3}+693 a \,d^{2} x^{2} c +231 b \,c^{3} x^{2}+1155 a d x \,c^{2}+1155 c^{3} a \right )}{1155 \sqrt {e x}}\) | \(81\) |
risch | \(\frac {2 x \left (105 b \,d^{3} x^{5}+385 b c \,d^{2} x^{4}+165 a \,x^{3} d^{3}+495 b \,c^{2} d \,x^{3}+693 a \,d^{2} x^{2} c +231 b \,c^{3} x^{2}+1155 a d x \,c^{2}+1155 c^{3} a \right )}{1155 \sqrt {e x}}\) | \(81\) |
orering | \(\frac {2 x \left (105 b \,d^{3} x^{5}+385 b c \,d^{2} x^{4}+165 a \,x^{3} d^{3}+495 b \,c^{2} d \,x^{3}+693 a \,d^{2} x^{2} c +231 b \,c^{3} x^{2}+1155 a d x \,c^{2}+1155 c^{3} a \right )}{1155 \sqrt {e x}}\) | \(81\) |
trager | \(\frac {\left (\frac {2}{11} b \,d^{3} x^{5}+\frac {2}{3} b c \,d^{2} x^{4}+\frac {2}{7} a \,x^{3} d^{3}+\frac {6}{7} b \,c^{2} d \,x^{3}+\frac {6}{5} a \,d^{2} x^{2} c +\frac {2}{5} b \,c^{3} x^{2}+2 a d x \,c^{2}+2 c^{3} a \right ) \sqrt {e x}}{e}\) | \(82\) |
derivativedivides | \(\frac {\frac {2 b \,d^{3} \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 c e \,d^{2} b \left (e x \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a \,d^{3} e^{2}+3 b \,c^{2} d \,e^{2}\right ) \left (e x \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 a c \,d^{2} e^{3}+b \,c^{3} e^{3}\right ) \left (e x \right )^{\frac {5}{2}}}{5}+2 c^{2} e^{4} d a \left (e x \right )^{\frac {3}{2}}+2 c^{3} e^{5} a \sqrt {e x}}{e^{6}}\) | \(110\) |
default | \(\frac {\frac {2 b \,d^{3} \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 c e \,d^{2} b \left (e x \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a \,d^{3} e^{2}+3 b \,c^{2} d \,e^{2}\right ) \left (e x \right )^{\frac {7}{2}}}{7}+\frac {2 \left (3 a c \,d^{2} e^{3}+b \,c^{3} e^{3}\right ) \left (e x \right )^{\frac {5}{2}}}{5}+2 c^{2} e^{4} d a \left (e x \right )^{\frac {3}{2}}+2 c^{3} e^{5} a \sqrt {e x}}{e^{6}}\) | \(110\) |
Input:
int((d*x+c)^3*(b*x^2+a)/(e*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*((1/5*b*x^2+a)*c^3+d*x*(3/7*b*x^2+a)*c^2+3/5*d^2*x^2*(5/9*b*x^2+a)*c+1/7 *d^3*x^3*(7/11*b*x^2+a))*(e*x)^(1/2)/e
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.64 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {2 \, {\left (105 \, b d^{3} x^{5} + 385 \, b c d^{2} x^{4} + 1155 \, a c^{2} d x + 1155 \, a c^{3} + 165 \, {\left (3 \, b c^{2} d + a d^{3}\right )} x^{3} + 231 \, {\left (b c^{3} + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt {e x}}{1155 \, e} \] Input:
integrate((d*x+c)^3*(b*x^2+a)/(e*x)^(1/2),x, algorithm="fricas")
Output:
2/1155*(105*b*d^3*x^5 + 385*b*c*d^2*x^4 + 1155*a*c^2*d*x + 1155*a*c^3 + 16 5*(3*b*c^2*d + a*d^3)*x^3 + 231*(b*c^3 + 3*a*c*d^2)*x^2)*sqrt(e*x)/e
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {2 a c^{3} x}{\sqrt {e x}} + \frac {2 a c^{2} d x^{2}}{\sqrt {e x}} + \frac {6 a c d^{2} x^{3}}{5 \sqrt {e x}} + \frac {2 a d^{3} x^{4}}{7 \sqrt {e x}} + \frac {2 b c^{3} x^{3}}{5 \sqrt {e x}} + \frac {6 b c^{2} d x^{4}}{7 \sqrt {e x}} + \frac {2 b c d^{2} x^{5}}{3 \sqrt {e x}} + \frac {2 b d^{3} x^{6}}{11 \sqrt {e x}} \] Input:
integrate((d*x+c)**3*(b*x**2+a)/(e*x)**(1/2),x)
