\(\int (d+e x)^3 (d^2-e^2 x^2)^{7/2} \, dx\) [60]

Optimal result

Integrand size = 24, antiderivative size = 189 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {91}{256} d^9 x \sqrt {d^2-e^2 x^2}+\frac {91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}-\frac {13 d (20 d+9 e x) \left (d^2-e^2 x^2\right )^{9/2}}{990 e}+\frac {91 d^{11} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e} \] Output:

91/256*d^9*x*(-e^2*x^2+d^2)^(1/2)+91/384*d^7*x*(-e^2*x^2+d^2)^(3/2)+91/480 
*d^5*x*(-e^2*x^2+d^2)^(5/2)+13/80*d^3*x*(-e^2*x^2+d^2)^(7/2)-1/11*(e*x+d)^ 
2*(-e^2*x^2+d^2)^(9/2)/e-13/990*d*(9*e*x+20*d)*(-e^2*x^2+d^2)^(9/2)/e+91/2 
56*d^11*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.94 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-44800 d^{10}+81675 d^9 e x+167680 d^8 e^2 x^2+12210 d^7 e^3 x^3-222720 d^6 e^4 x^4-142296 d^5 e^5 x^5+110080 d^4 e^6 x^6+131472 d^3 e^7 x^7+1280 d^2 e^8 x^8-38016 d e^9 x^9-11520 e^{10} x^{10}\right )}{126720 e}-\frac {91 d^{11} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{256 \sqrt {-e^2}} \] Input:

Integrate[(d + e*x)^3*(d^2 - e^2*x^2)^(7/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^2]*(-44800*d^10 + 81675*d^9*e*x + 167680*d^8*e^2*x^2 + 1 
2210*d^7*e^3*x^3 - 222720*d^6*e^4*x^4 - 142296*d^5*e^5*x^5 + 110080*d^4*e^ 
6*x^6 + 131472*d^3*e^7*x^7 + 1280*d^2*e^8*x^8 - 38016*d*e^9*x^9 - 11520*e^ 
10*x^10))/(126720*e) - (91*d^11*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]] 
)/(256*Sqrt[-e^2])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {469, 469, 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.

\(\displaystyle \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {13}{11} d \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}dx-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2}dx-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \left (d \int \left (d^2-e^2 x^2\right )^{7/2}dx-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \left (d \left (\frac {7}{8} d^2 \int \left (d^2-e^2 x^2\right )^{5/2}dx+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \left (d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \int \left (d^2-e^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \left (d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \left (d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \left (d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {13}{11} d \left (\frac {11}{10} d \left (d \left (\frac {7}{8} d^2 \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (d^2-e^2 x^2\right )^{7/2}\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}\right )-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}\right )-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}\)

Input:

Int[(d + e*x)^3*(d^2 - e^2*x^2)^(7/2),x]
 

Output:

-1/11*((d + e*x)^2*(d^2 - e^2*x^2)^(9/2))/e + (13*d*(-1/10*((d + e*x)*(d^2 
 - e^2*x^2)^(9/2))/e + (11*d*(-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)))/10))/11
 

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]
 

rule 469

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]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\left (11520 e^{10} x^{10}+38016 d \,e^{9} x^{9}-1280 d^{2} e^{8} x^{8}-131472 d^{3} e^{7} x^{7}-110080 d^{4} e^{6} x^{6}+142296 d^{5} e^{5} x^{5}+222720 d^{6} e^{4} x^{4}-12210 d^{7} e^{3} x^{3}-167680 d^{8} e^{2} x^{2}-81675 d^{9} e x +44800 d^{10}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{126720 e}+\frac {91 d^{11} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 \sqrt {e^{2}}}\)	\(160\)
default	\(d^{3} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )+e^{3} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{11 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{99 e^{4}}\right )+3 d \,e^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{10 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )}{10 e^{2}}\right )-\frac {d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{3 e}\)	\(350\)

Input:

int((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/126720*(11520*e^10*x^10+38016*d*e^9*x^9-1280*d^2*e^8*x^8-131472*d^3*e^7 
*x^7-110080*d^4*e^6*x^6+142296*d^5*e^5*x^5+222720*d^6*e^4*x^4-12210*d^7*e^ 
3*x^3-167680*d^8*e^2*x^2-81675*d^9*e*x+44800*d^10)/e*(-e^2*x^2+d^2)^(1/2)+ 
91/256*d^11/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.85 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=-\frac {90090 \, d^{11} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (11520 \, e^{10} x^{10} + 38016 \, d e^{9} x^{9} - 1280 \, d^{2} e^{8} x^{8} - 131472 \, d^{3} e^{7} x^{7} - 110080 \, d^{4} e^{6} x^{6} + 142296 \, d^{5} e^{5} x^{5} + 222720 \, d^{6} e^{4} x^{4} - 12210 \, d^{7} e^{3} x^{3} - 167680 \, d^{8} e^{2} x^{2} - 81675 \, d^{9} e x + 44800 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{126720 \, e} \] Input:

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
 

Output:

-1/126720*(90090*d^11*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (11520*e 
^10*x^10 + 38016*d*e^9*x^9 - 1280*d^2*e^8*x^8 - 131472*d^3*e^7*x^7 - 11008 
0*d^4*e^6*x^6 + 142296*d^5*e^5*x^5 + 222720*d^6*e^4*x^4 - 12210*d^7*e^3*x^ 
3 - 167680*d^8*e^2*x^2 - 81675*d^9*e*x + 44800*d^10)*sqrt(-e^2*x^2 + d^2)) 
/e
 

Sympy [A] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.22 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\begin {cases} \frac {91 d^{11} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{256} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {35 d^{10}}{99 e} + \frac {165 d^{9} x}{256} + \frac {131 d^{8} e x^{2}}{99} + \frac {37 d^{7} e^{2} x^{3}}{384} - \frac {58 d^{6} e^{3} x^{4}}{33} - \frac {539 d^{5} e^{4} x^{5}}{480} + \frac {86 d^{4} e^{5} x^{6}}{99} + \frac {83 d^{3} e^{6} x^{7}}{80} + \frac {d^{2} e^{7} x^{8}}{99} - \frac {3 d e^{8} x^{9}}{10} - \frac {e^{9} x^{10}}{11}\right ) & \text {for}\: e^{2} \neq 0 \\\left (d^{2}\right )^{\frac {7}{2}} \left (\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(7/2),x)
 

Output:

Piecewise((91*d**11*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))/2 
56 + sqrt(d**2 - e**2*x**2)*(-35*d**10/(99*e) + 165*d**9*x/256 + 131*d**8* 
e*x**2/99 + 37*d**7*e**2*x**3/384 - 58*d**6*e**3*x**4/33 - 539*d**5*e**4*x 
**5/480 + 86*d**4*e**5*x**6/99 + 83*d**3*e**6*x**7/80 + d**2*e**7*x**8/99 
- 3*d*e**8*x**9/10 - e**9*x**10/11), Ne(e**2, 0)), ((d**2)**(7/2)*Piecewis 
e((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True)), True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.87 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {91 \, d^{11} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{256 \, \sqrt {e^{2}}} + \frac {91}{256} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{9} x + \frac {91}{384} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} x + \frac {91}{480} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x + \frac {13}{80} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x - \frac {1}{11} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e x^{2} - \frac {3}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d x - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{2}}{99 \, e} \] Input:

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
 

Output:

91/256*d^11*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 91/256*sqrt(-e^2*x^2 + 
 d^2)*d^9*x + 91/384*(-e^2*x^2 + d^2)^(3/2)*d^7*x + 91/480*(-e^2*x^2 + d^2 
)^(5/2)*d^5*x + 13/80*(-e^2*x^2 + d^2)^(7/2)*d^3*x - 1/11*(-e^2*x^2 + d^2) 
^(9/2)*e*x^2 - 3/10*(-e^2*x^2 + d^2)^(9/2)*d*x - 35/99*(-e^2*x^2 + d^2)^(9 
/2)*d^2/e
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {91 \, d^{11} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{256 \, {\left | e \right |}} - \frac {1}{126720} \, {\left (\frac {44800 \, d^{10}}{e} - {\left (81675 \, d^{9} + 2 \, {\left (83840 \, d^{8} e + {\left (6105 \, d^{7} e^{2} - 4 \, {\left (27840 \, d^{6} e^{3} + {\left (17787 \, d^{5} e^{4} - 2 \, {\left (6880 \, d^{4} e^{5} + {\left (8217 \, d^{3} e^{6} + 8 \, {\left (10 \, d^{2} e^{7} - 9 \, {\left (10 \, e^{9} x + 33 \, d e^{8}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \] Input:

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
 

Output:

91/256*d^11*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/126720*(44800*d^10/e - 
(81675*d^9 + 2*(83840*d^8*e + (6105*d^7*e^2 - 4*(27840*d^6*e^3 + (17787*d^ 
5*e^4 - 2*(6880*d^4*e^5 + (8217*d^3*e^6 + 8*(10*d^2*e^7 - 9*(10*e^9*x + 33 
*d*e^8)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^3 \,d x \] Input:

int((d^2 - e^2*x^2)^(7/2)*(d + e*x)^3,x)
 

Output:

int((d^2 - e^2*x^2)^(7/2)*(d + e*x)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.44 \[ \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2} \, dx=\frac {45045 \mathit {asin} \left (\frac {e x}{d}\right ) d^{11}-44800 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{10}+81675 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{9} e x +167680 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} e^{2} x^{2}+12210 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} e^{3} x^{3}-222720 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6} e^{4} x^{4}-142296 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e^{5} x^{5}+110080 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{6} x^{6}+131472 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{7} x^{7}+1280 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{8} x^{8}-38016 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{9} x^{9}-11520 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{10} x^{10}+44800 d^{11}}{126720 e} \] Input:

int((e*x+d)^3*(-e^2*x^2+d^2)^(7/2),x)
 

Output:

(45045*asin((e*x)/d)*d**11 - 44800*sqrt(d**2 - e**2*x**2)*d**10 + 81675*sq 
rt(d**2 - e**2*x**2)*d**9*e*x + 167680*sqrt(d**2 - e**2*x**2)*d**8*e**2*x* 
*2 + 12210*sqrt(d**2 - e**2*x**2)*d**7*e**3*x**3 - 222720*sqrt(d**2 - e**2 
*x**2)*d**6*e**4*x**4 - 142296*sqrt(d**2 - e**2*x**2)*d**5*e**5*x**5 + 110 
080*sqrt(d**2 - e**2*x**2)*d**4*e**6*x**6 + 131472*sqrt(d**2 - e**2*x**2)* 
d**3*e**7*x**7 + 1280*sqrt(d**2 - e**2*x**2)*d**2*e**8*x**8 - 38016*sqrt(d 
**2 - e**2*x**2)*d*e**9*x**9 - 11520*sqrt(d**2 - e**2*x**2)*e**10*x**10 + 
44800*d**11)/(126720*e)