Integrand size = 24, antiderivative size = 147 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=-\frac {2 a^2 c^2}{e \sqrt {e x}}+\frac {4 a^2 c d \sqrt {e x}}{e^2}+\frac {2 a \left (2 b c^2+a d^2\right ) (e x)^{3/2}}{3 e^3}+\frac {8 a b c d (e x)^{5/2}}{5 e^4}+\frac {2 b \left (b c^2+2 a d^2\right ) (e x)^{7/2}}{7 e^5}+\frac {4 b^2 c d (e x)^{9/2}}{9 e^6}+\frac {2 b^2 d^2 (e x)^{11/2}}{11 e^7} \] Output:
-2*a^2*c^2/e/(e*x)^(1/2)+4*a^2*c*d*(e*x)^(1/2)/e^2+2/3*a*(a*d^2+2*b*c^2)*( e*x)^(3/2)/e^3+8/5*a*b*c*d*(e*x)^(5/2)/e^4+2/7*b*(2*a*d^2+b*c^2)*(e*x)^(7/ 2)/e^5+4/9*b^2*c*d*(e*x)^(9/2)/e^6+2/11*b^2*d^2*(e*x)^(11/2)/e^7
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.61 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {2 x \left (-1155 a^2 \left (3 c^2-6 c d x-d^2 x^2\right )+66 a b x^2 \left (35 c^2+42 c d x+15 d^2 x^2\right )+5 b^2 x^4 \left (99 c^2+154 c d x+63 d^2 x^2\right )\right )}{3465 (e x)^{3/2}} \] Input:
Integrate[((c + d*x)^2*(a + b*x^2)^2)/(e*x)^(3/2),x]
Output:
(2*x*(-1155*a^2*(3*c^2 - 6*c*d*x - d^2*x^2) + 66*a*b*x^2*(35*c^2 + 42*c*d* x + 15*d^2*x^2) + 5*b^2*x^4*(99*c^2 + 154*c*d*x + 63*d^2*x^2)))/(3465*(e*x )^(3/2))
Time = 0.44 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^2}{(e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {a^2 c^2}{(e x)^{3/2}}+\frac {2 a^2 c d}{e \sqrt {e x}}+\frac {b (e x)^{5/2} \left (2 a d^2+b c^2\right )}{e^4}+\frac {a \sqrt {e x} \left (a d^2+2 b c^2\right )}{e^2}+\frac {4 a b c d (e x)^{3/2}}{e^3}+\frac {2 b^2 c d (e x)^{7/2}}{e^5}+\frac {b^2 d^2 (e x)^{9/2}}{e^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^2 c^2}{e \sqrt {e x}}+\frac {4 a^2 c d \sqrt {e x}}{e^2}+\frac {2 b (e x)^{7/2} \left (2 a d^2+b c^2\right )}{7 e^5}+\frac {2 a (e x)^{3/2} \left (a d^2+2 b c^2\right )}{3 e^3}+\frac {8 a b c d (e x)^{5/2}}{5 e^4}+\frac {4 b^2 c d (e x)^{9/2}}{9 e^6}+\frac {2 b^2 d^2 (e x)^{11/2}}{11 e^7}\) |
Input:
Int[((c + d*x)^2*(a + b*x^2)^2)/(e*x)^(3/2),x]
Output:
(-2*a^2*c^2)/(e*Sqrt[e*x]) + (4*a^2*c*d*Sqrt[e*x])/e^2 + (2*a*(2*b*c^2 + a *d^2)*(e*x)^(3/2))/(3*e^3) + (8*a*b*c*d*(e*x)^(5/2))/(5*e^4) + (2*b*(b*c^2 + 2*a*d^2)*(e*x)^(7/2))/(7*e^5) + (4*b^2*c*d*(e*x)^(9/2))/(9*e^6) + (2*b^ 2*d^2*(e*x)^(11/2))/(11*e^7)
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 = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-\frac {b^{2} d^{2} x^{6}}{11}-\frac {2 b^{2} c d \,x^{5}}{9}+\frac {\left (-2 a b \,d^{2}-b^{2} c^{2}\right ) x^{4}}{7}-\frac {4 a b c d \,x^{3}}{5}-\frac {a \left (a \,d^{2}+2 b \,c^{2}\right ) x^{2}}{3}-2 a^{2} c d x +a^{2} c^{2}\right )}{\sqrt {e x}\, e}\) | \(96\) |
