3.9.46 \(\int \frac {(d+e x)^9}{(d^2-e^2 x^2)^{7/2}} \, dx\) [846]

Optimal. Leaf size=206 \[ \frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {231 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

2/5*(e*x+d)^8/e/(-e^2*x^2+d^2)^(5/2)-22/15*(e*x+d)^6/e/(-e^2*x^2+d^2)^(3/2)-231/2*d^3*arctan(e*x/(-e^2*x^2+d^2
)^(1/2))/e+66/5*(e*x+d)^4/e/(-e^2*x^2+d^2)^(1/2)+231/2*d^2*(-e^2*x^2+d^2)^(1/2)/e+77/2*d*(e*x+d)*(-e^2*x^2+d^2
)^(1/2)/e+77/5*(e*x+d)^2*(-e^2*x^2+d^2)^(1/2)/e

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {683, 685, 655, 223, 209} \begin {gather*} -\frac {231 d^3 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 \sqrt {d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac {77 d \sqrt {d^2-e^2 x^2} (d+e x)}{2 e}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^8)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (22*(d + e*x)^6)/(15*e*(d^2 - e^2*x^2)^(3/2)) + (66*(d + e*x)^4)
/(5*e*Sqrt[d^2 - e^2*x^2]) + (231*d^2*Sqrt[d^2 - e^2*x^2])/(2*e) + (77*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e)
+ (77*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(5*e) - (231*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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 683

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*(p + 1))), x] - Dist[e^2*((m + p)/(c*(p + 1))), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

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*} \int \frac {(d+e x)^9}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {11}{5} \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {33}{5} \int \frac {(d+e x)^5}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}-\frac {231}{5} \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-(77 d) \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^2\right ) \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {1}{2} \left (231 d^3\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 {2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {66 (d+e x)^4}{5 e \sqrt {d^2-e^2 x^2}}+\frac {231 d^2 \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 d (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}+\frac {77 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {231 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.50, size = 131, normalized size = 0.64 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-5446 d^5+12843 d^4 e x-8711 d^3 e^2 x^2+815 d^2 e^3 x^3+105 d e^4 x^4+10 e^5 x^5\right )}{30 e (-d+e x)^3}+\frac {231 d^3 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{2 \sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5446*d^5 + 12843*d^4*e*x - 8711*d^3*e^2*x^2 + 815*d^2*e^3*x^3 + 105*d*e^4*x^4 + 10*e^5*
x^5))/(30*e*(-d + e*x)^3) + (231*d^3*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(2*Sqrt[-e^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(985\) vs. \(2(178)=356\).
time = 0.52, size = 986, normalized size = 4.79

method	result	size
risch	\(\frac {\left (2 e^{2} x^{2}+27 d x e +238 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6 e}-\frac {231 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {832 d^{4} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{15 e^{3} \left (x -\frac {d}{e}\right )^{2}}-\frac {2768 d^{3} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{15 e^{2} \left (x -\frac {d}{e}\right )}-\frac {64 d^{5} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{4} \left (x -\frac {d}{e}\right )^{3}}\)	\(216\)
default	\(e^{9} \left (-\frac {x^{8}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {8 d^{2} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )}{3 e^{2}}\right )+9 d \,e^{8} \left (-\frac {x^{7}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d^{2} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )}{2 e^{2}}\right )+36 d^{2} e^{7} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )+84 d^{3} e^{6} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )+126 d^{4} e^{5} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+126 d^{5} e^{4} \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+84 d^{6} e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+36 d^{7} e^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {9 d^{8}}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{9} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\)	\(986\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

