3.9.38 \(\int \frac {(d+e x)^4}{(d^2-e^2 x^2)^{5/2}} \, dx\) [838]

Optimal. Leaf size=81 \[ \frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {683, 667, 223, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^3)/(3*e*(d^2 - e^2*x^2)^(3/2)) - (2*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) + 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 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 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*} \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\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)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 79, normalized size = 0.98 \begin {gather*} -\frac {4 (d-2 e x) \sqrt {d^2-e^2 x^2}}{3 e (d-e x)^2}-\frac {\log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(73)=146\).
time = 0.46, size = 281, normalized size = 3.47

method	result	size
default	\(e^{4} \left (\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )+4 d \,e^{3} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )+6 d^{2} e^{2} \left (\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}\right )+\frac {4 d^{3}}{3 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+d^{4} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 134, normalized size = 1.65 \begin {gather*} \frac {1}{3} \, {\left (\frac {3 \, x^{2} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, d^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}}\right )} x e^{4} + \frac {4 \, d x^{2} e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {4 \, d^{3} e^{\left (-1\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {7 \, d^{2} x}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {5 \, x}{3 \, \sqrt {-x^{2} e^{2} + d^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*x^2*e^(-2)/(-x^2*e^2 + d^2)^(3/2) - 2*d^2*e^(-4)/(-x^2*e^2 + d^2)^(3/2))*x*e^4 + 4*d*x^2*e/(-x^2*e^2 +
d^2)^(3/2) - 4/3*d^3*e^(-1)/(-x^2*e^2 + d^2)^(3/2) + arcsin(x*e/d)*e^(-1) + 7/3*d^2*x/(-x^2*e^2 + d^2)^(3/2) -
5/3*x/sqrt(-x^2*e^2 + d^2)

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Fricas [A]
time = 2.85, size = 109, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} e^{2} - 4 \, d x e + 2 \, d^{2} + 3 \, {\left (x^{2} e^{2} - 2 \, d x e + d^{2}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (2 \, x e - d\right )}\right )}}{3 \, {\left (x^{2} e^{3} - 2 \, d x e^{2} + d^{2} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.14, size = 79, normalized size = 0.98 \begin {gather*} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {8 \, {\left (\frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )} e^{\left (-1\right )}}{3 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(x*e/d)*e^(-1)*sgn(d) + 8/3*(3*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 1)*e^(-1)/((d*e + sqrt(-x^2*e^2
+ d^2)*e)*e^(-2)/x - 1)^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^4}{{\left (d^2-e^2\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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