\(\int (d+e x)^2 (d^2-e^2 x^2)^{7/2} \, dx\) [801]

Optimal result

Integrand size = 24, antiderivative size = 179 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e} \]

[Out]

77/384*d^6*x*(-e^2*x^2+d^2)^(3/2)+77/480*d^4*x*(-e^2*x^2+d^2)^(5/2)+11/80*d^2*x*(-e^2*x^2+d^2)^(7/2)-11/90*d*(
-e^2*x^2+d^2)^(9/2)/e-1/10*(e*x+d)*(-e^2*x^2+d^2)^(9/2)/e+77/256*d^10*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e+77/25
6*d^8*x*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {685, 655, 201, 223, 209} \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {77 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2} \]

[In]

Int[(d + e*x)^2*(d^2 - e^2*x^2)^(7/2),x]

[Out]

(77*d^8*x*Sqrt[d^2 - e^2*x^2])/256 + (77*d^6*x*(d^2 - e^2*x^2)^(3/2))/384 + (77*d^4*x*(d^2 - e^2*x^2)^(5/2))/4
80 + (11*d^2*x*(d^2 - e^2*x^2)^(7/2))/80 - (11*d*(d^2 - e^2*x^2)^(9/2))/(90*e) - ((d + e*x)*(d^2 - e^2*x^2)^(9
/2))/(10*e) + (77*d^10*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

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]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[2*c*d*((m + p)/(c*(m + 2*p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{10} (11 d) \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx \\ & = -\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{10} \left (11 d^2\right ) \int \left (d^2-e^2 x^2\right )^{7/2} \, dx \\ & = \frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{80} \left (77 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx \\ & = \frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{96} \left (77 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{128} \left (77 d^8\right ) \int \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{256} \left (77 d^{10}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{256} \left (77 d^{10}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = \frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (2560 d^9-8055 d^8 e x-10240 d^7 e^2 x^2+6150 d^6 e^3 x^3+15360 d^5 e^4 x^4+312 d^4 e^5 x^5-10240 d^3 e^6 x^6-3024 d^2 e^7 x^7+2560 d e^8 x^8+1152 e^9 x^9\right )+6930 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{11520 e} \]

[In]

Integrate[(d + e*x)^2*(d^2 - e^2*x^2)^(7/2),x]

[Out]

-1/11520*(Sqrt[d^2 - e^2*x^2]*(2560*d^9 - 8055*d^8*e*x - 10240*d^7*e^2*x^2 + 6150*d^6*e^3*x^3 + 15360*d^5*e^4*
x^4 + 312*d^4*e^5*x^5 - 10240*d^3*e^6*x^6 - 3024*d^2*e^7*x^7 + 2560*d*e^8*x^8 + 1152*e^9*x^9) + 6930*d^10*ArcT
an[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {\left (1152 e^{9} x^{9}+2560 d \,e^{8} x^{8}-3024 d^{2} e^{7} x^{7}-10240 d^{3} e^{6} x^{6}+312 d^{4} e^{5} x^{5}+15360 d^{5} e^{4} x^{4}+6150 d^{6} e^{3} x^{3}-10240 d^{7} e^{2} x^{2}-8055 d^{8} e x +2560 d^{9}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{11520 e}+\frac {77 d^{10} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{256 \sqrt {e^{2}}}\)	\(149\)
default	\(d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-x^{2} e^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )+e^{2} \left (-\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {9}{2}}}{10 e^{2}}+\frac {d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-x^{2} e^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )}{10 e^{2}}\right )-\frac {2 d \left (-x^{2} e^{2}+d^{2}\right )^{\frac {9}{2}}}{9 e}\)	\(297\)

[In]

int((e*x+d)^2*(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/11520*(1152*e^9*x^9+2560*d*e^8*x^8-3024*d^2*e^7*x^7-10240*d^3*e^6*x^6+312*d^4*e^5*x^5+15360*d^5*e^4*x^4+615
0*d^6*e^3*x^3-10240*d^7*e^2*x^2-8055*d^8*e*x+2560*d^9)/e*(-e^2*x^2+d^2)^(1/2)+77/256*d^10/(e^2)^(1/2)*arctan((
e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.83 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=-\frac {6930 \, d^{10} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (1152 \, e^{9} x^{9} + 2560 \, d e^{8} x^{8} - 3024 \, d^{2} e^{7} x^{7} - 10240 \, d^{3} e^{6} x^{6} + 312 \, d^{4} e^{5} x^{5} + 15360 \, d^{5} e^{4} x^{4} + 6150 \, d^{6} e^{3} x^{3} - 10240 \, d^{7} e^{2} x^{2} - 8055 \, d^{8} e x + 2560 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{11520 \, e} \]

