Integrand size = 24, antiderivative size = 188 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {55 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \]
55/192*d^5*x*(-e^2*x^2+d^2)^(3/2)+11/48*d^3*x*(-e^2*x^2+d^2)^(5/2)-11/56*d ^2*(-e^2*x^2+d^2)^(7/2)/e-11/72*d*(e*x+d)*(-e^2*x^2+d^2)^(7/2)/e-1/9*(e*x+ d)^2*(-e^2*x^2+d^2)^(7/2)/e+55/128*d^9*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e+ 55/128*d^7*x*(-e^2*x^2+d^2)^(1/2)
Time = 0.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.82 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-3712 d^8+4599 d^7 e x+10240 d^6 e^2 x^2+3066 d^5 e^3 x^3-8448 d^4 e^4 x^4-7224 d^3 e^5 x^5+1024 d^2 e^6 x^6+3024 d e^7 x^7+896 e^8 x^8\right )}{8064 e}-\frac {55 d^9 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {-e^2}} \]
(Sqrt[d^2 - e^2*x^2]*(-3712*d^8 + 4599*d^7*e*x + 10240*d^6*e^2*x^2 + 3066* d^5*e^3*x^3 - 8448*d^4*e^4*x^4 - 7224*d^3*e^5*x^5 + 1024*d^2*e^6*x^6 + 302 4*d*e^7*x^7 + 896*e^8*x^8))/(8064*e) - (55*d^9*Log[-(Sqrt[-e^2]*x) + Sqrt[ d^2 - e^2*x^2]])/(128*Sqrt[-e^2])
Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {469, 469, 455, 211, 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.
| \(\displaystyle \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle \frac {11}{9} d \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}dx-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle \frac {11}{9} d \left (\frac {9}{8} d \int (d+e x) \left (d^2-e^2 x^2\right )^{5/2}dx-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {11}{9} d \left (\frac {9}{8} d \left (d \int \left (d^2-e^2 x^2\right )^{5/2}dx-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {11}{9} d \left (\frac {9}{8} d \left (d \left (\frac {5}{6} d^2 \int \left (d^2-e^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {11}{9} d \left (\frac {9}{8} d \left (d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {11}{9} d \left (\frac {9}{8} d \left (d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {11}{9} d \left (\frac {9}{8} d \left (d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {11}{9} d \left (\frac {9}{8} d \left (d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}\) |
-1/9*((d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/e + (11*d*(-1/8*((d + e*x)*(d^2 - e^2*x^2)^(7/2))/e + (9*d*(-1/7*(d^2 - e^2*x^2)^(7/2)/e + d*((x*(d^2 - e^2 *x^2)^(5/2))/6 + (5*d^2*((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))/6)))/ 8))/9
3.1.70.3.1 Defintions of rubi rules used
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* ((n + p)/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* p + 1, 0] && IntegerQ[2*p]
Time = 0.40 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(-\frac {\left (-896 e^{8} x^{8}-3024 d \,e^{7} x^{7}-1024 d^{2} e^{6} x^{6}+7224 d^{3} e^{5} x^{5}+8448 d^{4} x^{4} e^{4}-3066 d^{5} e^{3} x^{3}-10240 d^{6} e^{2} x^{2}-4599 d^{7} e x +3712 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{8064 e}+\frac {55 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}}\) | \(138\) |
| default | \(d^{3} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )+e^{3} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )+3 d \,e^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )-\frac {3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e}\) | \(304\) |
-1/8064*(-896*e^8*x^8-3024*d*e^7*x^7-1024*d^2*e^6*x^6+7224*d^3*e^5*x^5+844 8*d^4*e^4*x^4-3066*d^5*e^3*x^3-10240*d^6*e^2*x^2-4599*d^7*e*x+3712*d^8)/e* (-e^2*x^2+d^2)^(1/2)+55/128*d^9/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2 +d^2)^(1/2))
Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.74 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {6930 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (896 \, e^{8} x^{8} + 3024 \, d e^{7} x^{7} + 1024 \, d^{2} e^{6} x^{6} - 7224 \, d^{3} e^{5} x^{5} - 8448 \, d^{4} e^{4} x^{4} + 3066 \, d^{5} e^{3} x^{3} + 10240 \, d^{6} e^{2} x^{2} + 4599 \, d^{7} e x - 3712 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8064 \, e} \]
-1/8064*(6930*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (896*e^8*x^8 + 3024*d*e^7*x^7 + 1024*d^2*e^6*x^6 - 7224*d^3*e^5*x^5 - 8448*d^4*e^4*x^4 + 3066*d^5*e^3*x^3 + 10240*d^6*e^2*x^2 + 4599*d^7*e*x - 3712*d^8)*sqrt(-e ^2*x^2 + d^2))/e
Time = 0.62 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.10 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {55 d^{9} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {29 d^{8}}{63 e} + \frac {73 d^{7} x}{128} + \frac {80 d^{6} e x^{2}}{63} + \frac {73 d^{5} e^{2} x^{3}}{192} - \frac {22 d^{4} e^{3} x^{4}}{21} - \frac {43 d^{3} e^{4} x^{5}}{48} + \frac {8 d^{2} e^{5} x^{6}}{63} + \frac {3 d e^{6} x^{7}}{8} + \frac {e^{7} x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\left (d^{2}\right )^{\frac {5}{2}} \left (\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((55*d**9*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 + sqrt(d**2 - e**2*x**2)*(-29*d**8/(63*e) + 73*d**7*x/128 + 80*d**6*e*x* *2/63 + 73*d**5*e**2*x**3/192 - 22*d**4*e**3*x**4/21 - 43*d**3*e**4*x**5/4 8 + 8*d**2*e**5*x**6/63 + 3*d*e**6*x**7/8 + e**7*x**8/9), Ne(e**2, 0)), (( d**2)**(5/2)*Piecewise((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True)), Tr ue))
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.77 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {55 \, d^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}}} + \frac {55}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x + \frac {55}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x + \frac {11}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x - \frac {1}{9} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{2} - \frac {3}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e} \]
55/128*d^9*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 55/128*sqrt(-e^2*x^2 + d^2)*d^7*x + 55/192*(-e^2*x^2 + d^2)^(3/2)*d^5*x + 11/48*(-e^2*x^2 + d^2)^ (5/2)*d^3*x - 1/9*(-e^2*x^2 + d^2)^(7/2)*e*x^2 - 3/8*(-e^2*x^2 + d^2)^(7/2 )*d*x - 29/63*(-e^2*x^2 + d^2)^(7/2)*d^2/e
Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.68 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {55 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, {\left | e \right |}} - \frac {1}{8064} \, {\left (\frac {3712 \, d^{8}}{e} - {\left (4599 \, d^{7} + 2 \, {\left (5120 \, d^{6} e + {\left (1533 \, d^{5} e^{2} - 4 \, {\left (1056 \, d^{4} e^{3} + {\left (903 \, d^{3} e^{4} - 2 \, {\left (64 \, d^{2} e^{5} + 7 \, {\left (8 \, e^{7} x + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
55/128*d^9*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/8064*(3712*d^8/e - (4599 *d^7 + 2*(5120*d^6*e + (1533*d^5*e^2 - 4*(1056*d^4*e^3 + (903*d^3*e^4 - 2* (64*d^2*e^5 + 7*(8*e^7*x + 27*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)
Timed out. \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]