\(\int (d+e x) (d^2-e^2 x^2)^{7/2} \, dx\) [62]

Optimal result

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

35/128*d^7*x*(-e^2*x^2+d^2)^(1/2)+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
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-128 d^8+837 d^7 e x+512 d^6 e^2 x^2-978 d^5 e^3 x^3-768 d^4 e^4 x^4+600 d^3 e^5 x^5+512 d^2 e^6 x^6-144 d e^7 x^7-128 e^8 x^8\right )}{1152 e}-\frac {35 d^9 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {-e^2}} \] Input:

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

Output:

(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])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {455, 211, 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.

\(\displaystyle \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx\)

\(\Big \downarrow \) 455

\(\displaystyle d \int \left (d^2-e^2 x^2\right )^{7/2}dx-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\)

\(\Big \downarrow \) 211

\(\displaystyle d \left (\frac {7}{8} d^2 \int \left (d^2-e^2 x^2\right )^{5/2}dx+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\)

\(\Big \downarrow \) 211

\(\displaystyle d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \int \left (d^2-e^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\)

\(\Big \downarrow \) 211

\(\displaystyle d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\)

\(\Big \downarrow \) 211

\(\displaystyle d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\)

\(\Big \downarrow \) 224

\(\displaystyle d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\)

\(\Big \downarrow \) 216

\(\displaystyle d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\)

Input:

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

Output:

-1/9*(d^2 - e^2*x^2)^(9/2)/e + d*((x*(d^2 - e^2*x^2)^(7/2))/8 + (7*d^2*((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)
 

Defintions of rubi rules used

rule 211

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])
 

rule 216

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])
 

rule 224

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 455

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]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93

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

Input:

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

Output:

-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)

Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=-\frac {630 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (128 \, e^{8} x^{8} + 144 \, d e^{7} x^{7} - 512 \, d^{2} e^{6} x^{6} - 600 \, d^{3} e^{5} x^{5} + 768 \, d^{4} e^{4} x^{4} + 978 \, d^{5} e^{3} x^{3} - 512 \, d^{6} e^{2} x^{2} - 837 \, d^{7} e x + 128 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1152 \, e} \] Input:

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

Output:

-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 + 97 
8*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 = 0.80 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.33 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\begin {cases} \frac {35 d^{9} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{8}}{9 e} + \frac {93 d^{7} x}{128} + \frac {4 d^{6} e x^{2}}{9} - \frac {163 d^{5} e^{2} x^{3}}{192} - \frac {2 d^{4} e^{3} x^{4}}{3} + \frac {25 d^{3} e^{4} x^{5}}{48} + \frac {4 d^{2} e^{5} x^{6}}{9} - \frac {d e^{6} x^{7}}{8} - \frac {e^{7} x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\left (d x + \frac {e x^{2}}{2}\right ) \left (d^{2}\right )^{\frac {7}{2}} & \text {otherwise} \end {cases} \] Input:

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

Output:

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))/12 
8 + 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)

Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {35 \, d^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}}} + \frac {35}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x + \frac {35}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x + \frac {7}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x + \frac {1}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}}{9 \, e} \] Input:

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

Output:

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)

Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {35 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, {\left | e \right |}} - \frac {1}{1152} \, {\left (\frac {128 \, d^{8}}{e} - {\left (837 \, d^{7} + 2 \, {\left (256 \, d^{6} e - {\left (489 \, d^{5} e^{2} + 4 \, {\left (96 \, d^{4} e^{3} - {\left (75 \, d^{3} e^{4} + 2 \, {\left (32 \, d^{2} e^{5} - {\left (8 \, e^{7} x + 9 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \] Input:

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

Output:

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 = 6.94 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.45 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {d\,x\,{\left (d^2-e^2\,x^2\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{2},\frac {1}{2};\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{7/2}}-\frac {{\left (d^2-e^2\,x^2\right )}^{9/2}}{9\,e} \] Input:

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

Output:

(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)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.51 \[ \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {315 \mathit {asin} \left (\frac {e x}{d}\right ) d^{9}-128 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8}+837 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} e x +512 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6} e^{2} x^{2}-978 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e^{3} x^{3}-768 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{4} x^{4}+600 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{5} x^{5}+512 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{6} x^{6}-144 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{7} x^{7}-128 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8} x^{8}+128 d^{9}}{1152 e} \] Input:

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

Output:

(315*asin((e*x)/d)*d**9 - 128*sqrt(d**2 - e**2*x**2)*d**8 + 837*sqrt(d**2 
- e**2*x**2)*d**7*e*x + 512*sqrt(d**2 - e**2*x**2)*d**6*e**2*x**2 - 978*sq 
rt(d**2 - e**2*x**2)*d**5*e**3*x**3 - 768*sqrt(d**2 - e**2*x**2)*d**4*e**4 
*x**4 + 600*sqrt(d**2 - e**2*x**2)*d**3*e**5*x**5 + 512*sqrt(d**2 - e**2*x 
**2)*d**2*e**6*x**6 - 144*sqrt(d**2 - e**2*x**2)*d*e**7*x**7 - 128*sqrt(d* 
*2 - e**2*x**2)*e**8*x**8 + 128*d**9)/(1152*e)