\(\int \frac {x^4 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\) [22]

Optimal result

Integrand size = 25, antiderivative size = 84 \[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {819, 272, 45} \[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^4 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}}-\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

Int[(x^4*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(
a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c*(p + 1))), x] - Dist[m*((c*d*f + a*e*g)/(2*a*c*(p + 1))), Int[(d + e*
x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplif
y[m + 2*p + 3], 0] && LtQ[p, -1]

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98 \[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (8 d^4-8 d^3 e x-12 d^2 e^2 x^2+12 d e^3 x^3+3 e^4 x^4\right )}{15 d e^5 (d-e x)^3 (d+e x)^2} \]

[In]

Integrate[(x^4*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(8*d^4 - 8*d^3*e*x - 12*d^2*e^2*x^2 + 12*d*e^3*x^3 + 3*e^4*x^4))/(15*d*e^5*(d - e*x)^3*(d
+ e*x)^2)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (3 e^{4} x^{4}+12 d \,e^{3} x^{3}-12 d^{2} e^{2} x^{2}-8 d^{3} e x +8 d^{4}\right )}{15 d \,e^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(77\)
trager	\(\frac {\left (3 e^{4} x^{4}+12 d \,e^{3} x^{3}-12 d^{2} e^{2} x^{2}-8 d^{3} e x +8 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 e^{5} d \left (-e x +d \right )^{3} \left (e x +d \right )^{2}}\)	\(79\)
default	\(e \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+d \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )\)	\(208\)

[In]

int(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/15*(-e*x+d)*(e*x+d)^2*(3*e^4*x^4+12*d*e^3*x^3-12*d^2*e^2*x^2-8*d^3*e*x+8*d^4)/d/e^5/(-e^2*x^2+d^2)^(7/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (72) = 144\).

Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.04 \[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {8 \, e^{5} x^{5} - 8 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 8 \, d^{5} - {\left (3 \, e^{4} x^{4} + 12 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} - 8 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d e^{10} x^{5} - d^{2} e^{9} x^{4} - 2 \, d^{3} e^{8} x^{3} + 2 \, d^{4} e^{7} x^{2} + d^{5} e^{6} x - d^{6} e^{5}\right )}} \]

[In]

integrate(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(8*e^5*x^5 - 8*d*e^4*x^4 - 16*d^2*e^3*x^3 + 16*d^3*e^2*x^2 + 8*d^4*e*x - 8*d^5 - (3*e^4*x^4 + 12*d*e^3*x^
3 - 12*d^2*e^2*x^2 - 8*d^3*e*x + 8*d^4)*sqrt(-e^2*x^2 + d^2))/(d*e^10*x^5 - d^2*e^9*x^4 - 2*d^3*e^8*x^3 + 2*d^
4*e^7*x^2 + d^5*e^6*x - d^6*e^5)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.36 (sec) , antiderivative size = 418, normalized size of antiderivative = 4.98 \[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=d \left (\begin {cases} - \frac {i x^{5}}{5 d^{7} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {x^{5}}{5 d^{7} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {8 d^{4}}{15 d^{4} e^{6} \sqrt {d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {20 d^{2} e^{2} x^{2}}{15 d^{4} e^{6} \sqrt {d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} + \frac {15 e^{4} x^{4}}{15 d^{4} e^{6} \sqrt {d^{2} - e^{2} x^{2}} - 30 d^{2} e^{8} x^{2} \sqrt {d^{2} - e^{2} x^{2}} + 15 e^{10} x^{4} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{6}}{6 \left (d^{2}\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate(x**4*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((-I*x**5/(5*d**7*sqrt(-1 + e**2*x**2/d**2) - 10*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 5*d**3*
e**4*x**4*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (x**5/(5*d**7*sqrt(1 - e**2*x**2/d**2) - 10*d*
*5*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 5*d**3*e**4*x**4*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise((8*d*
*4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2 -
e**2*x**2)) - 20*d**2*e**2*x**2/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**
2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)) + 15*e**4*x**4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*
x**2*sqrt(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**6/(6*(d**2)**(7/2)), True)
)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (72) = 144\).

Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.89 \[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {4 \, d^{2} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {3 \, d^{3} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}} + \frac {d x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} + \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{4}} \]

[In]

integrate(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

x^4/((-e^2*x^2 + d^2)^(5/2)*e) + 1/2*d*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 4/3*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*
e^3) - 3/10*d^3*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8/15*d^4/((-e^2*x^2 + d^2)^(5/2)*e^5) + 1/10*d*x/((-e^2*x^2 +
d^2)^(3/2)*e^4) + 1/5*x/(sqrt(-e^2*x^2 + d^2)*d*e^4)

Giac [F]

\[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(x^4*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate((e*x + d)*x^4/(-e^2*x^2 + d^2)^(7/2), x)

Mupad [B] (verification not implemented)

Time = 11.85 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (8\,d^4-8\,d^3\,e\,x-12\,d^2\,e^2\,x^2+12\,d\,e^3\,x^3+3\,e^4\,x^4\right )}{15\,d\,e^5\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^3} \]

[In]

int((x^4*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*(8*d^4 + 3*e^4*x^4 + 12*d*e^3*x^3 - 12*d^2*e^2*x^2 - 8*d^3*e*x))/(15*d*e^5*(d + e*x)^2*
(d - e*x)^3)