Integrand size = 19, antiderivative size = 196 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (c d^2+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {4 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^7 \sqrt {d+e x}}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) \sqrt {d+e x}}{e^7}+\frac {2 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {12 c^3 d (d+e x)^{5/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7} \]
-2/5*(a*e^2+c*d^2)^3/e^7/(e*x+d)^(5/2)+4*c*d*(a*e^2+c*d^2)^2/e^7/(e*x+d)^( 3/2)+2*c^2*(a*e^2+5*c*d^2)*(e*x+d)^(3/2)/e^7-12/5*c^3*d*(e*x+d)^(5/2)/e^7+ 2/7*c^3*(e*x+d)^(7/2)/e^7-6*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)/e^7/(e*x+d)^(1 /2)-8*c^2*d*(3*a*e^2+5*c*d^2)*(e*x+d)^(1/2)/e^7
Time = 0.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (7 a^3 e^6+7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a c^2 e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \]
(-2*(7*a^3*e^6 + 7*a^2*c*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 7*a*c^2*e^2 *(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) + c^ 3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2*e ^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)))/(35*e^7*(d + e*x)^(5/2))
Time = 0.32 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {476, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {3 c^2 \sqrt {d+e x} \left (a e^2+5 c d^2\right )}{e^6}-\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^6 \sqrt {d+e x}}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^{3/2}}-\frac {6 c d \left (a e^2+c d^2\right )^2}{e^6 (d+e x)^{5/2}}+\frac {\left (a e^2+c d^2\right )^3}{e^6 (d+e x)^{7/2}}+\frac {c^3 (d+e x)^{5/2}}{e^6}-\frac {6 c^3 d (d+e x)^{3/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac {8 c^2 d \sqrt {d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac {6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt {d+e x}}+\frac {4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac {12 c^3 d (d+e x)^{5/2}}{5 e^7}\) |
(-2*(c*d^2 + a*e^2)^3)/(5*e^7*(d + e*x)^(5/2)) + (4*c*d*(c*d^2 + a*e^2)^2) /(e^7*(d + e*x)^(3/2)) - (6*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(e^7*Sqrt [d + e*x]) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*Sqrt[d + e*x])/e^7 + (2*c^2*(5*c *d^2 + a*e^2)*(d + e*x)^(3/2))/e^7 - (12*c^3*d*(d + e*x)^(5/2))/(5*e^7) + (2*c^3*(d + e*x)^(7/2))/(7*e^7)
3.7.11.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 2.34 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {\frac {2 \left (5 e^{6} x^{6}-12 d \,e^{5} x^{5}+40 d^{2} e^{4} x^{4}-320 x^{3} d^{3} e^{3}-1920 d^{4} e^{2} x^{2}-2560 d^{5} e x -1024 d^{6}\right ) c^{3}}{35}-\frac {256 \left (-\frac {5}{128} e^{4} x^{4}+\frac {5}{16} d \,e^{3} x^{3}+\frac {15}{8} d^{2} e^{2} x^{2}+\frac {5}{2} d^{3} e x +d^{4}\right ) e^{2} a \,c^{2}}{5}-\frac {16 \left (\frac {15}{8} x^{2} e^{2}+\frac {5}{2} d e x +d^{2}\right ) e^{4} a^{2} c}{5}-\frac {2 e^{6} a^{3}}{5}}{\left (e x +d \right )^{\frac {5}{2}} e^{7}}\) | \(163\) |
| risch | \(-\frac {2 c^{2} \left (-5 c \,x^{3} e^{3}+27 c d \,x^{2} e^{2}-35 a \,e^{3} x -106 c \,d^{2} e x +385 a d \,e^{2}+562 d^{3} c \right ) \sqrt {e x +d}}{35 e^{7}}-\frac {2 \left (15 x^{2} a c \,e^{4}+75 x^{2} c^{2} d^{2} e^{2}+20 x a c d \,e^{3}+140 x \,c^{2} d^{3} e +a^{2} e^{4}+7 a c \,d^{2} e^{2}+66 c^{2} d^{4}\right ) \left (e^{2} a +c \,d^{2}\right )}{5 e^{7} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(175\) |
