\(\int \frac {(d+e x)^4}{(d^2-e^2 x^2)^{7/2}} \, dx\) [851]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}} \]

[In]

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

[Out]

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

Rule 665

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*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

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 {(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{3 d} \\ & = \frac {(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (4 d^2+3 d e x-e^2 x^2\right )}{15 d^2 e (d-e x)^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.85 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66

method	result	size
gosper	\(\frac {\left (e x +d \right )^{5} \left (-e x +d \right ) \left (-e x +4 d \right )}{15 d^{2} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(44\)
trager	\(\frac {\left (-x^{2} e^{2}+3 d e x +4 d^{2}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{15 d^{2} \left (-e x +d \right )^{3} e}\)	\(50\)
default	\(d^{4} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )+e^{4} \left (\frac {x^{3}}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+4 d \,e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+6 d^{2} e^{2} \left (\frac {x}{4 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {4 d^{3}}{5 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(381\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 \, e^{3} x^{3} - 12 \, d e^{2} x^{2} + 12 \, d^{2} e x - 4 \, d^{3} + {\left (e^{2} x^{2} - 3 \, d e x - 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{4} x^{3} - 3 \, d^{3} e^{3} x^{2} + 3 \, d^{4} e^{2} x - d^{5} e\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((d + e*x)**4/(-(-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. 123 vs. \(2 (59) = 118\).

Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.84 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {11 \, d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d^{3}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {x}{30 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}} \]

[In]

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

[Out]

1/2*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) + 4/3*d*e*x^2/(-e^2*x^2 + d^2)^(5/2) + 11/10*d^2*x/(-e^2*x^2 + d^2)^(5/2) +
4/15*d^3/((-e^2*x^2 + d^2)^(5/2)*e) - 1/30*x/(-e^2*x^2 + d^2)^(3/2) - 1/15*x/(sqrt(-e^2*x^2 + d^2)*d^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (59) = 118\).

Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.46 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (\frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - \frac {25 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} - 4\right )}}{15 \, d^{2} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]

[In]

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

[Out]

-2/15*(5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 25*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 15
*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^6*x^3) - 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^8*x^4) - 4)/(d^
2*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs(e))

Mupad [B] (verification not implemented)

Time = 10.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (4\,d^2+3\,d\,e\,x-e^2\,x^2\right )}{15\,d^2\,e\,{\left (d-e\,x\right )}^3} \]

[In]

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

[Out]

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