3.9.3 \(\int \frac {(d^2-e^2 x^2)^{7/2}}{d+e x} \, dx\) [803]

Optimal. Leaf size=124 \[ \frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e} \]

[Out]

5/24*d^3*x*(-e^2*x^2+d^2)^(3/2)+1/6*d*x*(-e^2*x^2+d^2)^(5/2)+1/7*(-e^2*x^2+d^2)^(7/2)/e+5/16*d^7*arctan(e*x/(-
e^2*x^2+d^2)^(1/2))/e+5/16*d^5*x*(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {679, 201, 223, 209} \begin {gather*} \frac {5 d^7 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d^5*x*Sqrt[d^2 - e^2*x^2])/16 + (5*d^3*x*(d^2 - e^2*x^2)^(3/2))/24 + (d*x*(d^2 - e^2*x^2)^(5/2))/6 + (d^2 -
e^2*x^2)^(7/2)/(7*e) + (5*d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 679

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

Rubi steps

\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx &=\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+d \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{6} \left (5 d^3\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{8} \left (5 d^5\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{16} \left (5 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{16} \left (5 d^7\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 133, normalized size = 1.07 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (48 d^6+231 d^5 e x-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+56 d e^5 x^5-48 e^6 x^6\right )}{336 e}-\frac {5 d^7 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{16 \sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(48*d^6 + 231*d^5*e*x - 144*d^4*e^2*x^2 - 182*d^3*e^3*x^3 + 144*d^2*e^4*x^4 + 56*d*e^5*x^
5 - 48*e^6*x^6))/(336*e) - (5*d^7*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(16*Sqrt[-e^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs. \(2(104)=208\).
time = 0.49, size = 248, normalized size = 2.00

method	result	size
risch	\(\frac {\left (-48 e^{6} x^{6}+56 d \,e^{5} x^{5}+144 d^{2} e^{4} x^{4}-182 d^{3} e^{3} x^{3}-144 d^{4} e^{2} x^{2}+231 d^{5} e x +48 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{336 e}+\frac {5 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}}\)	\(116\)
default	\(\frac {\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e}\)	\(248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(1/7*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(7/2)+d*e*(-1/12*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d
/e))^(5/2)+5/6*d^2*(-1/8*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2
*(x+d/e)+2*d*e)/e^2*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*(x+d/e
)^2+2*d*e*(x+d/e))^(1/2))))))

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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 124, normalized size = 1.00 \begin {gather*} -\frac {5}{16} i \, d^{7} \arcsin \left (\frac {x e}{d} + 2\right ) e^{\left (-1\right )} + \frac {5}{8} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{6} e^{\left (-1\right )} + \frac {5}{16} \, \sqrt {x^{2} e^{2} + 4 \, d x e + 3 \, d^{2}} d^{5} x + \frac {5}{24} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x + \frac {1}{6} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x + \frac {1}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*I*d^7*arcsin(x*e/d + 2)*e^(-1) + 5/8*sqrt(x^2*e^2 + 4*d*x*e + 3*d^2)*d^6*e^(-1) + 5/16*sqrt(x^2*e^2 + 4*
d*x*e + 3*d^2)*d^5*x + 5/24*(-x^2*e^2 + d^2)^(3/2)*d^3*x + 1/6*(-x^2*e^2 + d^2)^(5/2)*d*x + 1/7*(-x^2*e^2 + d^
2)^(7/2)*e^(-1)

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Fricas [A]
time = 2.27, size = 108, normalized size = 0.87 \begin {gather*} -\frac {1}{336} \, {\left (210 \, d^{7} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (48 \, x^{6} e^{6} - 56 \, d x^{5} e^{5} - 144 \, d^{2} x^{4} e^{4} + 182 \, d^{3} x^{3} e^{3} + 144 \, d^{4} x^{2} e^{2} - 231 \, d^{5} x e - 48 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/336*(210*d^7*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + (48*x^6*e^6 - 56*d*x^5*e^5 - 144*d^2*x^4*e^4 +
182*d^3*x^3*e^3 + 144*d^4*x^2*e^2 - 231*d^5*x*e - 48*d^6)*sqrt(-x^2*e^2 + d^2))*e^(-1)

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Sympy [C] Result contains complex when optimal does not.
time = 9.96, size = 813, normalized size = 6.56 \begin {gather*} d^{5} \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) - d^{4} e \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) - 2 d^{3} e^{2} \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + 2 d^{2} e^{3} \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) + d e^{4} \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - e^{5} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**5*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**3/(2*d*sqrt(-1 +
e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) -
d**4*e*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) - 2*d**3*e**2
*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 3*I*d*x**3/(8*sqrt(-
1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**4*asin(e*x/d
)/(8*e**3) - d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**2*x**5/(4*d
*sqrt(1 - e**2*x**2/d**2)), True)) + 2*d**2*e**3*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x*
*2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) + d
*e**4*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**3/(
48*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e*
*2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**2*x**2/d**
2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*s
qrt(1 - e**2*x**2/d**2)), True)) - e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sqr
t(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 - e**2*x**2)/7, N
e(e, 0)), (x**6*sqrt(d**2)/6, True))

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Giac [A]
time = 1.11, size = 95, normalized size = 0.77 \begin {gather*} \frac {5}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{336} \, {\left (48 \, d^{6} e^{\left (-1\right )} + {\left (231 \, d^{5} - 2 \, {\left (72 \, d^{4} e + {\left (91 \, d^{3} e^{2} - 4 \, {\left (18 \, d^{2} e^{3} - {\left (6 \, x e^{5} - 7 \, d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*d^7*arcsin(x*e/d)*e^(-1)*sgn(d) + 1/336*(48*d^6*e^(-1) + (231*d^5 - 2*(72*d^4*e + (91*d^3*e^2 - 4*(18*d^2
*e^3 - (6*x*e^5 - 7*d*e^4)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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