| gosper | \(-\frac {2 \left (-5 x^{6} c^{3} e^{6}+12 x^{5} c^{3} d \,e^{5}-35 x^{4} a \,c^{2} e^{6}-40 x^{4} c^{3} d^{2} e^{4}+280 x^{3} a \,c^{2} d \,e^{5}+320 x^{3} c^{3} d^{3} e^{3}+105 x^{2} a^{2} c \,e^{6}+1680 x^{2} a \,c^{2} d^{2} e^{4}+1920 x^{2} c^{3} d^{4} e^{2}+140 x \,a^{2} c d \,e^{5}+2240 x a \,c^{2} d^{3} e^{3}+2560 x \,c^{3} d^{5} e +7 e^{6} a^{3}+56 d^{2} e^{4} a^{2} c +896 d^{4} e^{2} c^{2} a +1024 c^{3} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}}\) | \(205\) |
| trager | \(-\frac {2 \left (-5 x^{6} c^{3} e^{6}+12 x^{5} c^{3} d \,e^{5}-35 x^{4} a \,c^{2} e^{6}-40 x^{4} c^{3} d^{2} e^{4}+280 x^{3} a \,c^{2} d \,e^{5}+320 x^{3} c^{3} d^{3} e^{3}+105 x^{2} a^{2} c \,e^{6}+1680 x^{2} a \,c^{2} d^{2} e^{4}+1920 x^{2} c^{3} d^{4} e^{2}+140 x \,a^{2} c d \,e^{5}+2240 x a \,c^{2} d^{3} e^{3}+2560 x \,c^{3} d^{5} e +7 e^{6} a^{3}+56 d^{2} e^{4} a^{2} c +896 d^{4} e^{2} c^{2} a +1024 c^{3} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}}\) | \(205\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}+2 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}-24 a \,c^{2} d \,e^{2} \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{\sqrt {e x +d}}+\frac {4 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(214\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}+2 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}-24 a \,c^{2} d \,e^{2} \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{\sqrt {e x +d}}+\frac {4 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(214\) |
2/35*((5*e^6*x^6-12*d*e^5*x^5+40*d^2*e^4*x^4-320*d^3*e^3*x^3-1920*d^4*e^2* x^2-2560*d^5*e*x-1024*d^6)*c^3-896*(-5/128*e^4*x^4+5/16*d*e^3*x^3+15/8*d^2 *e^2*x^2+5/2*d^3*e*x+d^4)*e^2*a*c^2-56*(15/8*x^2*e^2+5/2*d*e*x+d^2)*e^4*a^ 2*c-7*e^6*a^3)/(e*x+d)^(5/2)/e^7
Time = 0.68 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (5 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} - 1024 \, c^{3} d^{6} - 896 \, a c^{2} d^{4} e^{2} - 56 \, a^{2} c d^{2} e^{4} - 7 \, a^{3} e^{6} + 5 \, {\left (8 \, c^{3} d^{2} e^{4} + 7 \, a c^{2} e^{6}\right )} x^{4} - 40 \, {\left (8 \, c^{3} d^{3} e^{3} + 7 \, a c^{2} d e^{5}\right )} x^{3} - 15 \, {\left (128 \, c^{3} d^{4} e^{2} + 112 \, a c^{2} d^{2} e^{4} + 7 \, a^{2} c e^{6}\right )} x^{2} - 20 \, {\left (128 \, c^{3} d^{5} e + 112 \, a c^{2} d^{3} e^{3} + 7 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]
2/35*(5*c^3*e^6*x^6 - 12*c^3*d*e^5*x^5 - 1024*c^3*d^6 - 896*a*c^2*d^4*e^2 - 56*a^2*c*d^2*e^4 - 7*a^3*e^6 + 5*(8*c^3*d^2*e^4 + 7*a*c^2*e^6)*x^4 - 40* (8*c^3*d^3*e^3 + 7*a*c^2*d*e^5)*x^3 - 15*(128*c^3*d^4*e^2 + 112*a*c^2*d^2* e^4 + 7*a^2*c*e^6)*x^2 - 20*(128*c^3*d^5*e + 112*a*c^2*d^3*e^3 + 7*a^2*c*d *e^5)*x)*sqrt(e*x + d)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)
Time = 3.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {6 c^{3} d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {2 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{6} \left (d + e x\right )^{\frac {3}{2}}} - \frac {3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{e^{6}} - \frac {\left (a e^{2} + c d^{2}\right )^{3}}{5 e^{6} \left (d + e x\right )^{\frac {5}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(-6*c**3*d*(d + e*x)**(5/2)/(5*e**6) + c**3*(d + e*x)**(7/2)/ (7*e**6) + 2*c*d*(a*e**2 + c*d**2)**2/(e**6*(d + e*x)**(3/2)) - 3*c*(a*e** 2 + c*d**2)*(a*e**2 + 5*c*d**2)/(e**6*sqrt(d + e*x)) + (d + e*x)**(3/2)*(3 *a*c**2*e**2 + 15*c**3*d**2)/(3*e**6) + sqrt(d + e*x)*(-12*a*c**2*d*e**2 - 20*c**3*d**3)/e**6 - (a*e**2 + c*d**2)**3/(5*e**6*(d + e*x)**(5/2)))/e, N e(e, 0)), ((a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7)/d**(7/2) , True))
Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d + 35 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 140 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \sqrt {e x + d}}{e^{6}} - \frac {7 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 15 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{2} - 10 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{6}}\right )}}{35 \, e} \]
2/35*((5*(e*x + d)^(7/2)*c^3 - 42*(e*x + d)^(5/2)*c^3*d + 35*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(3/2) - 140*(5*c^3*d^3 + 3*a*c^2*d*e^2)*sqrt(e*x + d) )/e^6 - 7*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 15*(5*c ^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*x + d)^2 - 10*(c^3*d^5 + 2*a*c^2* d^3*e^2 + a^2*c*d*e^4)*(e*x + d))/((e*x + d)^(5/2)*e^6))/e
Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (75 \, {\left (e x + d\right )}^{2} c^{3} d^{4} - 10 \, {\left (e x + d\right )} c^{3} d^{5} + c^{3} d^{6} + 90 \, {\left (e x + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \, {\left (e x + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 15 \, {\left (e x + d\right )}^{2} a^{2} c e^{4} - 10 \, {\left (e x + d\right )} a^{2} c d e^{4} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}}{5 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{7}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} e^{42} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d e^{42} + 175 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt {e x + d} c^{3} d^{3} e^{42} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} e^{44} - 420 \, \sqrt {e x + d} a c^{2} d e^{44}\right )}}{35 \, e^{49}} \]
-2/5*(75*(e*x + d)^2*c^3*d^4 - 10*(e*x + d)*c^3*d^5 + c^3*d^6 + 90*(e*x + d)^2*a*c^2*d^2*e^2 - 20*(e*x + d)*a*c^2*d^3*e^2 + 3*a*c^2*d^4*e^2 + 15*(e* x + d)^2*a^2*c*e^4 - 10*(e*x + d)*a^2*c*d*e^4 + 3*a^2*c*d^2*e^4 + a^3*e^6) /((e*x + d)^(5/2)*e^7) + 2/35*(5*(e*x + d)^(7/2)*c^3*e^42 - 42*(e*x + d)^( 5/2)*c^3*d*e^42 + 175*(e*x + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt(e*x + d)*c^3 *d^3*e^42 + 35*(e*x + d)^(3/2)*a*c^2*e^44 - 420*sqrt(e*x + d)*a*c^2*d*e^44 )/e^49
Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )-\left (d+e\,x\right )\,\left (4\,a^2\,c\,d\,e^4+8\,a\,c^2\,d^3\,e^2+4\,c^3\,d^5\right )+\frac {2\,a^3\,e^6}{5}+\frac {2\,c^3\,d^6}{5}+\frac {6\,a\,c^2\,d^4\,e^2}{5}+\frac {6\,a^2\,c\,d^2\,e^4}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,\sqrt {d+e\,x}}{e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \]
((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(3/2))/(3*e^7) + (2*c^3*(d + e*x)^(7 /2))/(7*e^7) - ((d + e*x)^2*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2*d^2*e^2) - (d + e*x)*(4*c^3*d^5 + 8*a*c^2*d^3*e^2 + 4*a^2*c*d*e^4) + (2*a^3*e^6)/5 + (2*c^3*d^6)/5 + (6*a*c^2*d^4*e^2)/5 + (6*a^2*c*d^2*e^4)/5)/(e^7*(d + e*x )^(5/2)) - ((40*c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(1/2))/e^7 - (12*c^3*d *(d + e*x)^(5/2))/(5*e^7)