### 3.847 $$\int \frac{(d+e x)^8}{(d^2-e^2 x^2)^{7/2}} \, dx$$

Optimal. Leaf size=173 $\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}$

[Out]

(2*(d + e*x)^7)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (6*(d + e*x)^5)/(5*e*(d^2 - e^2*x^2)^(3/2)) + (42*(d + e*x)^3)/(
5*e*Sqrt[d^2 - e^2*x^2]) + (63*d*Sqrt[d^2 - e^2*x^2])/(2*e) + (21*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) - (63*d
^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi [A]  time = 0.0768588, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.208, Rules used = {669, 671, 641, 217, 203} $\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^8/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^7)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (6*(d + e*x)^5)/(5*e*(d^2 - e^2*x^2)^(3/2)) + (42*(d + e*x)^3)/(
5*e*Sqrt[d^2 - e^2*x^2]) + (63*d*Sqrt[d^2 - e^2*x^2])/(2*e) + (21*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e) - (63*d
^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

Rule 669

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]

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
]

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 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 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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{5} \int \frac{(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{21}{5} \int \frac{(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}-21 \int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{2} (63 d) \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{2} \left (63 d^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{2} \left (63 d^2\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{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.282841, size = 131, normalized size = 0.76 $\frac{(d+e x) \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (801 d^2 e^2 x^2-1163 d^3 e x+496 d^4-65 d e^3 x^3-5 e^4 x^4\right )-315 d (d-e x)^3 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{10 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^8/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(496*d^4 - 1163*d^3*e*x + 801*d^2*e^2*x^2 - 65*d*e^3*x^3 - 5*e^4*x^4) - 31
5*d*(d - e*x)^3*ArcSin[(e*x)/d]))/(10*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.148, size = 284, normalized size = 1.6 \begin{align*} -{\frac{76\,{d}^{6}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{e}^{6}{x}^{7}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{248\,{d}^{7}}{5\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{21\,{d}^{2}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+35\,{\frac{{d}^{4}{e}^{2}{x}^{3}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{63\,{e}^{4}{d}^{2}{x}^{5}}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-8\,{\frac{{e}^{5}d{x}^{6}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+104\,{\frac{{e}^{3}{d}^{3}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-120\,{\frac{e{d}^{5}{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{63\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{27\,{d}^{4}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{423\,{d}^{2}x}{10}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-76/5*d^6*x/(-e^2*x^2+d^2)^(5/2)-1/2*e^6*x^7/(-e^2*x^2+d^2)^(5/2)+248/5*d^7/e/(-e^2*x^2+d^2)^(5/2)-21/2*e^2*d^
2*x^3/(-e^2*x^2+d^2)^(3/2)+35*d^4*e^2*x^3/(-e^2*x^2+d^2)^(5/2)+63/10*e^4*d^2*x^5/(-e^2*x^2+d^2)^(5/2)-8*e^5*d*
x^6/(-e^2*x^2+d^2)^(5/2)+104*e^3*d^3*x^4/(-e^2*x^2+d^2)^(5/2)-120*e*d^5*x^2/(-e^2*x^2+d^2)^(5/2)-63/2*d^2/(e^2
)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+27/5*d^4*x/(-e^2*x^2+d^2)^(3/2)+423/10*d^2*x/(-e^2*x^2+d^2)
^(1/2)

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Maxima [B]  time = 1.81844, size = 485, normalized size = 2.8 \begin{align*} -\frac{e^{6} x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{21}{10} \, d^{2} e^{6} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{8 \, d e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{21}{2} \, d^{2} e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{104 \, d^{3} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{35 \, d^{4} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{120 \, d^{5} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{76 \, d^{6} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{248 \, d^{7}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{69 \, d^{4} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{39 \, d^{2} x}{10 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{63 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/2*e^6*x^7/(-e^2*x^2 + d^2)^(5/2) + 21/10*d^2*e^6*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2
*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 8*d*e^5*x^6/(-e^2*x^2 + d^2)^(5/2) - 21/2*d^2*e
^4*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4)) + 104*d^3*e^3*x^4/(-e^2*x^2 + d
^2)^(5/2) + 35*d^4*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) - 120*d^5*e*x^2/(-e^2*x^2 + d^2)^(5/2) - 76/5*d^6*x/(-e^2*x^
2 + d^2)^(5/2) + 248/5*d^7/((-e^2*x^2 + d^2)^(5/2)*e) + 69/5*d^4*x/(-e^2*x^2 + d^2)^(3/2) - 39/10*d^2*x/sqrt(-
e^2*x^2 + d^2) - 63/2*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2)

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Fricas [A]  time = 2.3162, size = 409, normalized size = 2.36 \begin{align*} \frac{496 \, d^{2} e^{3} x^{3} - 1488 \, d^{3} e^{2} x^{2} + 1488 \, d^{4} e x - 496 \, d^{5} + 630 \,{\left (d^{2} e^{3} x^{3} - 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - d^{5}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (5 \, e^{4} x^{4} + 65 \, d e^{3} x^{3} - 801 \, d^{2} e^{2} x^{2} + 1163 \, d^{3} e x - 496 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/10*(496*d^2*e^3*x^3 - 1488*d^3*e^2*x^2 + 1488*d^4*e*x - 496*d^5 + 630*(d^2*e^3*x^3 - 3*d^3*e^2*x^2 + 3*d^4*e
*x - d^5)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (5*e^4*x^4 + 65*d*e^3*x^3 - 801*d^2*e^2*x^2 + 1163*d^3*e
*x - 496*d^4)*sqrt(-e^2*x^2 + d^2))/(e^4*x^3 - 3*d*e^3*x^2 + 3*d^2*e^2*x - d^3*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{8}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**8/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**8/(-(-d + e*x)*(d + e*x))**(7/2), x)

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Giac [A]  time = 1.34107, size = 159, normalized size = 0.92 \begin{align*} -\frac{63}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (496 \, d^{7} e^{\left (-1\right )} +{\left (325 \, d^{6} -{\left (1200 \, d^{5} e +{\left (655 \, d^{4} e^{2} -{\left (1040 \, d^{3} e^{3} +{\left (591 \, d^{2} e^{4} - 5 \,{\left (x e^{6} + 16 \, d e^{5}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{10 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-63/2*d^2*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/10*(496*d^7*e^(-1) + (325*d^6 - (1200*d^5*e + (655*d^4*e^2 - (1040*d
^3*e^3 + (591*d^2*e^4 - 5*(x*e^6 + 16*d*e^5)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3