3.7.43 \(\int (d+e x) (d^2-e^2 x^2)^{7/2} \, dx\)

Optimal. Leaf size=148 \[ \frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}+\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]

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Rubi [A]  time = 0.05, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {641, 195, 217, 203} \begin {gather*} \frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35*d^7*x*Sqrt[d^2 - e^2*x^2])/128 + (35*d^5*x*(d^2 - e^2*x^2)^(3/2))/192 + (7*d^3*x*(d^2 - e^2*x^2)^(5/2))/48
+ (d*x*(d^2 - e^2*x^2)^(7/2))/8 - (d^2 - e^2*x^2)^(9/2)/(9*e) + (35*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1
28*e)

Rule 195

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

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 641

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

Rubi steps

\begin {align*} \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+d \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{8} \left (7 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{48} \left (35 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{64} \left (35 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{128} \left (35 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{128} \left (35 d^9\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 114, normalized size = 0.77 \begin {gather*} \frac {1}{384} d \sqrt {d^2-e^2 x^2} \left (279 d^6 x-326 d^4 e^2 x^3+200 d^2 e^4 x^5+\frac {105 d^7 \sin ^{-1}\left (\frac {e x}{d}\right )}{e \sqrt {1-\frac {e^2 x^2}{d^2}}}-48 e^6 x^7\right )-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(d^2 - e^2*x^2)^(9/2)/e + (d*Sqrt[d^2 - e^2*x^2]*(279*d^6*x - 326*d^4*e^2*x^3 + 200*d^2*e^4*x^5 - 48*e^6*
x^7 + (105*d^7*ArcSin[(e*x)/d])/(e*Sqrt[1 - (e^2*x^2)/d^2])))/384

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IntegrateAlgebraic [A]  time = 0.49, size = 158, normalized size = 1.07 \begin {gather*} \frac {35 d^9 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{128 e^2}+\frac {\sqrt {d^2-e^2 x^2} \left (-128 d^8+837 d^7 e x+512 d^6 e^2 x^2-978 d^5 e^3 x^3-768 d^4 e^4 x^4+600 d^3 e^5 x^5+512 d^2 e^6 x^6-144 d e^7 x^7-128 e^8 x^8\right )}{1152 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-128*d^8 + 837*d^7*e*x + 512*d^6*e^2*x^2 - 978*d^5*e^3*x^3 - 768*d^4*e^4*x^4 + 600*d^3*e
^5*x^5 + 512*d^2*e^6*x^6 - 144*d*e^7*x^7 - 128*e^8*x^8))/(1152*e) + (35*d^9*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + S
qrt[d^2 - e^2*x^2]])/(128*e^2)

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fricas [A]  time = 0.43, size = 138, normalized size = 0.93 \begin {gather*} -\frac {630 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (128 \, e^{8} x^{8} + 144 \, d e^{7} x^{7} - 512 \, d^{2} e^{6} x^{6} - 600 \, d^{3} e^{5} x^{5} + 768 \, d^{4} e^{4} x^{4} + 978 \, d^{5} e^{3} x^{3} - 512 \, d^{6} e^{2} x^{2} - 837 \, d^{7} e x + 128 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1152 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152*(630*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (128*e^8*x^8 + 144*d*e^7*x^7 - 512*d^2*e^6*x^6 -
600*d^3*e^5*x^5 + 768*d^4*e^4*x^4 + 978*d^5*e^3*x^3 - 512*d^6*e^2*x^2 - 837*d^7*e*x + 128*d^8)*sqrt(-e^2*x^2 +
d^2))/e

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giac [A]  time = 0.28, size = 119, normalized size = 0.80 \begin {gather*} \frac {35}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\relax (d) - \frac {1}{1152} \, {\left (128 \, d^{8} e^{\left (-1\right )} - {\left (837 \, d^{7} + 2 \, {\left (256 \, d^{6} e - {\left (489 \, d^{5} e^{2} + 4 \, {\left (96 \, d^{4} e^{3} - {\left (75 \, d^{3} e^{4} + 2 \, {\left (32 \, d^{2} e^{5} - {\left (8 \, x e^{7} + 9 \, d e^{6}\right )} x\right )} x\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*x+d)*(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

35/128*d^9*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/1152*(128*d^8*e^(-1) - (837*d^7 + 2*(256*d^6*e - (489*d^5*e^2 + 4*(
96*d^4*e^3 - (75*d^3*e^4 + 2*(32*d^2*e^5 - (8*x*e^7 + 9*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)

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maple [A]  time = 0.05, size = 131, normalized size = 0.89 \begin {gather*} \frac {35 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}}+\frac {35 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} x}{128}+\frac {35 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5} x}{192}+\frac {7 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} x}{48}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d x}{8}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{9 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(-e^2*x^2+d^2)^(9/2)/e+1/8*d*x*(-e^2*x^2+d^2)^(7/2)+7/48*d^3*x*(-e^2*x^2+d^2)^(5/2)+35/192*d^5*x*(-e^2*x^
2+d^2)^(3/2)+35/128*d^7*x*(-e^2*x^2+d^2)^(1/2)+35/128*d^9/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*
x)

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maxima [A]  time = 2.98, size = 113, normalized size = 0.76 \begin {gather*} \frac {35 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{128 \, e} + \frac {35}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x + \frac {35}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x + \frac {7}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x + \frac {1}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}}{9 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/128*d^9*arcsin(e*x/d)/e + 35/128*sqrt(-e^2*x^2 + d^2)*d^7*x + 35/192*(-e^2*x^2 + d^2)^(3/2)*d^5*x + 7/48*(-
e^2*x^2 + d^2)^(5/2)*d^3*x + 1/8*(-e^2*x^2 + d^2)^(7/2)*d*x - 1/9*(-e^2*x^2 + d^2)^(9/2)/e

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mupad [B]  time = 0.85, size = 67, normalized size = 0.45 \begin {gather*} \frac {d\,x\,{\left (d^2-e^2\,x^2\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{2},\frac {1}{2};\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{7/2}}-\frac {{\left (d^2-e^2\,x^2\right )}^{9/2}}{9\,e} \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]

(d*x*(d^2 - e^2*x^2)^(7/2)*hypergeom([-7/2, 1/2], 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^(7/2) - (d^2 - e^2*
x^2)^(9/2)/(9*e)

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sympy [C]  time = 27.73, size = 1284, normalized size = 8.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**7*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**6*e*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) - 3*d**5*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)) - 3*d**4*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)) + 3
*d**3*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*sqrt(1 - e**2*x**2/d**2)), True)) + 3*d**2*e**5*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d
**4*x**2*sqrt(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, Ne(e, 0)), (x**6*sqrt(d**2)/6, True)) - d*e**6*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d
**7*x/(128*e**6*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**5/
(192*e**2*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**9/(8*d*sqrt(-1 +
e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*
x**2/d**2)) + 5*d**5*x**3/(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**2*x**2/d**2))
+ 7*d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2)), True)) - e**7*Piecewise((
-16*d**8*sqrt(d**2 - e**2*x**2)/(315*e**8) - 8*d**6*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6) - 2*d**4*x**4*sqrt(
d**2 - e**2*x**2)/(105*e**4) - d**2*x**6*sqrt(d**2 - e**2*x**2)/(63*e**2) + x**8*sqrt(d**2 - e**2*x**2)/9, Ne(
e, 0)), (x**8*sqrt(d**2)/8, True))

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