gosper | \(-\frac {2 x \left (-315 b^{2} d^{2} x^{6}-770 b^{2} c d \,x^{5}-990 a b \,d^{2} x^{4}-495 b^{2} c^{2} x^{4}-2772 a b c d \,x^{3}-1155 a^{2} d^{2} x^{2}-2310 a b \,c^{2} x^{2}-6930 a^{2} c d x +3465 a^{2} c^{2}\right )}{3465 \left (e x \right )^{\frac {3}{2}}}\) | \(98\) |
orering | \(-\frac {2 x \left (-315 b^{2} d^{2} x^{6}-770 b^{2} c d \,x^{5}-990 a b \,d^{2} x^{4}-495 b^{2} c^{2} x^{4}-2772 a b c d \,x^{3}-1155 a^{2} d^{2} x^{2}-2310 a b \,c^{2} x^{2}-6930 a^{2} c d x +3465 a^{2} c^{2}\right )}{3465 \left (e x \right )^{\frac {3}{2}}}\) | \(98\) |
risch | \(-\frac {2 \left (-315 b^{2} d^{2} x^{6}-770 b^{2} c d \,x^{5}-990 a b \,d^{2} x^{4}-495 b^{2} c^{2} x^{4}-2772 a b c d \,x^{3}-1155 a^{2} d^{2} x^{2}-2310 a b \,c^{2} x^{2}-6930 a^{2} c d x +3465 a^{2} c^{2}\right )}{3465 e \sqrt {e x}}\) | \(100\) |
trager | \(-\frac {2 \left (-315 b^{2} d^{2} x^{6}-770 b^{2} c d \,x^{5}-990 a b \,d^{2} x^{4}-495 b^{2} c^{2} x^{4}-2772 a b c d \,x^{3}-1155 a^{2} d^{2} x^{2}-2310 a b \,c^{2} x^{2}-6930 a^{2} c d x +3465 a^{2} c^{2}\right ) \sqrt {e x}}{3465 e^{2} x}\) | \(103\) |
derivativedivides | \(\frac {\frac {2 b^{2} d^{2} \left (e x \right )^{\frac {11}{2}}}{11}+\frac {4 b^{2} c d e \left (e x \right )^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} e^{2} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {2 b^{2} c^{2} e^{2} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {8 a b c d \,e^{3} \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} d^{2} e^{4} \left (e x \right )^{\frac {3}{2}}}{3}+\frac {4 a b \,c^{2} e^{4} \left (e x \right )^{\frac {3}{2}}}{3}+4 a^{2} c d \,e^{5} \sqrt {e x}-\frac {2 a^{2} c^{2} e^{6}}{\sqrt {e x}}}{e^{7}}\) | \(140\) |
default | \(\frac {\frac {2 b^{2} d^{2} \left (e x \right )^{\frac {11}{2}}}{11}+\frac {4 b^{2} c d e \left (e x \right )^{\frac {9}{2}}}{9}+\frac {4 a b \,d^{2} e^{2} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {2 b^{2} c^{2} e^{2} \left (e x \right )^{\frac {7}{2}}}{7}+\frac {8 a b c d \,e^{3} \left (e x \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} d^{2} e^{4} \left (e x \right )^{\frac {3}{2}}}{3}+\frac {4 a b \,c^{2} e^{4} \left (e x \right )^{\frac {3}{2}}}{3}+4 a^{2} c d \,e^{5} \sqrt {e x}-\frac {2 a^{2} c^{2} e^{6}}{\sqrt {e x}}}{e^{7}}\) | \(140\) |
Input:
