3.9.47 \(\int \frac {(d+e x)^8}{(d^2-e^2 x^2)^{7/2}} \, dx\) [847]

Optimal. Leaf size=173 \[ \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}+\frac {63 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {21 (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}-\frac {63 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

2/5*(e*x+d)^7/e/(-e^2*x^2+d^2)^(5/2)-6/5*(e*x+d)^5/e/(-e^2*x^2+d^2)^(3/2)-63/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(
1/2))/e+42/5*(e*x+d)^3/e/(-e^2*x^2+d^2)^(1/2)+63/2*d*(-e^2*x^2+d^2)^(1/2)/e+21/2*(e*x+d)*(-e^2*x^2+d^2)^(1/2)/
e

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Rubi [A]
time = 0.05, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {63 d^2 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}+\frac {21 \sqrt {d^2-e^2 x^2} (d+e x)}{2 e}+\frac {63 d \sqrt {d^2-e^2 x^2}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^8/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^7)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (6*(d + e*x)^5)/(5*e*(d^2 - e^2*x^2)^(3/2)) + (42*(d + e*x)^3)/(
5*e*Sqrt[d^2 - e^2*x^2]) + (63*d*Sqrt[d^2 - e^2*x^2])/(2*e) + (21*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) - (63*d
^2*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)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {21}{5} \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}-21 \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}+\frac {21 (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{2} (63 d) \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}+\frac {63 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {21 (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{2} \left (63 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}+\frac {63 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {21 (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{2} \left (63 d^2\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)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}+\frac {63 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {21 (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}-\frac {63 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 120, normalized size = 0.69 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-496 d^4+1163 d^3 e x-801 d^2 e^2 x^2+65 d e^3 x^3+5 e^4 x^4\right )}{10 e (-d+e x)^3}+\frac {63 d^2 \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)^8/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-496*d^4 + 1163*d^3*e*x - 801*d^2*e^2*x^2 + 65*d*e^3*x^3 + 5*e^4*x^4))/(10*e*(-d + e*x)^
3) + (63*d^2*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(2*Sqrt[-e^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(840\) vs. \(2(149)=298\).
time = 0.53, size = 841, normalized size = 4.86

method	result	size
risch	\(\frac {\left (e x +16 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e}-\frac {63 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {112 d^{3} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{3} \left (x -\frac {d}{e}\right )^{2}}-\frac {288 d^{2} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{2} \left (x -\frac {d}{e}\right )}-\frac {32 d^{4} \sqrt {-e^{2} \left (x -\frac {d}{e}\right )^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{4} \left (x -\frac {d}{e}\right )^{3}}\)	\(204\)
default	\(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 )+8 d \,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 )+28 d^{2} 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 )+56 d^{3} 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 )+70 d^{4} 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 )+56 d^{5} 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 )+28 d^{6} 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 {8 d^{7}}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{8} \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 )\)	\(841\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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)))
)))+8*d*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))))+28*d^2*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)))))+56*d^3*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)))+70*d^4*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)))))+56*d^5*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))+28*d^6*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))))+8/5*d^7/e/(-e^2*x^2+d^2)^(5/2)+d^8*(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)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (142) = 284\).
time = 0.52, size = 324, normalized size = 1.87 \begin {gather*} -\frac {x^{7} e^{6}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {8 \, d x^{6} e^{5}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {104 \, d^{3} x^{4} e^{3}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {35 \, d^{4} x^{3} e^{2}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {120 \, d^{5} x^{2} e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {248 \, d^{7} e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {21}{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^{2} x e^{6} - \frac {21}{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^{2} x e^{4} - \frac {76 \, d^{6} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {63}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {69 \, d^{4} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {39 \, d^{2} x}{10 \, \sqrt {-x^{2} e^{2} + d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/2*x^7*e^6/(-x^2*e^2 + d^2)^(5/2) - 8*d*x^6*e^5/(-x^2*e^2 + d^2)^(5/2) + 104*d^3*x^4*e^3/(-x^2*e^2 + d^2)^(5
/2) + 35*d^4*x^3*e^2/(-x^2*e^2 + d^2)^(5/2) - 120*d^5*x^2*e/(-x^2*e^2 + d^2)^(5/2) + 248/5*d^7*e^(-1)/(-x^2*e^
2 + d^2)^(5/2) + 21/10*(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^2*x*e^6 - 21/2*(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^2*x*e^4 - 76/5*d^6*x/(-x^2*e^2 + d^2)^(5/2) - 63/2*d^2*arcsin(x*e/d)*e^(-1) + 69/5*d^4*x/(-x
^2*e^2 + d^2)^(3/2) - 39/10*d^2*x/sqrt(-x^2*e^2 + d^2)

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Fricas [A]
time = 2.18, size = 181, normalized size = 1.05 \begin {gather*} \frac {496 \, d^{2} x^{3} e^{3} - 1488 \, d^{3} x^{2} e^{2} + 1488 \, d^{4} x e - 496 \, d^{5} + 630 \, {\left (d^{2} x^{3} e^{3} - 3 \, d^{3} x^{2} e^{2} + 3 \, d^{4} x e - d^{5}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (5 \, x^{4} e^{4} + 65 \, d x^{3} e^{3} - 801 \, d^{2} x^{2} e^{2} + 1163 \, d^{3} x e - 496 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{10 \, {\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)^8/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/10*(496*d^2*x^3*e^3 - 1488*d^3*x^2*e^2 + 1488*d^4*x*e - 496*d^5 + 630*(d^2*x^3*e^3 - 3*d^3*x^2*e^2 + 3*d^4*x
*e - d^5)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (5*x^4*e^4 + 65*d*x^3*e^3 - 801*d^2*x^2*e^2 + 1163*d^
3*x*e - 496*d^4)*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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{8}}{\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)**8/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**8/(-(-d + e*x)*(d + e*x))**(7/2), x)

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Giac [A]
time = 1.15, size = 211, normalized size = 1.22 \begin {gather*} -\frac {63}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (16 \, d e^{\left (-1\right )} + x\right )} - \frac {32 \, {\left (\frac {55 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{\left (-2\right )}}{x} - \frac {85 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{\left (-4\right )}}{x^{2}} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} e^{\left (-6\right )}}{x^{3}} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{\left (-8\right )}}{x^{4}} - 13 \, d^{2}\right )} e^{\left (-1\right )}}{5 \, {\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)^8/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-63/2*d^2*arcsin(x*e/d)*e^(-1)*sgn(d) + 1/2*sqrt(-x^2*e^2 + d^2)*(16*d*e^(-1) + x) - 32/5*(55*(d*e + sqrt(-x^2
*e^2 + d^2)*e)*d^2*e^(-2)/x - 85*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*e^(-4)/x^2 + 45*(d*e + sqrt(-x^2*e^2 + d
^2)*e)^3*d^2*e^(-6)/x^3 - 10*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^2*e^(-8)/x^4 - 13*d^2)*e^(-1)/((d*e + sqrt(-x^
2*e^2 + d^2)*e)*e^(-2)/x - 1)^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^8}{{\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)^8/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int((d + e*x)^8/(d^2 - e^2*x^2)^(7/2), x)

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