\(\int \frac {1}{(d+e x)^4 (d^2-e^2 x^2)^{5/2}} \, dx\) [845]

Optimal result

Integrand size = 24, antiderivative size = 181 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}} \]

[Out]

8/99*x/d^6/(-e^2*x^2+d^2)^(3/2)-1/11/d/e/(e*x+d)^4/(-e^2*x^2+d^2)^(3/2)-7/99/d^2/e/(e*x+d)^3/(-e^2*x^2+d^2)^(3
/2)-2/33/d^3/e/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2)-2/33/d^4/e/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+16/99*x/d^8/(-e^2*x^2+d^
2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(8*x)/(99*d^6*(d^2 - e^2*x^2)^(3/2)) - 1/(11*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(3/2)) - 7/(99*d^2*e*(d + e*x)^3*
(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^4*e*(d + e*x)*(d^2 - e^2*x^2
)^(3/2)) + (16*x)/(99*d^8*Sqrt[d^2 - e^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{11 d} \\ & = -\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^2} \\ & = -\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^3} \\ & = -\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{33 d^4} \\ & = \frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{99 d^6} \\ & = \frac {8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{99 d^8 \sqrt {d^2-e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-28 d^7-13 d^6 e x+72 d^5 e^2 x^2+122 d^4 e^3 x^3+32 d^3 e^4 x^4-72 d^2 e^5 x^5-64 d e^6 x^6-16 e^7 x^7\right )}{99 d^8 e (d-e x)^2 (d+e x)^6} \]

[In]

Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-28*d^7 - 13*d^6*e*x + 72*d^5*e^2*x^2 + 122*d^4*e^3*x^3 + 32*d^3*e^4*x^4 - 72*d^2*e^5*x^
5 - 64*d*e^6*x^6 - 16*e^7*x^7))/(99*d^8*e*(d - e*x)^2*(d + e*x)^6)

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.61

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (16 e^{7} x^{7}+64 e^{6} x^{6} d +72 d^{2} e^{5} x^{5}-32 d^{3} e^{4} x^{4}-122 d^{4} e^{3} x^{3}-72 e^{2} x^{2} d^{5}+13 x \,d^{6} e +28 d^{7}\right )}{99 \left (e x +d \right )^{3} d^{8} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(110\)
trager	\(-\frac {\left (16 e^{7} x^{7}+64 e^{6} x^{6} d +72 d^{2} e^{5} x^{5}-32 d^{3} e^{4} x^{4}-122 d^{4} e^{3} x^{3}-72 e^{2} x^{2} d^{5}+13 x \,d^{6} e +28 d^{7}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{99 d^{8} \left (e x +d \right )^{6} \left (-e x +d \right )^{2} e}\)	\(112\)
default	\(\frac {-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{4} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {7 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {2 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {5 e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{7 d}\right )}{3 d}\right )}{11 d}}{e^{4}}\)	\(320\)

[In]

int(1/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/99*(-e*x+d)*(16*e^7*x^7+64*d*e^6*x^6+72*d^2*e^5*x^5-32*d^3*e^4*x^4-122*d^4*e^3*x^3-72*d^5*e^2*x^2+13*d^6*e*
x+28*d^7)/(e*x+d)^3/d^8/e/(-e^2*x^2+d^2)^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {28 \, e^{8} x^{8} + 112 \, d e^{7} x^{7} + 112 \, d^{2} e^{6} x^{6} - 112 \, d^{3} e^{5} x^{5} - 280 \, d^{4} e^{4} x^{4} - 112 \, d^{5} e^{3} x^{3} + 112 \, d^{6} e^{2} x^{2} + 112 \, d^{7} e x + 28 \, d^{8} + {\left (16 \, e^{7} x^{7} + 64 \, d e^{6} x^{6} + 72 \, d^{2} e^{5} x^{5} - 32 \, d^{3} e^{4} x^{4} - 122 \, d^{4} e^{3} x^{3} - 72 \, d^{5} e^{2} x^{2} + 13 \, d^{6} e x + 28 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d^{8} e^{9} x^{8} + 4 \, d^{9} e^{8} x^{7} + 4 \, d^{10} e^{7} x^{6} - 4 \, d^{11} e^{6} x^{5} - 10 \, d^{12} e^{5} x^{4} - 4 \, d^{13} e^{4} x^{3} + 4 \, d^{14} e^{3} x^{2} + 4 \, d^{15} e^{2} x + d^{16} e\right )}} \]

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/99*(28*e^8*x^8 + 112*d*e^7*x^7 + 112*d^2*e^6*x^6 - 112*d^3*e^5*x^5 - 280*d^4*e^4*x^4 - 112*d^5*e^3*x^3 + 11
2*d^6*e^2*x^2 + 112*d^7*e*x + 28*d^8 + (16*e^7*x^7 + 64*d*e^6*x^6 + 72*d^2*e^5*x^5 - 32*d^3*e^4*x^4 - 122*d^4*
e^3*x^3 - 72*d^5*e^2*x^2 + 13*d^6*e*x + 28*d^7)*sqrt(-e^2*x^2 + d^2))/(d^8*e^9*x^8 + 4*d^9*e^8*x^7 + 4*d^10*e^
7*x^6 - 4*d^11*e^6*x^5 - 10*d^12*e^5*x^4 - 4*d^13*e^4*x^3 + 4*d^14*e^3*x^2 + 4*d^15*e^2*x + d^16*e)

Sympy [F]

\[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{4}}\, dx \]

[In]

integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**4), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (157) = 314\).

Time = 0.20 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.06 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{11 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{5} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{4} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{3} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {7}{99 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {2}{33 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} - \frac {2}{33 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e\right )}} + \frac {8 \, x}{99 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {16 \, x}{99 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8}} \]

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

-1/11/((-e^2*x^2 + d^2)^(3/2)*d*e^5*x^4 + 4*(-e^2*x^2 + d^2)^(3/2)*d^2*e^4*x^3 + 6*(-e^2*x^2 + d^2)^(3/2)*d^3*
e^3*x^2 + 4*(-e^2*x^2 + d^2)^(3/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(3/2)*d^5*e) - 7/99/((-e^2*x^2 + d^2)^(3/2)*d^
2*e^4*x^3 + 3*(-e^2*x^2 + d^2)^(3/2)*d^3*e^3*x^2 + 3*(-e^2*x^2 + d^2)^(3/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(3/2)
*d^5*e) - 2/33/((-e^2*x^2 + d^2)^(3/2)*d^3*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(3/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(3/
2)*d^5*e) - 2/33/((-e^2*x^2 + d^2)^(3/2)*d^4*e^2*x + (-e^2*x^2 + d^2)^(3/2)*d^5*e) + 8/99*x/((-e^2*x^2 + d^2)^
(3/2)*d^6) + 16/99*x/(sqrt(-e^2*x^2 + d^2)*d^8)

Giac [F]

\[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(1/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^4), x)

Mupad [B] (verification not implemented)

Time = 10.21 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {215\,x}{1584\,d^6}-\frac {91}{792\,d^5\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{44\,d^3\,e\,{\left (d+e\,x\right )}^6}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{99\,d^4\,e\,{\left (d+e\,x\right )}^5}-\frac {79\,\sqrt {d^2-e^2\,x^2}}{1584\,d^5\,e\,{\left (d+e\,x\right )}^4}-\frac {29\,\sqrt {d^2-e^2\,x^2}}{528\,d^6\,e\,{\left (d+e\,x\right )}^3}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{99\,d^8\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]

[In]

int(1/((d^2 - e^2*x^2)^(5/2)*(d + e*x)^4),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*((215*x)/(1584*d^6) - 91/(792*d^5*e)))/((d + e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1/2
)/(44*d^3*e*(d + e*x)^6) - (4*(d^2 - e^2*x^2)^(1/2))/(99*d^4*e*(d + e*x)^5) - (79*(d^2 - e^2*x^2)^(1/2))/(1584
*d^5*e*(d + e*x)^4) - (29*(d^2 - e^2*x^2)^(1/2))/(528*d^6*e*(d + e*x)^3) + (16*x*(d^2 - e^2*x^2)^(1/2))/(99*d^
8*(d + e*x)*(d - e*x))