Integrand size = 22, antiderivative size = 147 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2 a^3 c \sqrt {e x}}{e}+\frac {2 a^3 d (e x)^{3/2}}{3 e^2}+\frac {6 a^2 b c (e x)^{5/2}}{5 e^3}+\frac {6 a^2 b d (e x)^{7/2}}{7 e^4}+\frac {2 a b^2 c (e x)^{9/2}}{3 e^5}+\frac {6 a b^2 d (e x)^{11/2}}{11 e^6}+\frac {2 b^3 c (e x)^{13/2}}{13 e^7}+\frac {2 b^3 d (e x)^{15/2}}{15 e^8} \] Output:
2*a^3*c*(e*x)^(1/2)/e+2/3*a^3*d*(e*x)^(3/2)/e^2+6/5*a^2*b*c*(e*x)^(5/2)/e^ 3+6/7*a^2*b*d*(e*x)^(7/2)/e^4+2/3*a*b^2*c*(e*x)^(9/2)/e^5+6/11*a*b^2*d*(e* x)^(11/2)/e^6+2/13*b^3*c*(e*x)^(13/2)/e^7+2/15*b^3*d*(e*x)^(15/2)/e^8
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2 x \left (5005 a^3 (3 c+d x)+1287 a^2 b x^2 (7 c+5 d x)+455 a b^2 x^4 (11 c+9 d x)+77 b^3 x^6 (15 c+13 d x)\right )}{15015 \sqrt {e x}} \] Input:
Integrate[((c + d*x)*(a + b*x^2)^3)/Sqrt[e*x],x]
Output:
(2*x*(5005*a^3*(3*c + d*x) + 1287*a^2*b*x^2*(7*c + 5*d*x) + 455*a*b^2*x^4* (11*c + 9*d*x) + 77*b^3*x^6*(15*c + 13*d*x)))/(15015*Sqrt[e*x])
Time = 0.45 (sec) , antiderivative size = 147, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^3 (c+d x)}{\sqrt {e x}} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {a^3 c}{\sqrt {e x}}+\frac {a^3 d \sqrt {e x}}{e}+\frac {3 a^2 b c (e x)^{3/2}}{e^2}+\frac {3 a^2 b d (e x)^{5/2}}{e^3}+\frac {3 a b^2 c (e x)^{7/2}}{e^4}+\frac {3 a b^2 d (e x)^{9/2}}{e^5}+\frac {b^3 c (e x)^{11/2}}{e^6}+\frac {b^3 d (e x)^{13/2}}{e^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^3 c \sqrt {e x}}{e}+\frac {2 a^3 d (e x)^{3/2}}{3 e^2}+\frac {6 a^2 b c (e x)^{5/2}}{5 e^3}+\frac {6 a^2 b d (e x)^{7/2}}{7 e^4}+\frac {2 a b^2 c (e x)^{9/2}}{3 e^5}+\frac {6 a b^2 d (e x)^{11/2}}{11 e^6}+\frac {2 b^3 c (e x)^{13/2}}{13 e^7}+\frac {2 b^3 d (e x)^{15/2}}{15 e^8}\) |
Input:
Int[((c + d*x)*(a + b*x^2)^3)/Sqrt[e*x],x]
Output:
(2*a^3*c*Sqrt[e*x])/e + (2*a^3*d*(e*x)^(3/2))/(3*e^2) + (6*a^2*b*c*(e*x)^( 5/2))/(5*e^3) + (6*a^2*b*d*(e*x)^(7/2))/(7*e^4) + (2*a*b^2*c*(e*x)^(9/2))/ (3*e^5) + (6*a*b^2*d*(e*x)^(11/2))/(11*e^6) + (2*b^3*c*(e*x)^(13/2))/(13*e ^7) + (2*b^3*d*(e*x)^(15/2))/(15*e^8)