int((d*x+c)^2*(b*x^2+a)^2/(e*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*(-1/11*b^2*d^2*x^6-2/9*b^2*c*d*x^5+1/7*(-2*a*b*d^2-b^2*c^2)*x^4-4/5*a*b *c*d*x^3-1/3*a*(a*d^2+2*b*c^2)*x^2-2*a^2*c*d*x+a^2*c^2)/(e*x)^(1/2)/e
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {2 \, {\left (315 \, b^{2} d^{2} x^{6} + 770 \, b^{2} c d x^{5} + 2772 \, a b c d x^{3} + 6930 \, a^{2} c d x + 495 \, {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} x^{4} - 3465 \, a^{2} c^{2} + 1155 \, {\left (2 \, a b c^{2} + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {e x}}{3465 \, e^{2} x} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^2/(e*x)^(3/2),x, algorithm="fricas")
Output:
2/3465*(315*b^2*d^2*x^6 + 770*b^2*c*d*x^5 + 2772*a*b*c*d*x^3 + 6930*a^2*c* d*x + 495*(b^2*c^2 + 2*a*b*d^2)*x^4 - 3465*a^2*c^2 + 1155*(2*a*b*c^2 + a^2 *d^2)*x^2)*sqrt(e*x)/(e^2*x)
Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=- \frac {2 a^{2} c^{2} x}{\left (e x\right )^{\frac {3}{2}}} + \frac {4 a^{2} c d x^{2}}{\left (e x\right )^{\frac {3}{2}}} + \frac {2 a^{2} d^{2} x^{3}}{3 \left (e x\right )^{\frac {3}{2}}} + \frac {4 a b c^{2} x^{3}}{3 \left (e x\right )^{\frac {3}{2}}} + \frac {8 a b c d x^{4}}{5 \left (e x\right )^{\frac {3}{2}}} + \frac {4 a b d^{2} x^{5}}{7 \left (e x\right )^{\frac {3}{2}}} + \frac {2 b^{2} c^{2} x^{5}}{7 \left (e x\right )^{\frac {3}{2}}} + \frac {4 b^{2} c d x^{6}}{9 \left (e x\right )^{\frac {3}{2}}} + \frac {2 b^{2} d^{2} x^{7}}{11 \left (e x\right )^{\frac {3}{2}}} \] Input:
integrate((d*x+c)**2*(b*x**2+a)**2/(e*x)**(3/2),x)
Output:
-2*a**2*c**2*x/(e*x)**(3/2) + 4*a**2*c*d*x**2/(e*x)**(3/2) + 2*a**2*d**2*x **3/(3*(e*x)**(3/2)) + 4*a*b*c**2*x**3/(3*(e*x)**(3/2)) + 8*a*b*c*d*x**4/( 5*(e*x)**(3/2)) + 4*a*b*d**2*x**5/(7*(e*x)**(3/2)) + 2*b**2*c**2*x**5/(7*( e*x)**(3/2)) + 4*b**2*c*d*x**6/(9*(e*x)**(3/2)) + 2*b**2*d**2*x**7/(11*(e* x)**(3/2))
Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {3465 \, a^{2} c^{2}}{\sqrt {e x}} - \frac {315 \, \left (e x\right )^{\frac {11}{2}} b^{2} d^{2} + 770 \, \left (e x\right )^{\frac {9}{2}} b^{2} c d e + 2772 \, \left (e x\right )^{\frac {5}{2}} a b c d e^{3} + 6930 \, \sqrt {e x} a^{2} c d e^{5} + 495 \, {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} \left (e x\right )^{\frac {7}{2}} e^{2} + 1155 \, {\left (2 \, a b c^{2} + a^{2} d^{2}\right )} \left (e x\right )^{\frac {3}{2}} e^{4}}{e^{6}}\right )}}{3465 \, e} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^2/(e*x)^(3/2),x, algorithm="maxima")
Output:
