Integrand size = 22, antiderivative size = 103 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=-\frac {2 a^2 c}{e \sqrt {e x}}+\frac {2 a^2 d \sqrt {e x}}{e^2}+\frac {4 a b c (e x)^{3/2}}{3 e^3}+\frac {4 a b d (e x)^{5/2}}{5 e^4}+\frac {2 b^2 c (e x)^{7/2}}{7 e^5}+\frac {2 b^2 d (e x)^{9/2}}{9 e^6} \] Output:
-2*a^2*c/e/(e*x)^(1/2)+2*a^2*d*(e*x)^(1/2)/e^2+4/3*a*b*c*(e*x)^(3/2)/e^3+4 /5*a*b*d*(e*x)^(5/2)/e^4+2/7*b^2*c*(e*x)^(7/2)/e^5+2/9*b^2*d*(e*x)^(9/2)/e ^6
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.53 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {x \left (-630 a^2 (c-d x)+84 a b x^2 (5 c+3 d x)+10 b^2 x^4 (9 c+7 d x)\right )}{315 (e x)^{3/2}} \] Input:
Integrate[((c + d*x)*(a + b*x^2)^2)/(e*x)^(3/2),x]
Output:
(x*(-630*a^2*(c - d*x) + 84*a*b*x^2*(5*c + 3*d*x) + 10*b^2*x^4*(9*c + 7*d* x)))/(315*(e*x)^(3/2))
Time = 0.38 (sec) , antiderivative size = 103, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)}{(e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {a^2 c}{(e x)^{3/2}}+\frac {a^2 d}{e \sqrt {e x}}+\frac {2 a b c \sqrt {e x}}{e^2}+\frac {2 a b d (e x)^{3/2}}{e^3}+\frac {b^2 c (e x)^{5/2}}{e^4}+\frac {b^2 d (e x)^{7/2}}{e^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^2 c}{e \sqrt {e x}}+\frac {2 a^2 d \sqrt {e x}}{e^2}+\frac {4 a b c (e x)^{3/2}}{3 e^3}+\frac {4 a b d (e x)^{5/2}}{5 e^4}+\frac {2 b^2 c (e x)^{7/2}}{7 e^5}+\frac {2 b^2 d (e x)^{9/2}}{9 e^6}\) |
Input:
Int[((c + d*x)*(a + b*x^2)^2)/(e*x)^(3/2),x]
Output:
(-2*a^2*c)/(e*Sqrt[e*x]) + (2*a^2*d*Sqrt[e*x])/e^2 + (4*a*b*c*(e*x)^(3/2)) /(3*e^3) + (4*a*b*d*(e*x)^(5/2))/(5*e^4) + (2*b^2*c*(e*x)^(7/2))/(7*e^5) + (2*b^2*d*(e*x)^(9/2))/(9*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 = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {2 x \left (-35 b^{2} d \,x^{5}-45 b^{2} c \,x^{4}-126 a b d \,x^{3}-210 a b c \,x^{2}-315 a^{2} d x +315 a^{2} c \right )}{315 \left (e x \right )^{\frac {3}{2}}}\) | \(57\) |
orering | \(-\frac {2 x \left (-35 b^{2} d \,x^{5}-45 b^{2} c \,x^{4}-126 a b d \,x^{3}-210 a b c \,x^{2}-315 a^{2} d x +315 a^{2} c \right )}{315 \left (e x \right )^{\frac {3}{2}}}\) | \(57\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {1}{9} b^{2} d \,x^{5}-\frac {1}{7} b^{2} c \,x^{4}-\frac {2}{5} a b d \,x^{3}-\frac {2}{3} a b c \,x^{2}-a^{2} d x +a^{2} c \right )}{\sqrt {e x}\, e}\) | \(58\) |
