\(\int \frac {(d+e x)^7}{(d^2-e^2 x^2)^{7/2}} \, dx\) [848]

Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {7 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

[Out]

2/5*(e*x+d)^6/e/(-e^2*x^2+d^2)^(5/2)-14/15*(e*x+d)^4/e/(-e^2*x^2+d^2)^(3/2)-7*d*arctan(e*x/(-e^2*x^2+d^2)^(1/2
))/e+14/3*(e*x+d)^2/e/(-e^2*x^2+d^2)^(1/2)+7*(-e^2*x^2+d^2)^(1/2)/e

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {683, 655, 223, 209} \[ \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e} \]

[In]

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

[Out]

(2*(d + e*x)^6)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (14*(d + e*x)^4)/(15*e*(d^2 - e^2*x^2)^(3/2)) + (14*(d + e*x)^2)
/(3*e*Sqrt[d^2 - e^2*x^2]) + (7*Sqrt[d^2 - e^2*x^2])/e - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{5} \int \frac {(d+e x)^5}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx \\ & = \frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7}{3} \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = \frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}-7 \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-(7 d) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-(7 d) \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)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-167 d^3+381 d^2 e x-277 d e^2 x^2+15 e^3 x^3\right )}{15 e (-d+e x)^3}+\frac {7 d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-167*d^3 + 381*d^2*e*x - 277*d*e^2*x^2 + 15*e^3*x^3))/(15*e*(-d + e*x)^3) + (7*d*Log[-(S
qrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/Sqrt[-e^2]

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.39

method	result	size
risch	\(\frac {\sqrt {-x^{2} e^{2}+d^{2}}}{e}-\frac {7 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {232 d \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{15 e^{2} \left (x -\frac {d}{e}\right )}-\frac {16 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{4} \left (x -\frac {d}{e}\right )^{3}}-\frac {128 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{15 e^{3} \left (x -\frac {d}{e}\right )^{2}}\)	\(192\)
default	\(d^{7} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )+e^{7} \left (-\frac {x^{6}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )+7 d \,e^{6} \left (\frac {x^{5}}{5 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-x^{2} e^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )+\frac {7 d^{6}}{5 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+21 d^{2} e^{5} \left (\frac {x^{4}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+35 d^{3} e^{4} \left (\frac {x^{3}}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+35 d^{4} e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+21 d^{5} e^{2} \left (\frac {x}{4 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\)	\(694\)

[In]

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

[Out]

(-e^2*x^2+d^2)^(1/2)/e-7*d/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-232/15*d/e^2/(x-d/e)*(-(x-d/
e)^2*e^2-2*(x-d/e)*d*e)^(1/2)-16/5*d^3/e^4/(x-d/e)^3*(-(x-d/e)^2*e^2-2*(x-d/e)*d*e)^(1/2)-128/15*d^2/e^3/(x-d/
e)^2*(-(x-d/e)^2*e^2-2*(x-d/e)*d*e)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.27 \[ \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {167 \, d e^{3} x^{3} - 501 \, d^{2} e^{2} x^{2} + 501 \, d^{3} e x - 167 \, d^{4} + 210 \, {\left (d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{3} x^{3} - 277 \, d e^{2} x^{2} + 381 \, d^{2} e x - 167 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} e\right )}} \]

[In]

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

[Out]

1/15*(167*d*e^3*x^3 - 501*d^2*e^2*x^2 + 501*d^3*e*x - 167*d^4 + 210*(d*e^3*x^3 - 3*d^2*e^2*x^2 + 3*d^3*e*x - d
^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*e^3*x^3 - 277*d*e^2*x^2 + 381*d^2*e*x - 167*d^3)*sqrt(-e^2
*x^2 + d^2))/(e^4*x^3 - 3*d*e^3*x^2 + 3*d^2*e^2*x - d^3*e)

Sympy [F]

\[ \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{7}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (122) = 244\).

Time = 0.31 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.37 \[ \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {7}{15} \, d e^{6} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {7}{3} \, d e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {27 \, d^{2} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {35 \, d^{3} e^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {73 \, d^{4} e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {61 \, d^{5} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {167 \, d^{6}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {127 \, d^{3} x}{30 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {22 \, d x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}}} - \frac {7 \, d \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} \]

[In]

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

[Out]

7/15*d*e^6*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2
+ d^2)^(5/2)*e^6)) - e^5*x^6/(-e^2*x^2 + d^2)^(5/2) - 7/3*d*e^4*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2
/((-e^2*x^2 + d^2)^(3/2)*e^4)) + 27*d^2*e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 35/2*d^3*e^2*x^3/(-e^2*x^2 + d^2)^(5/
2) - 73/3*d^4*e*x^2/(-e^2*x^2 + d^2)^(5/2) - 61/10*d^5*x/(-e^2*x^2 + d^2)^(5/2) + 167/15*d^6/((-e^2*x^2 + d^2)
^(5/2)*e) + 127/30*d^3*x/(-e^2*x^2 + d^2)^(3/2) + 22/15*d*x/sqrt(-e^2*x^2 + d^2) - 7*d*arcsin(e^2*x/(d*sqrt(e^
2)))/sqrt(e^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 \, d \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e} + \frac {16 \, {\left (19 \, d - \frac {80 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d}{e^{2} x} + \frac {130 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d}{e^{4} x^{2}} - \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d}{e^{8} x^{4}}\right )}}{15 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]

[In]

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

[Out]

-7*d*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + sqrt(-e^2*x^2 + d^2)/e + 16/15*(19*d - 80*(d*e + sqrt(-e^2*x^2 + d^2
)*abs(e))*d/(e^2*x) + 130*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d/(e^4*x^2) - 60*(d*e + sqrt(-e^2*x^2 + d^2)*a
bs(e))^3*d/(e^6*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d/(e^8*x^4))/(((d*e + sqrt(-e^2*x^2 + d^2)*abs
(e))/(e^2*x) - 1)^5*abs(e))

Mupad [F(-1)]

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

[In]

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

[Out]

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