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

Optimal. Leaf size=179 \[ \frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}+\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2} \]

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Rubi [A]  time = 0.07, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {671, 641, 195, 217, 203} \begin {gather*} \frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(77*d^8*x*Sqrt[d^2 - e^2*x^2])/256 + (77*d^6*x*(d^2 - e^2*x^2)^(3/2))/384 + (77*d^4*x*(d^2 - e^2*x^2)^(5/2))/4
80 + (11*d^2*x*(d^2 - e^2*x^2)^(7/2))/80 - (11*d*(d^2 - e^2*x^2)^(9/2))/(90*e) - ((d + e*x)*(d^2 - e^2*x^2)^(9
/2))/(10*e) + (77*d^10*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*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]

Rule 671

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

Rubi steps

\begin {align*} \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{10} (11 d) \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{10} \left (11 d^2\right ) \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{80} \left (77 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{96} \left (77 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{128} \left (77 d^8\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{256} \left (77 d^{10}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{256} \left (77 d^{10}\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 {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 118, normalized size = 0.66 \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^{9/2} \left (\frac {33 d^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 )}{\left (d^2-e^2 x^2\right )^4}-\frac {2560 d}{e}-1152 x\right )}{11520} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d^2 - e^2*x^2)^(9/2)*((-2560*d)/e - 1152*x + (33*d^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])))/(d^2 - e^2*x^2)^4))/11520

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IntegrateAlgebraic [A]  time = 0.50, size = 169, normalized size = 0.94 \begin {gather*} \frac {77 d^{10} \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{256 e^2}+\frac {\sqrt {d^2-e^2 x^2} \left (-2560 d^9+8055 d^8 e x+10240 d^7 e^2 x^2-6150 d^6 e^3 x^3-15360 d^5 e^4 x^4-312 d^4 e^5 x^5+10240 d^3 e^6 x^6+3024 d^2 e^7 x^7-2560 d e^8 x^8-1152 e^9 x^9\right )}{11520 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-2560*d^9 + 8055*d^8*e*x + 10240*d^7*e^2*x^2 - 6150*d^6*e^3*x^3 - 15360*d^5*e^4*x^4 - 31
2*d^4*e^5*x^5 + 10240*d^3*e^6*x^6 + 3024*d^2*e^7*x^7 - 2560*d*e^8*x^8 - 1152*e^9*x^9))/(11520*e) + (77*d^10*Sq
rt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(256*e^2)

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fricas [A]  time = 0.43, size = 149, normalized size = 0.83 \begin {gather*} -\frac {6930 \, d^{10} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (1152 \, e^{9} x^{9} + 2560 \, d e^{8} x^{8} - 3024 \, d^{2} e^{7} x^{7} - 10240 \, d^{3} e^{6} x^{6} + 312 \, d^{4} e^{5} x^{5} + 15360 \, d^{5} e^{4} x^{4} + 6150 \, d^{6} e^{3} x^{3} - 10240 \, d^{7} e^{2} x^{2} - 8055 \, d^{8} e x + 2560 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{11520 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11520*(6930*d^10*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (1152*e^9*x^9 + 2560*d*e^8*x^8 - 3024*d^2*e^7*
x^7 - 10240*d^3*e^6*x^6 + 312*d^4*e^5*x^5 + 15360*d^5*e^4*x^4 + 6150*d^6*e^3*x^3 - 10240*d^7*e^2*x^2 - 8055*d^
8*e*x + 2560*d^9)*sqrt(-e^2*x^2 + d^2))/e

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giac [A]  time = 0.25, size = 128, normalized size = 0.72 \begin {gather*} \frac {77}{256} \, d^{10} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\relax (d) - \frac {1}{11520} \, {\left (2560 \, d^{9} e^{\left (-1\right )} - {\left (8055 \, d^{8} + 2 \, {\left (5120 \, d^{7} e - {\left (3075 \, d^{6} e^{2} + 4 \, {\left (1920 \, d^{5} e^{3} + {\left (39 \, d^{4} e^{4} - 2 \, {\left (640 \, d^{3} e^{5} + {\left (189 \, d^{2} e^{6} - 8 \, {\left (9 \, x e^{8} + 20 \, d e^{7}\right )} x\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)^2*(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