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 = 81, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(\frac {2 x \left (1001 b^{3} d \,x^{7}+1155 b^{3} c \,x^{6}+4095 a \,b^{2} d \,x^{5}+5005 a \,b^{2} c \,x^{4}+6435 a^{2} b d \,x^{3}+9009 a^{2} b c \,x^{2}+5005 a^{3} d x +15015 c \,a^{3}\right )}{15015 \sqrt {e x}}\) | \(81\) |
risch | \(\frac {2 x \left (1001 b^{3} d \,x^{7}+1155 b^{3} c \,x^{6}+4095 a \,b^{2} d \,x^{5}+5005 a \,b^{2} c \,x^{4}+6435 a^{2} b d \,x^{3}+9009 a^{2} b c \,x^{2}+5005 a^{3} d x +15015 c \,a^{3}\right )}{15015 \sqrt {e x}}\) | \(81\) |
orering | \(\frac {2 x \left (1001 b^{3} d \,x^{7}+1155 b^{3} c \,x^{6}+4095 a \,b^{2} d \,x^{5}+5005 a \,b^{2} c \,x^{4}+6435 a^{2} b d \,x^{3}+9009 a^{2} b c \,x^{2}+5005 a^{3} d x +15015 c \,a^{3}\right )}{15015 \sqrt {e x}}\) | \(81\) |
trager | \(\frac {\left (\frac {2}{15} b^{3} d \,x^{7}+\frac {2}{13} b^{3} c \,x^{6}+\frac {6}{11} a \,b^{2} d \,x^{5}+\frac {2}{3} a \,b^{2} c \,x^{4}+\frac {6}{7} a^{2} b d \,x^{3}+\frac {6}{5} a^{2} b c \,x^{2}+\frac {2}{3} a^{3} d x +2 c \,a^{3}\right ) \sqrt {e x}}{e}\) | \(82\) |
pseudoelliptic | \(\frac {2 \sqrt {e x}\, \left (\frac {1}{15} b^{3} d \,x^{7}+\frac {1}{13} b^{3} c \,x^{6}+\frac {3}{11} a \,b^{2} d \,x^{5}+\frac {1}{3} a \,b^{2} c \,x^{4}+\frac {3}{7} a^{2} b d \,x^{3}+\frac {3}{5} a^{2} b c \,x^{2}+\frac {1}{3} a^{3} d x +c \,a^{3}\right )}{e}\) | \(82\) |
derivativedivides | \(\frac {\frac {2 b^{3} d \left (e x \right )^{\frac {15}{2}}}{15}+\frac {2 c e \,b^{3} \left (e x \right )^{\frac {13}{2}}}{13}+\frac {6 d a \,e^{2} b^{2} \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 c \,e^{3} a \,b^{2} \left (e x \right )^{\frac {9}{2}}}{3}+\frac {6 d \,a^{2} e^{4} b \left (e x \right )^{\frac {7}{2}}}{7}+\frac {6 c \,e^{5} a^{2} b \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 d \,a^{3} e^{6} \left (e x \right )^{\frac {3}{2}}}{3}+2 c \,e^{7} a^{3} \sqrt {e x}}{e^{8}}\) | \(117\) |
default | \(\frac {\frac {2 b^{3} d \left (e x \right )^{\frac {15}{2}}}{15}+\frac {2 c e \,b^{3} \left (e x \right )^{\frac {13}{2}}}{13}+\frac {6 d a \,e^{2} b^{2} \left (e x \right )^{\frac {11}{2}}}{11}+\frac {2 c \,e^{3} a \,b^{2} \left (e x \right )^{\frac {9}{2}}}{3}+\frac {6 d \,a^{2} e^{4} b \left (e x \right )^{\frac {7}{2}}}{7}+\frac {6 c \,e^{5} a^{2} b \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 d \,a^{3} e^{6} \left (e x \right )^{\frac {3}{2}}}{3}+2 c \,e^{7} a^{3} \sqrt {e x}}{e^{8}}\) | \(117\) |