e^9*(-1/3*x^8/e^2/(-e^2*x^2+d^2)^(5/2)+8/3*d^2/e^2*(-x^6/e^2/(-e^2*x^2+d^2)^(5/2)+6*d^2/e^2*(x^4/e^2/(-e^2*x^2
+d^2)^(5/2)-4*d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/2)))))+9*d*e^8*(-1/2*x^
7/e^2/(-e^2*x^2+d^2)^(5/2)+7/2*d^2/e^2*(1/5*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1/e^2*(1/3*x^3/e^2/(-e^2*x^2+d^2)^(3/
2)-1/e^2*(x/e^2/(-e^2*x^2+d^2)^(1/2)-1/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))))+36*d^2*e
^7*(-x^6/e^2/(-e^2*x^2+d^2)^(5/2)+6*d^2/e^2*(x^4/e^2/(-e^2*x^2+d^2)^(5/2)-4*d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2
)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/2))))+84*d^3*e^6*(1/5*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1/e^2*(1/3*x^3/e^2/(
-e^2*x^2+d^2)^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^2)^(1/2)-1/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1
/2)))))+126*d^4*e^5*(x^4/e^2/(-e^2*x^2+d^2)^(5/2)-4*d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e
^2*x^2+d^2)^(5/2)))+126*d^5*e^4*(1/2*x^3/e^2/(-e^2*x^2+d^2)^(5/2)-3/2*d^2/e^2*(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-
1/4*d^2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(
1/2)))))+84*d^6*e^3*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/2))+36*d^7*e^2*(1/4*x/e^2
/(-e^2*x^2+d^2)^(5/2)-1/4*d^2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*
x/d^4/(-e^2*x^2+d^2)^(1/2))))+9/5*d^8/e/(-e^2*x^2+d^2)^(5/2)+d^9*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*
x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2)))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (170) = 340\).
time = 0.52, size = 347, normalized size = 1.68 \begin {gather*} -\frac {x^{8} e^{7}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {9 \, d x^{7} e^{6}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {116 \, d^{2} x^{6} e^{5}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {358 \, d^{4} x^{4} e^{3}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {63 \, d^{5} x^{3} e^{2}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {1348 \, d^{6} x^{2} e}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2723 \, d^{8} e^{\left (-1\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {77}{10} \, {\left (\frac {15 \, x^{4} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {20 \, d^{2} x^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, d^{4} e^{\left (-6\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}\right )} d^{3} x e^{6} - \frac {77}{2} \, {\left (\frac {3 \, x^{2} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, d^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}}\right )} d^{3} x e^{4} - \frac {152 \, d^{7} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {231}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {619 \, d^{5} x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {989 \, d^{3} x}{30 \, \sqrt {-x^{2} e^{2} + d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/3*x^8*e^7/(-x^2*e^2 + d^2)^(5/2) - 9/2*d*x^7*e^6/(-x^2*e^2 + d^2)^(5/2) - 116/3*d^2*x^6*e^5/(-x^2*e^2 + d^2
)^(5/2) + 358*d^4*x^4*e^3/(-x^2*e^2 + d^2)^(5/2) + 63*d^5*x^3*e^2/(-x^2*e^2 + d^2)^(5/2) - 1348/3*d^6*x^2*e/(-
x^2*e^2 + d^2)^(5/2) + 2723/15*d^8*e^(-1)/(-x^2*e^2 + d^2)^(5/2) + 77/10*(15*x^4*e^(-2)/(-x^2*e^2 + d^2)^(5/2)
- 20*d^2*x^2*e^(-4)/(-x^2*e^2 + d^2)^(5/2) + 8*d^4*e^(-6)/(-x^2*e^2 + d^2)^(5/2))*d^3*x*e^6 - 77/2*(3*x^2*e^(
-2)/(-x^2*e^2 + d^2)^(3/2) - 2*d^2*e^(-4)/(-x^2*e^2 + d^2)^(3/2))*d^3*x*e^4 - 152/5*d^7*x/(-x^2*e^2 + d^2)^(5/
2) - 231/2*d^3*arcsin(x*e/d)*e^(-1) + 619/15*d^5*x/(-x^2*e^2 + d^2)^(3/2) - 989/30*d^3*x/sqrt(-x^2*e^2 + d^2)

________________________________________________________________________________________

Fricas [A]
time = 2.56, size = 191, normalized size = 0.93 \begin {gather*} \frac {5446 \, d^{3} x^{3} e^{3} - 16338 \, d^{4} x^{2} e^{2} + 16338 \, d^{5} x e - 5446 \, d^{6} + 6930 \, {\left (d^{3} x^{3} e^{3} - 3 \, d^{4} x^{2} e^{2} + 3 \, d^{5} x e - d^{6}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (10 \, x^{5} e^{5} + 105 \, d x^{4} e^{4} + 815 \, d^{2} x^{3} e^{3} - 8711 \, d^{3} x^{2} e^{2} + 12843 \, d^{4} x e - 5446 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{3} e^{4} - 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} - d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(5446*d^3*x^3*e^3 - 16338*d^4*x^2*e^2 + 16338*d^5*x*e - 5446*d^6 + 6930*(d^3*x^3*e^3 - 3*d^4*x^2*e^2 + 3*
d^5*x*e - d^6)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (10*x^5*e^5 + 105*d*x^4*e^4 + 815*d^2*x^3*e^3 -
8711*d^3*x^2*e^2 + 12843*d^4*x*e - 5446*d^5)*sqrt(-x^2*e^2 + d^2))/(x^3*e^4 - 3*d*x^2*e^3 + 3*d^2*x*e^2 - d^3*
e)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{9}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**9/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**9/(-(-d + e*x)*(d + e*x))**(7/2), x)

________________________________________________________________________________________

Giac [A]
time = 1.20, size = 223, normalized size = 1.08 \begin {gather*} -\frac {231}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (238 \, d^{2} e^{\left (-1\right )} + {\left (2 \, x e + 27 \, d\right )} x\right )} - \frac {32 \, {\left (\frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-2\right )}}{x} - \frac {850 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e^{\left (-4\right )}}{x^{2}} + \frac {480 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-6\right )}}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-8\right )}}{x^{4}} - 133 \, d^{3}\right )} e^{\left (-1\right )}}{15 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-231/2*d^3*arcsin(x*e/d)*e^(-1)*sgn(d) + 1/6*sqrt(-x^2*e^2 + d^2)*(238*d^2*e^(-1) + (2*x*e + 27*d)*x) - 32/15*
(560*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^3*e^(-2)/x - 850*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^3*e^(-4)/x^2 + 480*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^3*e^(-6)/x^3 - 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^3*e^(-8)/x^4 - 133*d^3
)*e^(-1)/((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 1)^5

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^9}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int((d + e*x)^9/(d^2 - e^2*x^2)^(7/2), x)

________________________________________________________________________________________