Output:
2*a*c**3*x/sqrt(e*x) + 2*a*c**2*d*x**2/sqrt(e*x) + 6*a*c*d**2*x**3/(5*sqrt (e*x)) + 2*a*d**3*x**4/(7*sqrt(e*x)) + 2*b*c**3*x**3/(5*sqrt(e*x)) + 6*b*c **2*d*x**4/(7*sqrt(e*x)) + 2*b*c*d**2*x**5/(3*sqrt(e*x)) + 2*b*d**3*x**6/( 11*sqrt(e*x))
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {2 \, {\left (105 \, \left (e x\right )^{\frac {11}{2}} b d^{3} + 385 \, \left (e x\right )^{\frac {9}{2}} b c d^{2} e + 1155 \, \left (e x\right )^{\frac {3}{2}} a c^{2} d e^{4} + 1155 \, \sqrt {e x} a c^{3} e^{5} + 165 \, {\left (3 \, b c^{2} d + a d^{3}\right )} \left (e x\right )^{\frac {7}{2}} e^{2} + 231 \, {\left (b c^{3} + 3 \, a c d^{2}\right )} \left (e x\right )^{\frac {5}{2}} e^{3}\right )}}{1155 \, e^{6}} \] Input:
integrate((d*x+c)^3*(b*x^2+a)/(e*x)^(1/2),x, algorithm="maxima")
Output:
2/1155*(105*(e*x)^(11/2)*b*d^3 + 385*(e*x)^(9/2)*b*c*d^2*e + 1155*(e*x)^(3 /2)*a*c^2*d*e^4 + 1155*sqrt(e*x)*a*c^3*e^5 + 165*(3*b*c^2*d + a*d^3)*(e*x) ^(7/2)*e^2 + 231*(b*c^3 + 3*a*c*d^2)*(e*x)^(5/2)*e^3)/e^6
Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {2 \, {\left (105 \, \sqrt {e x} b d^{3} x^{5} + 385 \, \sqrt {e x} b c d^{2} x^{4} + 495 \, \sqrt {e x} b c^{2} d x^{3} + 165 \, \sqrt {e x} a d^{3} x^{3} + 231 \, \sqrt {e x} b c^{3} x^{2} + 693 \, \sqrt {e x} a c d^{2} x^{2} + 1155 \, \sqrt {e x} a c^{2} d x + 1155 \, \sqrt {e x} a c^{3}\right )}}{1155 \, e} \] Input:
integrate((d*x+c)^3*(b*x^2+a)/(e*x)^(1/2),x, algorithm="giac")
Output:
2/1155*(105*sqrt(e*x)*b*d^3*x^5 + 385*sqrt(e*x)*b*c*d^2*x^4 + 495*sqrt(e*x )*b*c^2*d*x^3 + 165*sqrt(e*x)*a*d^3*x^3 + 231*sqrt(e*x)*b*c^3*x^2 + 693*sq rt(e*x)*a*c*d^2*x^2 + 1155*sqrt(e*x)*a*c^2*d*x + 1155*sqrt(e*x)*a*c^3)/e
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {2\,c\,{\left (e\,x\right )}^{5/2}\,\left (b\,c^2+3\,a\,d^2\right )}{5\,e^3}+\frac {2\,d\,{\left (e\,x\right )}^{7/2}\,\left (3\,b\,c^2+a\,d^2\right )}{7\,e^4}+\frac {2\,a\,c^3\,\sqrt {e\,x}}{e}+\frac {2\,b\,d^3\,{\left (e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,a\,c^2\,d\,{\left (e\,x\right )}^{3/2}}{e^2}+\frac {2\,b\,c\,d^2\,{\left (e\,x\right )}^{9/2}}{3\,e^5} \] Input:
int(((a + b*x^2)*(c + d*x)^3)/(e*x)^(1/2),x)
Output:
(2*c*(e*x)^(5/2)*(3*a*d^2 + b*c^2))/(5*e^3) + (2*d*(e*x)^(7/2)*(a*d^2 + 3* b*c^2))/(7*e^4) + (2*a*c^3*(e*x)^(1/2))/e + (2*b*d^3*(e*x)^(11/2))/(11*e^6 ) + (2*a*c^2*d*(e*x)^(3/2))/e^2 + (2*b*c*d^2*(e*x)^(9/2))/(3*e^5)
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.65 \[ \int \frac {(c+d x)^3 \left (a+b x^2\right )}{\sqrt {e x}} \, dx=\frac {2 \sqrt {x}\, \sqrt {e}\, \left (105 b \,d^{3} x^{5}+385 b c \,d^{2} x^{4}+165 a \,d^{3} x^{3}+495 b \,c^{2} d \,x^{3}+693 a c \,d^{2} x^{2}+231 b \,c^{3} x^{2}+1155 a \,c^{2} d x +1155 a \,c^{3}\right )}{1155 e} \] Input:
int((d*x+c)^3*(b*x^2+a)/(e*x)^(1/2),x)
Output:
(2*sqrt(x)*sqrt(e)*(1155*a*c**3 + 1155*a*c**2*d*x + 693*a*c*d**2*x**2 + 16 5*a*d**3*x**3 + 231*b*c**3*x**2 + 495*b*c**2*d*x**3 + 385*b*c*d**2*x**4 + 105*b*d**3*x**5))/(1155*e)