risch | \(-\frac {2 \left (-35 b^{2} d \,x^{5}-45 b^{2} c \,x^{4}-126 a b d \,x^{3}-210 a b c \,x^{2}-315 a^{2} d x +315 a^{2} c \right )}{315 e \sqrt {e x}}\) | \(59\) |
trager | \(-\frac {2 \left (-35 b^{2} d \,x^{5}-45 b^{2} c \,x^{4}-126 a b d \,x^{3}-210 a b c \,x^{2}-315 a^{2} d x +315 a^{2} c \right ) \sqrt {e x}}{315 e^{2} x}\) | \(62\) |
derivativedivides | \(\frac {\frac {2 b^{2} d \left (e x \right )^{\frac {9}{2}}}{9}+\frac {2 b^{2} c e \left (e x \right )^{\frac {7}{2}}}{7}+\frac {4 a b d \,e^{2} \left (e x \right )^{\frac {5}{2}}}{5}+\frac {4 a b c \,e^{3} \left (e x \right )^{\frac {3}{2}}}{3}+2 a^{2} d \,e^{4} \sqrt {e x}-\frac {2 a^{2} c \,e^{5}}{\sqrt {e x}}}{e^{6}}\) | \(83\) |
default | \(\frac {\frac {2 b^{2} d \left (e x \right )^{\frac {9}{2}}}{9}+\frac {2 b^{2} c e \left (e x \right )^{\frac {7}{2}}}{7}+\frac {4 a b d \,e^{2} \left (e x \right )^{\frac {5}{2}}}{5}+\frac {4 a b c \,e^{3} \left (e x \right )^{\frac {3}{2}}}{3}+2 a^{2} d \,e^{4} \sqrt {e x}-\frac {2 a^{2} c \,e^{5}}{\sqrt {e x}}}{e^{6}}\) | \(83\) |
Input:
int((d*x+c)*(b*x^2+a)^2/(e*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/315*x*(-35*b^2*d*x^5-45*b^2*c*x^4-126*a*b*d*x^3-210*a*b*c*x^2-315*a^2*d *x+315*a^2*c)/(e*x)^(3/2)
Time = 0.17 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.59 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, b^{2} d x^{5} + 45 \, b^{2} c x^{4} + 126 \, a b d x^{3} + 210 \, a b c x^{2} + 315 \, a^{2} d x - 315 \, a^{2} c\right )} \sqrt {e x}}{315 \, e^{2} x} \] Input:
integrate((d*x+c)*(b*x^2+a)^2/(e*x)^(3/2),x, algorithm="fricas")
Output:
2/315*(35*b^2*d*x^5 + 45*b^2*c*x^4 + 126*a*b*d*x^3 + 210*a*b*c*x^2 + 315*a ^2*d*x - 315*a^2*c)*sqrt(e*x)/(e^2*x)
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=- \frac {2 a^{2} c x}{\left (e x\right )^{\frac {3}{2}}} + \frac {2 a^{2} d x^{2}}{\left (e x\right )^{\frac {3}{2}}} + \frac {4 a b c x^{3}}{3 \left (e x\right )^{\frac {3}{2}}} + \frac {4 a b d x^{4}}{5 \left (e x\right )^{\frac {3}{2}}} + \frac {2 b^{2} c x^{5}}{7 \left (e x\right )^{\frac {3}{2}}} + \frac {2 b^{2} d x^{6}}{9 \left (e x\right )^{\frac {3}{2}}} \] Input:
integrate((d*x+c)*(b*x**2+a)**2/(e*x)**(3/2),x)
Output:
-2*a**2*c*x/(e*x)**(3/2) + 2*a**2*d*x**2/(e*x)**(3/2) + 4*a*b*c*x**3/(3*(e *x)**(3/2)) + 4*a*b*d*x**4/(5*(e*x)**(3/2)) + 2*b**2*c*x**5/(7*(e*x)**(3/2 )) + 2*b**2*d*x**6/(9*(e*x)**(3/2))
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {315 \, a^{2} c}{\sqrt {e x}} - \frac {35 \, \left (e x\right )^{\frac {9}{2}} b^{2} d + 45 \, \left (e x\right )^{\frac {7}{2}} b^{2} c e + 126 \, \left (e x\right )^{\frac {5}{2}} a b d e^{2} + 210 \, \left (e x\right )^{\frac {3}{2}} a b c e^{3} + 315 \, \sqrt {e x} a^{2} d e^{4}}{e^{5}}\right )}}{315 \, e} \] Input:
integrate((d*x+c)*(b*x^2+a)^2/(e*x)^(3/2),x, algorithm="maxima")
Output:
-2/315*(315*a^2*c/sqrt(e*x) - (35*(e*x)^(9/2)*b^2*d + 45*(e*x)^(7/2)*b^2*c *e + 126*(e*x)^(5/2)*a*b*d*e^2 + 210*(e*x)^(3/2)*a*b*c*e^3 + 315*sqrt(e*x) *a^2*d*e^4)/e^5)/e
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {315 \, a^{2} c}{\sqrt {e x}} - \frac {35 \, \sqrt {e x} b^{2} d e^{44} x^{4} + 45 \, \sqrt {e x} b^{2} c e^{44} x^{3} + 126 \, \sqrt {e x} a b d e^{44} x^{2} + 210 \, \sqrt {e x} a b c e^{44} x + 315 \, \sqrt {e x} a^{2} d e^{44}}{e^{45}}\right )}}{315 \, e} \] Input:
integrate((d*x+c)*(b*x^2+a)^2/(e*x)^(3/2),x, algorithm="giac")
Output:
-2/315*(315*a^2*c/sqrt(e*x) - (35*sqrt(e*x)*b^2*d*e^44*x^4 + 45*sqrt(e*x)* b^2*c*e^44*x^3 + 126*sqrt(e*x)*a*b*d*e^44*x^2 + 210*sqrt(e*x)*a*b*c*e^44*x + 315*sqrt(e*x)*a^2*d*e^44)/e^45)/e
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {2\,a^2\,d\,\sqrt {e\,x}}{e^2}-\frac {2\,a^2\,c}{e\,\sqrt {e\,x}}+\frac {2\,b^2\,c\,{\left (e\,x\right )}^{7/2}}{7\,e^5}+\frac {2\,b^2\,d\,{\left (e\,x\right )}^{9/2}}{9\,e^6}+\frac {4\,a\,b\,c\,{\left (e\,x\right )}^{3/2}}{3\,e^3}+\frac {4\,a\,b\,d\,{\left (e\,x\right )}^{5/2}}{5\,e^4} \] Input:
int(((a + b*x^2)^2*(c + d*x))/(e*x)^(3/2),x)
Output:
(2*a^2*d*(e*x)^(1/2))/e^2 - (2*a^2*c)/(e*(e*x)^(1/2)) + (2*b^2*c*(e*x)^(7/ 2))/(7*e^5) + (2*b^2*d*(e*x)^(9/2))/(9*e^6) + (4*a*b*c*(e*x)^(3/2))/(3*e^3 ) + (4*a*b*d*(e*x)^(5/2))/(5*e^4)
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.57 \[ \int \frac {(c+d x) \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {2 \sqrt {e}\, \left (35 b^{2} d \,x^{5}+45 b^{2} c \,x^{4}+126 a b d \,x^{3}+210 a b c \,x^{2}+315 a^{2} d x -315 a^{2} c \right )}{315 \sqrt {x}\, e^{2}} \] Input:
int((d*x+c)*(b*x^2+a)^2/(e*x)^(3/2),x)
Output:
(2*sqrt(e)*( - 315*a**2*c + 315*a**2*d*x + 210*a*b*c*x**2 + 126*a*b*d*x**3 + 45*b**2*c*x**4 + 35*b**2*d*x**5))/(315*sqrt(x)*e**2)