-2/3465*(3465*a^2*c^2/sqrt(e*x) - (315*(e*x)^(11/2)*b^2*d^2 + 770*(e*x)^(9 /2)*b^2*c*d*e + 2772*(e*x)^(5/2)*a*b*c*d*e^3 + 6930*sqrt(e*x)*a^2*c*d*e^5 + 495*(b^2*c^2 + 2*a*b*d^2)*(e*x)^(7/2)*e^2 + 1155*(2*a*b*c^2 + a^2*d^2)*( e*x)^(3/2)*e^4)/e^6)/e
Time = 0.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.12 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {3465 \, a^{2} c^{2}}{\sqrt {e x}} - \frac {315 \, \sqrt {e x} b^{2} d^{2} e^{65} x^{5} + 770 \, \sqrt {e x} b^{2} c d e^{65} x^{4} + 495 \, \sqrt {e x} b^{2} c^{2} e^{65} x^{3} + 990 \, \sqrt {e x} a b d^{2} e^{65} x^{3} + 2772 \, \sqrt {e x} a b c d e^{65} x^{2} + 2310 \, \sqrt {e x} a b c^{2} e^{65} x + 1155 \, \sqrt {e x} a^{2} d^{2} e^{65} x + 6930 \, \sqrt {e x} a^{2} c d e^{65}}{e^{66}}\right )}}{3465 \, e} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^2/(e*x)^(3/2),x, algorithm="giac")
Output:
-2/3465*(3465*a^2*c^2/sqrt(e*x) - (315*sqrt(e*x)*b^2*d^2*e^65*x^5 + 770*sq rt(e*x)*b^2*c*d*e^65*x^4 + 495*sqrt(e*x)*b^2*c^2*e^65*x^3 + 990*sqrt(e*x)* a*b*d^2*e^65*x^3 + 2772*sqrt(e*x)*a*b*c*d*e^65*x^2 + 2310*sqrt(e*x)*a*b*c^ 2*e^65*x + 1155*sqrt(e*x)*a^2*d^2*e^65*x + 6930*sqrt(e*x)*a^2*c*d*e^65)/e^ 66)/e
Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {2\,b^2\,d^2\,{\left (e\,x\right )}^{11/2}}{11\,e^7}-\frac {2\,a^2\,c^2}{e\,\sqrt {e\,x}}+\frac {2\,a\,{\left (e\,x\right )}^{3/2}\,\left (2\,b\,c^2+a\,d^2\right )}{3\,e^3}+\frac {2\,b\,{\left (e\,x\right )}^{7/2}\,\left (b\,c^2+2\,a\,d^2\right )}{7\,e^5}+\frac {4\,a^2\,c\,d\,\sqrt {e\,x}}{e^2}+\frac {4\,b^2\,c\,d\,{\left (e\,x\right )}^{9/2}}{9\,e^6}+\frac {8\,a\,b\,c\,d\,{\left (e\,x\right )}^{5/2}}{5\,e^4} \] Input:
int(((a + b*x^2)^2*(c + d*x)^2)/(e*x)^(3/2),x)
Output:
(2*b^2*d^2*(e*x)^(11/2))/(11*e^7) - (2*a^2*c^2)/(e*(e*x)^(1/2)) + (2*a*(e* x)^(3/2)*(a*d^2 + 2*b*c^2))/(3*e^3) + (2*b*(e*x)^(7/2)*(2*a*d^2 + b*c^2))/ (7*e^5) + (4*a^2*c*d*(e*x)^(1/2))/e^2 + (4*b^2*c*d*(e*x)^(9/2))/(9*e^6) + (8*a*b*c*d*(e*x)^(5/2))/(5*e^4)
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {2 \sqrt {e}\, \left (315 b^{2} d^{2} x^{6}+770 b^{2} c d \,x^{5}+990 a b \,d^{2} x^{4}+495 b^{2} c^{2} x^{4}+2772 a b c d \,x^{3}+1155 a^{2} d^{2} x^{2}+2310 a b \,c^{2} x^{2}+6930 a^{2} c d x -3465 a^{2} c^{2}\right )}{3465 \sqrt {x}\, e^{2}} \] Input:
int((d*x+c)^2*(b*x^2+a)^2/(e*x)^(3/2),x)
Output:
(2*sqrt(e)*( - 3465*a**2*c**2 + 6930*a**2*c*d*x + 1155*a**2*d**2*x**2 + 23 10*a*b*c**2*x**2 + 2772*a*b*c*d*x**3 + 990*a*b*d**2*x**4 + 495*b**2*c**2*x **4 + 770*b**2*c*d*x**5 + 315*b**2*d**2*x**6))/(3465*sqrt(x)*e**2)