Input:
int((d*x+c)*(b*x^2+a)^3/(e*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/15015*x*(1001*b^3*d*x^7+1155*b^3*c*x^6+4095*a*b^2*d*x^5+5005*a*b^2*c*x^4 +6435*a^2*b*d*x^3+9009*a^2*b*c*x^2+5005*a^3*d*x+15015*a^3*c)/(e*x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.56 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2 \, {\left (1001 \, b^{3} d x^{7} + 1155 \, b^{3} c x^{6} + 4095 \, a b^{2} d x^{5} + 5005 \, a b^{2} c x^{4} + 6435 \, a^{2} b d x^{3} + 9009 \, a^{2} b c x^{2} + 5005 \, a^{3} d x + 15015 \, a^{3} c\right )} \sqrt {e x}}{15015 \, e} \] Input:
integrate((d*x+c)*(b*x^2+a)^3/(e*x)^(1/2),x, algorithm="fricas")
Output:
2/15015*(1001*b^3*d*x^7 + 1155*b^3*c*x^6 + 4095*a*b^2*d*x^5 + 5005*a*b^2*c *x^4 + 6435*a^2*b*d*x^3 + 9009*a^2*b*c*x^2 + 5005*a^3*d*x + 15015*a^3*c)*s qrt(e*x)/e
Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2 a^{3} c x}{\sqrt {e x}} + \frac {2 a^{3} d x^{2}}{3 \sqrt {e x}} + \frac {6 a^{2} b c x^{3}}{5 \sqrt {e x}} + \frac {6 a^{2} b d x^{4}}{7 \sqrt {e x}} + \frac {2 a b^{2} c x^{5}}{3 \sqrt {e x}} + \frac {6 a b^{2} d x^{6}}{11 \sqrt {e x}} + \frac {2 b^{3} c x^{7}}{13 \sqrt {e x}} + \frac {2 b^{3} d x^{8}}{15 \sqrt {e x}} \] Input:
integrate((d*x+c)*(b*x**2+a)**3/(e*x)**(1/2),x)
Output:
2*a**3*c*x/sqrt(e*x) + 2*a**3*d*x**2/(3*sqrt(e*x)) + 6*a**2*b*c*x**3/(5*sq rt(e*x)) + 6*a**2*b*d*x**4/(7*sqrt(e*x)) + 2*a*b**2*c*x**5/(3*sqrt(e*x)) + 6*a*b**2*d*x**6/(11*sqrt(e*x)) + 2*b**3*c*x**7/(13*sqrt(e*x)) + 2*b**3*d* x**8/(15*sqrt(e*x))
Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2 \, {\left (1001 \, \left (e x\right )^{\frac {15}{2}} b^{3} d + 1155 \, \left (e x\right )^{\frac {13}{2}} b^{3} c e + 4095 \, \left (e x\right )^{\frac {11}{2}} a b^{2} d e^{2} + 5005 \, \left (e x\right )^{\frac {9}{2}} a b^{2} c e^{3} + 6435 \, \left (e x\right )^{\frac {7}{2}} a^{2} b d e^{4} + 9009 \, \left (e x\right )^{\frac {5}{2}} a^{2} b c e^{5} + 5005 \, \left (e x\right )^{\frac {3}{2}} a^{3} d e^{6} + 15015 \, \sqrt {e x} a^{3} c e^{7}\right )}}{15015 \, e^{8}} \] Input:
integrate((d*x+c)*(b*x^2+a)^3/(e*x)^(1/2),x, algorithm="maxima")
Output:
2/15015*(1001*(e*x)^(15/2)*b^3*d + 1155*(e*x)^(13/2)*b^3*c*e + 4095*(e*x)^ (11/2)*a*b^2*d*e^2 + 5005*(e*x)^(9/2)*a*b^2*c*e^3 + 6435*(e*x)^(7/2)*a^2*b *d*e^4 + 9009*(e*x)^(5/2)*a^2*b*c*e^5 + 5005*(e*x)^(3/2)*a^3*d*e^6 + 15015 *sqrt(e*x)*a^3*c*e^7)/e^8
Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2 \, {\left (1001 \, \sqrt {e x} b^{3} d x^{7} + 1155 \, \sqrt {e x} b^{3} c x^{6} + 4095 \, \sqrt {e x} a b^{2} d x^{5} + 5005 \, \sqrt {e x} a b^{2} c x^{4} + 6435 \, \sqrt {e x} a^{2} b d x^{3} + 9009 \, \sqrt {e x} a^{2} b c x^{2} + 5005 \, \sqrt {e x} a^{3} d x + 15015 \, \sqrt {e x} a^{3} c\right )}}{15015 \, e} \] Input:
integrate((d*x+c)*(b*x^2+a)^3/(e*x)^(1/2),x, algorithm="giac")
Output:
2/15015*(1001*sqrt(e*x)*b^3*d*x^7 + 1155*sqrt(e*x)*b^3*c*x^6 + 4095*sqrt(e *x)*a*b^2*d*x^5 + 5005*sqrt(e*x)*a*b^2*c*x^4 + 6435*sqrt(e*x)*a^2*b*d*x^3 + 9009*sqrt(e*x)*a^2*b*c*x^2 + 5005*sqrt(e*x)*a^3*d*x + 15015*sqrt(e*x)*a^ 3*c)/e
Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2\,a^3\,c\,\sqrt {e\,x}}{e}+\frac {2\,a^3\,d\,{\left (e\,x\right )}^{3/2}}{3\,e^2}+\frac {2\,b^3\,c\,{\left (e\,x\right )}^{13/2}}{13\,e^7}+\frac {2\,b^3\,d\,{\left (e\,x\right )}^{15/2}}{15\,e^8}+\frac {6\,a^2\,b\,c\,{\left (e\,x\right )}^{5/2}}{5\,e^3}+\frac {2\,a\,b^2\,c\,{\left (e\,x\right )}^{9/2}}{3\,e^5}+\frac {6\,a^2\,b\,d\,{\left (e\,x\right )}^{7/2}}{7\,e^4}+\frac {6\,a\,b^2\,d\,{\left (e\,x\right )}^{11/2}}{11\,e^6} \] Input:
int(((a + b*x^2)^3*(c + d*x))/(e*x)^(1/2),x)
Output:
(2*a^3*c*(e*x)^(1/2))/e + (2*a^3*d*(e*x)^(3/2))/(3*e^2) + (2*b^3*c*(e*x)^( 13/2))/(13*e^7) + (2*b^3*d*(e*x)^(15/2))/(15*e^8) + (6*a^2*b*c*(e*x)^(5/2) )/(5*e^3) + (2*a*b^2*c*(e*x)^(9/2))/(3*e^5) + (6*a^2*b*d*(e*x)^(7/2))/(7*e ^4) + (6*a*b^2*d*(e*x)^(11/2))/(11*e^6)
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.55 \[ \int \frac {(c+d x) \left (a+b x^2\right )^3}{\sqrt {e x}} \, dx=\frac {2 \sqrt {x}\, \sqrt {e}\, \left (1001 b^{3} d \,x^{7}+1155 b^{3} c \,x^{6}+4095 a \,b^{2} d \,x^{5}+5005 a \,b^{2} c \,x^{4}+6435 a^{2} b d \,x^{3}+9009 a^{2} b c \,x^{2}+5005 a^{3} d x +15015 a^{3} c \right )}{15015 e} \] Input:
int((d*x+c)*(b*x^2+a)^3/(e*x)^(1/2),x)
Output:
(2*sqrt(x)*sqrt(e)*(15015*a**3*c + 5005*a**3*d*x + 9009*a**2*b*c*x**2 + 64 35*a**2*b*d*x**3 + 5005*a*b**2*c*x**4 + 4095*a*b**2*d*x**5 + 1155*b**3*c*x **6 + 1001*b**3*d*x**7))/(15015*e)