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/11520*(6930*d^10*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (1152*e^9*x^9 + 2560*d*e^8*x^8 - 3024*d^2*e^7*
x^7 - 10240*d^3*e^6*x^6 + 312*d^4*e^5*x^5 + 15360*d^5*e^4*x^4 + 6150*d^6*e^3*x^3 - 10240*d^7*e^2*x^2 - 8055*d^
8*e*x + 2560*d^9)*sqrt(-e^2*x^2 + d^2))/e

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\begin {cases} \frac {77 d^{10} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{256} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{9}}{9 e} + \frac {179 d^{8} x}{256} + \frac {8 d^{7} e x^{2}}{9} - \frac {205 d^{6} e^{2} x^{3}}{384} - \frac {4 d^{5} e^{3} x^{4}}{3} - \frac {13 d^{4} e^{4} x^{5}}{480} + \frac {8 d^{3} e^{5} x^{6}}{9} + \frac {21 d^{2} e^{6} x^{7}}{80} - \frac {2 d e^{7} x^{8}}{9} - \frac {e^{8} x^{9}}{10}\right ) & \text {for}\: e^{2} \neq 0 \\\left (d^{2}\right )^{\frac {7}{2}} \left (\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)**2*(-e**2*x**2+d**2)**(7/2),x)

[Out]

Piecewise((77*d**10*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))/256 + sqrt(d**2 - e**2*x**2)*(-2*d**9/(9*e) + 179*d**8*x/256 + 8*d**7*e*x
**2/9 - 205*d**6*e**2*x**3/384 - 4*d**5*e**3*x**4/3 - 13*d**4*e**4*x**5/480 + 8*d**3*e**5*x**6/9 + 21*d**2*e**
6*x**7/80 - 2*d*e**7*x**8/9 - e**8*x**9/10), Ne(e**2, 0)), ((d**2)**(7/2)*Piecewise((d**2*x, Eq(e, 0)), ((d +
e*x)**3/(3*e), True)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.79 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {77 \, d^{10} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{256 \, \sqrt {e^{2}}} + \frac {77}{256} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8} x + \frac {77}{384} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} x + \frac {77}{480} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x + \frac {11}{80} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x - \frac {1}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} x - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d}{9 \, e} \]

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

77/256*d^10*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 77/256*sqrt(-e^2*x^2 + d^2)*d^8*x + 77/384*(-e^2*x^2 + d^2
)^(3/2)*d^6*x + 77/480*(-e^2*x^2 + d^2)^(5/2)*d^4*x + 11/80*(-e^2*x^2 + d^2)^(7/2)*d^2*x - 1/10*(-e^2*x^2 + d^
2)^(9/2)*x - 2/9*(-e^2*x^2 + d^2)^(9/2)*d/e

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.78 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {77 \, d^{10} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{256 \, {\left | e \right |}} - \frac {1}{11520} \, {\left (\frac {2560 \, d^{9}}{e} - {\left (8055 \, d^{8} + 2 \, {\left (5120 \, d^{7} e - {\left (3075 \, d^{6} e^{2} + 4 \, {\left (1920 \, d^{5} e^{3} + {\left (39 \, d^{4} e^{4} - 2 \, {\left (640 \, d^{3} e^{5} + {\left (189 \, d^{2} e^{6} - 8 \, {\left (9 \, e^{8} x + 20 \, d e^{7}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

integrate((e*x+d)^2*(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

77/256*d^10*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/11520*(2560*d^9/e - (8055*d^8 + 2*(5120*d^7*e - (3075*d^6*e
^2 + 4*(1920*d^5*e^3 + (39*d^4*e^4 - 2*(640*d^3*e^5 + (189*d^2*e^6 - 8*(9*e^8*x + 20*d*e^7)*x)*x)*x)*x)*x)*x)*
x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^2 \,d x \]

[In]

int((d^2 - e^2*x^2)^(7/2)*(d + e*x)^2,x)

[Out]

int((d^2 - e^2*x^2)^(7/2)*(d + e*x)^2, x)