77/256*d^10*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/11520*(2560*d^9*e^(-1) - (8055*d^8 + 2*(5120*d^7*e - (3075*d^6*e^2
+ 4*(1920*d^5*e^3 + (39*d^4*e^4 - 2*(640*d^3*e^5 + (189*d^2*e^6 - 8*(9*x*e^8 + 20*d*e^7)*x)*x)*x)*x)*x)*x)*x)
*x)*sqrt(-x^2*e^2 + d^2)

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maple [A]  time = 0.06, size = 151, normalized size = 0.84 \begin {gather*} \frac {77 d^{10} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 \sqrt {e^{2}}}+\frac {77 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} x}{256}+\frac {77 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{6} x}{384}+\frac {77 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{4} x}{480}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2} x}{80}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}} x}{10}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}} d}{9 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*x*(-e^2*x^2+d^2)^(9/2)+11/80*d^2*x*(-e^2*x^2+d^2)^(7/2)+77/480*d^4*x*(-e^2*x^2+d^2)^(5/2)+77/384*d^6*x*(
-e^2*x^2+d^2)^(3/2)+77/256*d^8*x*(-e^2*x^2+d^2)^(1/2)+77/256*d^10/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2
)^(1/2)*x)-2/9*d*(-e^2*x^2+d^2)^(9/2)/e

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maxima [A]  time = 2.96, size = 133, normalized size = 0.74 \begin {gather*} \frac {77 \, d^{10} \arcsin \left (\frac {e x}{d}\right )}{256 \, e} + \frac {77}{256} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8} x + \frac {77}{384} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} x + \frac {77}{480} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x + \frac {11}{80} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x - \frac {1}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} x - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d}{9 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

77/256*d^10*arcsin(e*x/d)/e + 77/256*sqrt(-e^2*x^2 + d^2)*d^8*x + 77/384*(-e^2*x^2 + d^2)^(3/2)*d^6*x + 77/480
*(-e^2*x^2 + d^2)^(5/2)*d^4*x + 11/80*(-e^2*x^2 + d^2)^(7/2)*d^2*x - 1/10*(-e^2*x^2 + d^2)^(9/2)*x - 2/9*(-e^2
*x^2 + d^2)^(9/2)*d/e

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

[Out]

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

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sympy [C]  time = 35.74, size = 1413, normalized size = 7.89

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**8*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)) +
2*d**7*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**6*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)) - 6*d**5*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)) +
6*d**3*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)) + 2*d**2*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)) - 2*d*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, Tr
ue)) - e**8*Piecewise((-7*I*d**10*acosh(e*x/d)/(256*e**9) + 7*I*d**9*x/(256*e**8*sqrt(-1 + e**2*x**2/d**2)) -
7*I*d**7*x**3/(768*e**6*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**5*x**5/(1920*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d
**3*x**7/(480*e**2*sqrt(-1 + e**2*x**2/d**2)) - 9*I*d*x**9/(80*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**11/(10*d
*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (7*d**10*asin(e*x/d)/(256*e**9) - 7*d**9*x/(256*e**8*sq
rt(1 - e**2*x**2/d**2)) + 7*d**7*x**3/(768*e**6*sqrt(1 - e**2*x**2/d**2)) + 7*d**5*x**5/(1920*e**4*sqrt(1 - e*
*2*x**2/d**2)) + d**3*x**7/(480*e**2*sqrt(1 - e**2*x**2/d**2)) + 9*d*x**9/(80*sqrt(1 - e**2*x**2/d**2)) - e**2
*x**11/(10*d*sqrt(1 - e**2*x**2/d**2)), True))

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