Integrand size = 19, antiderivative size = 123 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2+a e^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {8 c d \left (c d^2+a e^2\right ) \sqrt {d+e x}}{e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \]
4/3*c*(a*e^2+3*c*d^2)*(e*x+d)^(3/2)/e^5-8/5*c^2*d*(e*x+d)^(5/2)/e^5+2/7*c^ 2*(e*x+d)^(7/2)/e^5-2*(a*e^2+c*d^2)^2/e^5/(e*x+d)^(1/2)-8*c*d*(a*e^2+c*d^2 )*(e*x+d)^(1/2)/e^5
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (105 a^2 e^4+70 a c e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+3 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )}{105 e^5 \sqrt {d+e x}} \]
(-2*(105*a^2*e^4 + 70*a*c*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) + 3*c^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)))/(105*e^5*Sqrt[ d + e*x])
Time = 0.25 (sec) , antiderivative size = 123, 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.
\(\displaystyle \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \int \left (\frac {2 c \sqrt {d+e x} \left (a e^2+3 c d^2\right )}{e^4}-\frac {4 c d \left (a e^2+c d^2\right )}{e^4 \sqrt {d+e x}}+\frac {\left (a e^2+c d^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {c^2 (d+e x)^{5/2}}{e^4}-\frac {4 c^2 d (d+e x)^{3/2}}{e^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 c (d+e x)^{3/2} \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {8 c d \sqrt {d+e x} \left (a e^2+c d^2\right )}{e^5}-\frac {2 \left (a e^2+c d^2\right )^2}{e^5 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c^2 d (d+e x)^{5/2}}{5 e^5}\) |
(-2*(c*d^2 + a*e^2)^2)/(e^5*Sqrt[d + e*x]) - (8*c*d*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^5 + (4*c*(3*c*d^2 + a*e^2)*(d + e*x)^(3/2))/(3*e^5) - (8*c^2*d* (d + e*x)^(5/2))/(5*e^5) + (2*c^2*(d + e*x)^(7/2))/(7*e^5)
3.7.2.3.1 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 = 2.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {1}{7} x^{4} c^{2}-\frac {2}{3} a c \,x^{2}+a^{2}\right ) e^{4}+\frac {8 x \left (\frac {3 c \,x^{2}}{35}+a \right ) c d \,e^{3}}{3}+\frac {16 \left (-\frac {3 c \,x^{2}}{35}+a \right ) c \,d^{2} e^{2}}{3}+\frac {64 x \,c^{2} d^{3} e}{35}+\frac {128 c^{2} d^{4}}{35}\right )}{\sqrt {e x +d}\, e^{5}}\) | \(88\) |
risch | \(-\frac {2 c \left (-15 c \,x^{3} e^{3}+39 c d \,x^{2} e^{2}-70 a \,e^{3} x -87 c \,d^{2} e x +350 a d \,e^{2}+279 d^{3} c \right ) \sqrt {e x +d}}{105 e^{5}}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{5} \sqrt {e x +d}}\) | \(100\) |
gosper | \(-\frac {2 \left (-15 c^{2} x^{4} e^{4}+24 x^{3} c^{2} d \,e^{3}-70 x^{2} a c \,e^{4}-48 x^{2} c^{2} d^{2} e^{2}+280 x a c d \,e^{3}+192 x \,c^{2} d^{3} e +105 a^{2} e^{4}+560 a c \,d^{2} e^{2}+384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\) | \(106\) |
trager | \(-\frac {2 \left (-15 c^{2} x^{4} e^{4}+24 x^{3} c^{2} d \,e^{3}-70 x^{2} a c \,e^{4}-48 x^{2} c^{2} d^{2} e^{2}+280 x a c d \,e^{3}+192 x \,c^{2} d^{3} e +105 a^{2} e^{4}+560 a c \,d^{2} e^{2}+384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\) | \(106\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}-8 a c d \,e^{2} \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(125\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}-8 a c d \,e^{2} \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(125\) |
-2*((-1/7*x^4*c^2-2/3*a*c*x^2+a^2)*e^4+8/3*x*(3/35*c*x^2+a)*c*d*e^3+16/3*( -3/35*c*x^2+a)*c*d^2*e^2+64/35*x*c^2*d^3*e+128/35*c^2*d^4)/(e*x+d)^(1/2)/e ^5
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (15 \, c^{2} e^{4} x^{4} - 24 \, c^{2} d e^{3} x^{3} - 384 \, c^{2} d^{4} - 560 \, a c d^{2} e^{2} - 105 \, a^{2} e^{4} + 2 \, {\left (24 \, c^{2} d^{2} e^{2} + 35 \, a c e^{4}\right )} x^{2} - 8 \, {\left (24 \, c^{2} d^{3} e + 35 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{6} x + d e^{5}\right )}} \]
2/105*(15*c^2*e^4*x^4 - 24*c^2*d*e^3*x^3 - 384*c^2*d^4 - 560*a*c*d^2*e^2 - 105*a^2*e^4 + 2*(24*c^2*d^2*e^2 + 35*a*c*e^4)*x^2 - 8*(24*c^2*d^3*e + 35* a*c*d*e^3)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)
Time = 1.62 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {4 c^{2} d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} + \frac {c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 a c e^{2} + 6 c^{2} d^{2}\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (- 4 a c d e^{2} - 4 c^{2} d^{3}\right )}{e^{4}} - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{e^{4} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + \frac {2 a c x^{3}}{3} + \frac {c^{2} x^{5}}{5}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(-4*c**2*d*(d + e*x)**(5/2)/(5*e**4) + c**2*(d + e*x)**(7/2)/ (7*e**4) + (d + e*x)**(3/2)*(2*a*c*e**2 + 6*c**2*d**2)/(3*e**4) + sqrt(d + e*x)*(-4*a*c*d*e**2 - 4*c**2*d**3)/e**4 - (a*e**2 + c*d**2)**2/(e**4*sqrt (d + e*x)))/e, Ne(e, 0)), ((a**2*x + 2*a*c*x**3/3 + c**2*x**5/5)/d**(3/2), True))
Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} - 84 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d + 70 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 420 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {105 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{105 \, e} \]
2/105*((15*(e*x + d)^(7/2)*c^2 - 84*(e*x + d)^(5/2)*c^2*d + 70*(3*c^2*d^2 + a*c*e^2)*(e*x + d)^(3/2) - 420*(c^2*d^3 + a*c*d*e^2)*sqrt(e*x + d))/e^4 - 105*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)/(sqrt(e*x + d)*e^4))/e
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{5}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} e^{30} - 84 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d e^{30} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt {e x + d} c^{2} d^{3} e^{30} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a c e^{32} - 420 \, \sqrt {e x + d} a c d e^{32}\right )}}{105 \, e^{35}} \]
-2*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)/(sqrt(e*x + d)*e^5) + 2/105*(15*(e* x + d)^(7/2)*c^2*e^30 - 84*(e*x + d)^(5/2)*c^2*d*e^30 + 210*(e*x + d)^(3/2 )*c^2*d^2*e^30 - 420*sqrt(e*x + d)*c^2*d^3*e^30 + 70*(e*x + d)^(3/2)*a*c*e ^32 - 420*sqrt(e*x + d)*a*c*d*e^32)/e^35
Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,\sqrt {d+e\,x}}{e^5}-\frac {2\,a^2\,e^4+4\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{e^5\,\sqrt {d+e\,x}}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \]