\(\int (d+e x) (d^2-e^2 x^2)^{7/2} \, dx\) [802]

Optimal result

Integrand size = 22, antiderivative size = 148 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \]

[Out]

35/192*d^5*x*(-e^2*x^2+d^2)^(3/2)+7/48*d^3*x*(-e^2*x^2+d^2)^(5/2)+1/8*d*x*(-e^2*x^2+d^2)^(7/2)-1/9*(-e^2*x^2+d
^2)^(9/2)/e+35/128*d^9*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e+35/128*d^7*x*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {655, 201, 223, 209} \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {35 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]

[In]

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

[Out]

(35*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (35*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (7*d^3*x*(d^2 - e^2*x^2)^(5/2))/48
+ (d*x*(d^2 - e^2*x^2)^(7/2))/8 - (d^2 - e^2*x^2)^(9/2)/(9*e) + (35*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1
28*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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+d \int \left (d^2-e^2 x^2\right )^{7/2} \, dx \\ & = \frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{8} \left (7 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx \\ & = \frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{48} \left (35 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{64} \left (35 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{128} \left (35 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{128} \left (35 d^9\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 {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-128 d^8+837 d^7 e x+512 d^6 e^2 x^2-978 d^5 e^3 x^3-768 d^4 e^4 x^4+600 d^3 e^5 x^5+512 d^2 e^6 x^6-144 d e^7 x^7-128 e^8 x^8\right )}{1152 e}-\frac {35 d^9 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {-e^2}} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-128*d^8 + 837*d^7*e*x + 512*d^6*e^2*x^2 - 978*d^5*e^3*x^3 - 768*d^4*e^4*x^4 + 600*d^3*e
^5*x^5 + 512*d^2*e^6*x^6 - 144*d*e^7*x^7 - 128*e^8*x^8))/(1152*e) - (35*d^9*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e
^2*x^2]])/(128*Sqrt[-e^2])

Maple [A] (verified)

Time = 2.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {\left (128 e^{8} x^{8}+144 d \,e^{7} x^{7}-512 d^{2} e^{6} x^{6}-600 d^{3} e^{5} x^{5}+768 d^{4} e^{4} x^{4}+978 d^{5} e^{3} x^{3}-512 d^{6} e^{2} x^{2}-837 d^{7} e x +128 d^{8}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{1152 e}+\frac {35 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}}\)	\(138\)
default	\(d \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-x^{2} e^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )-\frac {\left (-x^{2} e^{2}+d^{2}\right )^{\frac {9}{2}}}{9 e}\)	\(142\)

[In]

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

[Out]

-1/1152*(128*e^8*x^8+144*d*e^7*x^7-512*d^2*e^6*x^6-600*d^3*e^5*x^5+768*d^4*e^4*x^4+978*d^5*e^3*x^3-512*d^6*e^2
*x^2-837*d^7*e*x+128*d^8)/e*(-e^2*x^2+d^2)^(1/2)+35/128*d^9/(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 = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=-\frac {630 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (128 \, e^{8} x^{8} + 144 \, d e^{7} x^{7} - 512 \, d^{2} e^{6} x^{6} - 600 \, d^{3} e^{5} x^{5} + 768 \, d^{4} e^{4} x^{4} + 978 \, d^{5} e^{3} x^{3} - 512 \, d^{6} e^{2} x^{2} - 837 \, d^{7} e x + 128 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1152 \, e} \]

[In]

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

[Out]

-1/1152*(630*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (128*e^8*x^8 + 144*d*e^7*x^7 - 512*d^2*e^6*x^6 -
600*d^3*e^5*x^5 + 768*d^4*e^4*x^4 + 978*d^5*e^3*x^3 - 512*d^6*e^2*x^2 - 837*d^7*e*x + 128*d^8)*sqrt(-e^2*x^2 +
d^2))/e

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.33 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\begin {cases} \frac {35 d^{9} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{8}}{9 e} + \frac {93 d^{7} x}{128} + \frac {4 d^{6} e x^{2}}{9} - \frac {163 d^{5} e^{2} x^{3}}{192} - \frac {2 d^{4} e^{3} x^{4}}{3} + \frac {25 d^{3} e^{4} x^{5}}{48} + \frac {4 d^{2} e^{5} x^{6}}{9} - \frac {d e^{6} x^{7}}{8} - \frac {e^{7} x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\left (d x + \frac {e x^{2}}{2}\right ) \left (d^{2}\right )^{\frac {7}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((35*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))/128 + sqrt(d**2 - e**2*x**2)*(-d**8/(9*e) + 93*d**7*x/128 + 4*d**6*e*x**2/
9 - 163*d**5*e**2*x**3/192 - 2*d**4*e**3*x**4/3 + 25*d**3*e**4*x**5/48 + 4*d**2*e**5*x**6/9 - d*e**6*x**7/8 -
e**7*x**8/9), Ne(e**2, 0)), ((d*x + e*x**2/2)*(d**2)**(7/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {35 \, d^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}}} + \frac {35}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x + \frac {35}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x + \frac {7}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x + \frac {1}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}}{9 \, e} \]

[In]

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

[Out]

35/128*d^9*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 35/128*sqrt(-e^2*x^2 + d^2)*d^7*x + 35/192*(-e^2*x^2 + d^2)
^(3/2)*d^5*x + 7/48*(-e^2*x^2 + d^2)^(5/2)*d^3*x + 1/8*(-e^2*x^2 + d^2)^(7/2)*d*x - 1/9*(-e^2*x^2 + d^2)^(9/2)
/e

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {35 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, {\left | e \right |}} - \frac {1}{1152} \, {\left (\frac {128 \, d^{8}}{e} - {\left (837 \, d^{7} + 2 \, {\left (256 \, d^{6} e - {\left (489 \, d^{5} e^{2} + 4 \, {\left (96 \, d^{4} e^{3} - {\left (75 \, d^{3} e^{4} + 2 \, {\left (32 \, d^{2} e^{5} - {\left (8 \, e^{7} x + 9 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

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

[Out]

35/128*d^9*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/1152*(128*d^8/e - (837*d^7 + 2*(256*d^6*e - (489*d^5*e^2 + 4
*(96*d^4*e^3 - (75*d^3*e^4 + 2*(32*d^2*e^5 - (8*e^7*x + 9*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.45 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {d\,x\,{\left (d^2-e^2\,x^2\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{2},\frac {1}{2};\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{7/2}}-\frac {{\left (d^2-e^2\,x^2\right )}^{9/2}}{9\,e} \]

[In]

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

[Out]

(d*x*(d^2 - e^2*x^2)^(7/2)*hypergeom([-7/2, 1/2], 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^(7/2) - (d^2 - e^2*
x^2)^(9/2)/(9*e)