Integrand size = 22, antiderivative size = 105 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2 a^2 c \sqrt {e x}}{e}+\frac {2 a^2 d (e x)^{3/2}}{3 e^2}+\frac {4 a b c (e x)^{5/2}}{5 e^3}+\frac {4 a b d (e x)^{7/2}}{7 e^4}+\frac {2 b^2 c (e x)^{9/2}}{9 e^5}+\frac {2 b^2 d (e x)^{11/2}}{11 e^6} \] Output:
2*a^2*c*(e*x)^(1/2)/e+2/3*a^2*d*(e*x)^(3/2)/e^2+4/5*a*b*c*(e*x)^(5/2)/e^3+ 4/7*a*b*d*(e*x)^(7/2)/e^4+2/9*b^2*c*(e*x)^(9/2)/e^5+2/11*b^2*d*(e*x)^(11/2 )/e^6
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.53 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2310 a^2 x (3 c+d x)+396 a b x^3 (7 c+5 d x)+70 b^2 x^5 (11 c+9 d x)}{3465 \sqrt {e x}} \] Input:
Integrate[((c + d*x)*(a + b*x^2)^2)/Sqrt[e*x],x]
Output:
(2310*a^2*x*(3*c + d*x) + 396*a*b*x^3*(7*c + 5*d*x) + 70*b^2*x^5*(11*c + 9 *d*x))/(3465*Sqrt[e*x])
Time = 0.38 (sec) , antiderivative size = 105, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)}{\sqrt {e x}} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {a^2 c}{\sqrt {e x}}+\frac {a^2 d \sqrt {e x}}{e}+\frac {2 a b c (e x)^{3/2}}{e^2}+\frac {2 a b d (e x)^{5/2}}{e^3}+\frac {b^2 c (e x)^{7/2}}{e^4}+\frac {b^2 d (e x)^{9/2}}{e^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^2 c \sqrt {e x}}{e}+\frac {2 a^2 d (e x)^{3/2}}{3 e^2}+\frac {4 a b c (e x)^{5/2}}{5 e^3}+\frac {4 a b d (e x)^{7/2}}{7 e^4}+\frac {2 b^2 c (e x)^{9/2}}{9 e^5}+\frac {2 b^2 d (e x)^{11/2}}{11 e^6}\) |
Input:
Int[((c + d*x)*(a + b*x^2)^2)/Sqrt[e*x],x]
Output:
(2*a^2*c*Sqrt[e*x])/e + (2*a^2*d*(e*x)^(3/2))/(3*e^2) + (4*a*b*c*(e*x)^(5/ 2))/(5*e^3) + (4*a*b*d*(e*x)^(7/2))/(7*e^4) + (2*b^2*c*(e*x)^(9/2))/(9*e^5 ) + (2*b^2*d*(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 = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(\frac {2 x \left (315 b^{2} d \,x^{5}+385 b^{2} c \,x^{4}+990 a b d \,x^{3}+1386 a b c \,x^{2}+1155 a^{2} d x +3465 a^{2} c \right )}{3465 \sqrt {e x}}\) | \(57\) |
risch | \(\frac {2 x \left (315 b^{2} d \,x^{5}+385 b^{2} c \,x^{4}+990 a b d \,x^{3}+1386 a b c \,x^{2}+1155 a^{2} d x +3465 a^{2} c \right )}{3465 \sqrt {e x}}\) | \(57\) |
orering | \(\frac {2 x \left (315 b^{2} d \,x^{5}+385 b^{2} c \,x^{4}+990 a b d \,x^{3}+1386 a b c \,x^{2}+1155 a^{2} d x +3465 a^{2} c \right )}{3465 \sqrt {e x}}\) | \(57\) |
trager | \(\frac {\left (\frac {2}{11} b^{2} d \,x^{5}+\frac {2}{9} b^{2} c \,x^{4}+\frac {4}{7} a b d \,x^{3}+\frac {4}{5} a b c \,x^{2}+\frac {2}{3} a^{2} d x +2 a^{2} c \right ) \sqrt {e x}}{e}\) | \(58\) |
pseudoelliptic | \(\frac {2 \left (\frac {1}{11} b^{2} d \,x^{5}+\frac {1}{9} b^{2} c \,x^{4}+\frac {2}{7} a b d \,x^{3}+\frac {2}{5} a b c \,x^{2}+\frac {1}{3} a^{2} d x +a^{2} c \right ) \sqrt {e x}}{e}\) | \(58\) |
derivativedivides | \(\frac {\frac {2 b^{2} d \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 c e \,b^{2} \left (e x \right )^{\frac {9}{2}}}{9}+\frac {4 d a \,e^{2} b \left (e x \right )^{\frac {7}{2}}}{7}+\frac {4 c \,e^{3} a b \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 d \,a^{2} e^{4} \left (e x \right )^{\frac {3}{2}}}{3}+2 c \,e^{5} a^{2} \sqrt {e x}}{e^{6}}\) | \(83\) |
default | \(\frac {\frac {2 b^{2} d \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 c e \,b^{2} \left (e x \right )^{\frac {9}{2}}}{9}+\frac {4 d a \,e^{2} b \left (e x \right )^{\frac {7}{2}}}{7}+\frac {4 c \,e^{3} a b \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 d \,a^{2} e^{4} \left (e x \right )^{\frac {3}{2}}}{3}+2 c \,e^{5} a^{2} \sqrt {e x}}{e^{6}}\) | \(83\) |
