Integrand size = 25, antiderivative size = 172 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}+\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {3 d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4} \]
1/64*d^4*x*(-e^2*x^2+d^2)^(3/2)/e^3-1/7*d*x^2*(-e^2*x^2+d^2)^(5/2)/e^2-1/8 *x^3*(-e^2*x^2+d^2)^(5/2)/e-1/560*d^2*(35*e*x+32*d)*(-e^2*x^2+d^2)^(5/2)/e ^4+3/128*d^8*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4+3/128*d^6*x*(-e^2*x^2+d^ 2)^(1/2)/e^3
Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.79 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (256 d^7+105 d^6 e x+128 d^5 e^2 x^2+70 d^4 e^3 x^3-1024 d^3 e^4 x^4-840 d^2 e^5 x^5+640 d e^6 x^6+560 e^7 x^7\right )+210 d^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{4480 e^4} \]
-1/4480*(Sqrt[d^2 - e^2*x^2]*(256*d^7 + 105*d^6*e*x + 128*d^5*e^2*x^2 + 70 *d^4*e^3*x^3 - 1024*d^3*e^4*x^4 - 840*d^2*e^5*x^5 + 640*d*e^6*x^6 + 560*e^ 7*x^7) + 210*d^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e^4
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {533, 27, 533, 27, 533, 27, 455, 211, 211, 224, 216}
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.
| \(\displaystyle \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {\int d e x^2 (3 d+8 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{8 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int x^2 (3 d+8 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {d \left (\frac {\int d e x (16 d+21 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{7 e^2}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \int x (16 d+21 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\int 3 d e (7 d+32 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{6 e^2}-\frac {7 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \int (7 d+32 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{2 e}-\frac {7 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (7 d \int \left (d^2-e^2 x^2\right )^{3/2}dx-\frac {32 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {7 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (7 d \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {32 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {7 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (7 d \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {32 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {7 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (7 d \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {32 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {7 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (7 d \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {32 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {7 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {8 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\) |
-1/8*(x^3*(d^2 - e^2*x^2)^(5/2))/e + (d*((-8*x^2*(d^2 - e^2*x^2)^(5/2))/(7 *e) + (d*((-7*x*(d^2 - e^2*x^2)^(5/2))/(2*e) + (d*((-32*(d^2 - e^2*x^2)^(5 /2))/(5*e) + 7*d*((x*(d^2 - e^2*x^2)^(3/2))/4 + (3*d^2*((x*Sqrt[d^2 - e^2* x^2])/2 + (d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)))/4)))/(2*e)))/(7* e)))/(8*e)
3.1.3.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Time = 0.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(-\frac {\left (560 e^{7} x^{7}+640 d \,e^{6} x^{6}-840 d^{2} e^{5} x^{5}-1024 d^{3} e^{4} x^{4}+70 d^{4} e^{3} x^{3}+128 d^{5} e^{2} x^{2}+105 d^{6} e x +256 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{4480 e^{4}}+\frac {3 d^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{3} \sqrt {e^{2}}}\) | \(130\) |
| default | \(e \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6 e^{2}}\right )}{8 e^{2}}\right )+d \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 e^{4}}\right )\) | \(184\) |
-1/4480*(560*e^7*x^7+640*d*e^6*x^6-840*d^2*e^5*x^5-1024*d^3*e^4*x^4+70*d^4 *e^3*x^3+128*d^5*e^2*x^2+105*d^6*e*x+256*d^7)/e^4*(-e^2*x^2+d^2)^(1/2)+3/1 28*d^8/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.74 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {210 \, d^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (560 \, e^{7} x^{7} + 640 \, d e^{6} x^{6} - 840 \, d^{2} e^{5} x^{5} - 1024 \, d^{3} e^{4} x^{4} + 70 \, d^{4} e^{3} x^{3} + 128 \, d^{5} e^{2} x^{2} + 105 \, d^{6} e x + 256 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4480 \, e^{4}} \]
-1/4480*(210*d^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (560*e^7*x^7 + 640*d*e^6*x^6 - 840*d^2*e^5*x^5 - 1024*d^3*e^4*x^4 + 70*d^4*e^3*x^3 + 12 8*d^5*e^2*x^2 + 105*d^6*e*x + 256*d^7)*sqrt(-e^2*x^2 + d^2))/e^4
Time = 0.53 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.10 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 d^{8} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{3}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{7}}{35 e^{4}} - \frac {3 d^{6} x}{128 e^{3}} - \frac {d^{5} x^{2}}{35 e^{2}} - \frac {d^{4} x^{3}}{64 e} + \frac {8 d^{3} x^{4}}{35} + \frac {3 d^{2} e x^{5}}{16} - \frac {d e^{2} x^{6}}{7} - \frac {e^{3} x^{7}}{8}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d x^{4}}{4} + \frac {e x^{5}}{5}\right ) \left (d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]
Piecewise((3*d**8*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e** 2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(12 8*e**3) + sqrt(d**2 - e**2*x**2)*(-2*d**7/(35*e**4) - 3*d**6*x/(128*e**3) - d**5*x**2/(35*e**2) - d**4*x**3/(64*e) + 8*d**3*x**4/35 + 3*d**2*e*x**5/ 16 - d*e**2*x**6/7 - e**3*x**7/8), Ne(e**2, 0)), ((d*x**4/4 + e*x**5/5)*(d **2)**(3/2), True))
Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.95 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 \, d^{8} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}} e^{3}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6} x}{128 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{3}}{8 \, e} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x}{64 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{2}}{7 \, e^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x}{16 \, e^{3}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{35 \, e^{4}} \]
3/128*d^8*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^3) + 3/128*sqrt(-e^2*x^ 2 + d^2)*d^6*x/e^3 - 1/8*(-e^2*x^2 + d^2)^(5/2)*x^3/e + 1/64*(-e^2*x^2 + d ^2)^(3/2)*d^4*x/e^3 - 1/7*(-e^2*x^2 + d^2)^(5/2)*d*x^2/e^2 - 1/16*(-e^2*x^ 2 + d^2)^(5/2)*d^2*x/e^3 - 2/35*(-e^2*x^2 + d^2)^(5/2)*d^3/e^4
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.69 \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 \, d^{8} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, e^{3} {\left | e \right |}} - \frac {1}{4480} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {256 \, d^{7}}{e^{4}} + {\left (\frac {105 \, d^{6}}{e^{3}} + 2 \, {\left (\frac {64 \, d^{5}}{e^{2}} + {\left (\frac {35 \, d^{4}}{e} - 4 \, {\left (128 \, d^{3} + 5 \, {\left (21 \, d^{2} e - 2 \, {\left (7 \, e^{3} x + 8 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
3/128*d^8*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^3*abs(e)) - 1/4480*sqrt(-e^2*x^2 + d^2)*(256*d^7/e^4 + (105*d^6/e^3 + 2*(64*d^5/e^2 + (35*d^4/e - 4*(128*d^ 3 + 5*(21*d^2*e - 2*(7*e^3*x + 8*d*e^2)*x)*x)*x)*x)*x)*x)
Timed out. \[ \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\int x^3\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \]