\(\int \frac {x (d+e x)^3}{(d^2-e^2 x^2)^{7/2}} \, dx\) [87]

Optimal result

Integrand size = 25, antiderivative size = 86 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e \sqrt {d^2-e^2 x^2}} \]

[Out]

1/5*(e*x+d)^3/e^2/(-e^2*x^2+d^2)^(5/2)-2/5*(e*x+d)/e^2/(-e^2*x^2+d^2)^(3/2)-1/5*x/d^2/e/(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {803, 667, 197} \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e \sqrt {d^2-e^2 x^2}} \]

[In]

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

[Out]

(d + e*x)^3/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (2*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(3/2)) - x/(5*d^2*e*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 667

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

Rule 803

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

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.58 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (d^2-3 d e x+e^2 x^2\right )}{5 d^2 e^2 (d-e x)^3} \]

[In]

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

[Out]

-1/5*(Sqrt[d^2 - e^2*x^2]*(d^2 - 3*d*e*x + e^2*x^2))/(d^2*e^2*(d - e*x)^3)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.55

method	result	size
trager	\(-\frac {\left (e^{2} x^{2}-3 d e x +d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{2} \left (-e x +d \right )^{3} e^{2}}\)	\(47\)
gosper	\(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (e^{2} x^{2}-3 d e x +d^{2}\right )}{5 d^{2} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(52\)
default	\(e^{3} \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+\frac {d^{3}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+3 d \,e^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+3 d^{2} e \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\)	\(308\)

[In]

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

[Out]

-1/5*(e^2*x^2-3*d*e*x+d^2)/d^2/(-e*x+d)^3/e^2*(-e^2*x^2+d^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3} - {\left (e^{2} x^{2} - 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{2} e^{5} x^{3} - 3 \, d^{3} e^{4} x^{2} + 3 \, d^{4} e^{3} x - d^{5} e^{2}\right )}} \]

[In]

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

[Out]

-1/5*(e^3*x^3 - 3*d*e^2*x^2 + 3*d^2*e*x - d^3 - (e^2*x^2 - 3*d*e*x + d^2)*sqrt(-e^2*x^2 + d^2))/(d^2*e^5*x^3 -
3*d^3*e^4*x^2 + 3*d^4*e^3*x - d^5*e^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.49 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {d^{3}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e} \]

[In]

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

[Out]

1/2*e*x^3/(-e^2*x^2 + d^2)^(5/2) + d*x^2/(-e^2*x^2 + d^2)^(5/2) + 3/10*d^2*x/((-e^2*x^2 + d^2)^(5/2)*e) - 1/5*
d^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 1/10*x/((-e^2*x^2 + d^2)^(3/2)*e) - 1/5*x/(sqrt(-e^2*x^2 + d^2)*d^2*e)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.59 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} - 1\right )}}{5 \, d^{2} e {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]

[In]

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

[Out]

2/5*(5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 5*(d*
e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^6*x^3) - 1)/(d^2*e*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*a
bs(e))

Mupad [B] (verification not implemented)

Time = 11.73 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.53 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2-3\,d\,e\,x+e^2\,x^2\right )}{5\,d^2\,e^2\,{\left (d-e\,x\right )}^3} \]

[In]

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

[Out]

-((d^2 - e^2*x^2)^(1/2)*(d^2 + e^2*x^2 - 3*d*e*x))/(5*d^2*e^2*(d - e*x)^3)