Input:
int((d*x+c)*(b*x^2+a)^2/(e*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3465*x*(315*b^2*d*x^5+385*b^2*c*x^4+990*a*b*d*x^3+1386*a*b*c*x^2+1155*a^ 2*d*x+3465*a^2*c)/(e*x)^(1/2)
Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.55 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2 \, {\left (315 \, b^{2} d x^{5} + 385 \, b^{2} c x^{4} + 990 \, a b d x^{3} + 1386 \, a b c x^{2} + 1155 \, a^{2} d x + 3465 \, a^{2} c\right )} \sqrt {e x}}{3465 \, e} \] Input:
integrate((d*x+c)*(b*x^2+a)^2/(e*x)^(1/2),x, algorithm="fricas")
Output:
2/3465*(315*b^2*d*x^5 + 385*b^2*c*x^4 + 990*a*b*d*x^3 + 1386*a*b*c*x^2 + 1 155*a^2*d*x + 3465*a^2*c)*sqrt(e*x)/e
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2 a^{2} c x}{\sqrt {e x}} + \frac {2 a^{2} d x^{2}}{3 \sqrt {e x}} + \frac {4 a b c x^{3}}{5 \sqrt {e x}} + \frac {4 a b d x^{4}}{7 \sqrt {e x}} + \frac {2 b^{2} c x^{5}}{9 \sqrt {e x}} + \frac {2 b^{2} d x^{6}}{11 \sqrt {e x}} \] Input:
integrate((d*x+c)*(b*x**2+a)**2/(e*x)**(1/2),x)
Output:
2*a**2*c*x/sqrt(e*x) + 2*a**2*d*x**2/(3*sqrt(e*x)) + 4*a*b*c*x**3/(5*sqrt( e*x)) + 4*a*b*d*x**4/(7*sqrt(e*x)) + 2*b**2*c*x**5/(9*sqrt(e*x)) + 2*b**2* d*x**6/(11*sqrt(e*x))
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2 \, {\left (315 \, \left (e x\right )^{\frac {11}{2}} b^{2} d + 385 \, \left (e x\right )^{\frac {9}{2}} b^{2} c e + 990 \, \left (e x\right )^{\frac {7}{2}} a b d e^{2} + 1386 \, \left (e x\right )^{\frac {5}{2}} a b c e^{3} + 1155 \, \left (e x\right )^{\frac {3}{2}} a^{2} d e^{4} + 3465 \, \sqrt {e x} a^{2} c e^{5}\right )}}{3465 \, e^{6}} \] Input:
integrate((d*x+c)*(b*x^2+a)^2/(e*x)^(1/2),x, algorithm="maxima")
Output:
2/3465*(315*(e*x)^(11/2)*b^2*d + 385*(e*x)^(9/2)*b^2*c*e + 990*(e*x)^(7/2) *a*b*d*e^2 + 1386*(e*x)^(5/2)*a*b*c*e^3 + 1155*(e*x)^(3/2)*a^2*d*e^4 + 346 5*sqrt(e*x)*a^2*c*e^5)/e^6
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x} b^{2} d x^{5} + 385 \, \sqrt {e x} b^{2} c x^{4} + 990 \, \sqrt {e x} a b d x^{3} + 1386 \, \sqrt {e x} a b c x^{2} + 1155 \, \sqrt {e x} a^{2} d x + 3465 \, \sqrt {e x} a^{2} c\right )}}{3465 \, e} \] Input:
integrate((d*x+c)*(b*x^2+a)^2/(e*x)^(1/2),x, algorithm="giac")
Output:
2/3465*(315*sqrt(e*x)*b^2*d*x^5 + 385*sqrt(e*x)*b^2*c*x^4 + 990*sqrt(e*x)* a*b*d*x^3 + 1386*sqrt(e*x)*a*b*c*x^2 + 1155*sqrt(e*x)*a^2*d*x + 3465*sqrt( e*x)*a^2*c)/e
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2\,a^2\,c\,\sqrt {e\,x}}{e}+\frac {2\,a^2\,d\,{\left (e\,x\right )}^{3/2}}{3\,e^2}+\frac {2\,b^2\,c\,{\left (e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,b^2\,d\,{\left (e\,x\right )}^{11/2}}{11\,e^6}+\frac {4\,a\,b\,c\,{\left (e\,x\right )}^{5/2}}{5\,e^3}+\frac {4\,a\,b\,d\,{\left (e\,x\right )}^{7/2}}{7\,e^4} \] Input:
int(((a + b*x^2)^2*(c + d*x))/(e*x)^(1/2),x)
Output:
(2*a^2*c*(e*x)^(1/2))/e + (2*a^2*d*(e*x)^(3/2))/(3*e^2) + (2*b^2*c*(e*x)^( 9/2))/(9*e^5) + (2*b^2*d*(e*x)^(11/2))/(11*e^6) + (4*a*b*c*(e*x)^(5/2))/(5 *e^3) + (4*a*b*d*(e*x)^(7/2))/(7*e^4)
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.54 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {2 \sqrt {x}\, \sqrt {e}\, \left (315 b^{2} d \,x^{5}+385 b^{2} c \,x^{4}+990 a b d \,x^{3}+1386 a b c \,x^{2}+1155 a^{2} d x +3465 a^{2} c \right )}{3465 e} \] Input:
int((d*x+c)*(b*x^2+a)^2/(e*x)^(1/2),x)
Output:
(2*sqrt(x)*sqrt(e)*(3465*a**2*c + 1155*a**2*d*x + 1386*a*b*c*x**2 + 990*a* b*d*x**3 + 385*b**2*c*x**4 + 315*b**2*d*x**5))/(3465*e)