### 3.846 $$\int \frac{(d+e x)^9}{(d^2-e^2 x^2)^{7/2}} \, dx$$

Optimal. Leaf size=206 $\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 \sqrt{d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac{77 d \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}$

[Out]

(2*(d + e*x)^8)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (22*(d + e*x)^6)/(15*e*(d^2 - e^2*x^2)^(3/2)) + (66*(d + e*x)^4)
/(5*e*Sqrt[d^2 - e^2*x^2]) + (231*d^2*Sqrt[d^2 - e^2*x^2])/(2*e) + (77*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e)
+ (77*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(5*e) - (231*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)

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Rubi [A]  time = 0.101482, antiderivative size = 206, normalized size of antiderivative = 1., 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 = {669, 671, 641, 217, 203} $\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 \sqrt{d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac{77 d \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^8)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (22*(d + e*x)^6)/(15*e*(d^2 - e^2*x^2)^(3/2)) + (66*(d + e*x)^4)
/(5*e*Sqrt[d^2 - e^2*x^2]) + (231*d^2*Sqrt[d^2 - e^2*x^2])/(2*e) + (77*d*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(2*e)
+ (77*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])/(5*e) - (231*d^3*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)^9}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{11}{5} \int \frac{(d+e x)^7}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{33}{5} \int \frac{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}-\frac{231}{5} \int \frac{(d+e x)^3}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-(77 d) \int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{2} \left (231 d^2\right ) \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{2} \left (231 d^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{2} \left (231 d^3\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)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.342552, size = 144, normalized size = 0.7 $\frac{(d+e x) \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (8711 d^3 e^2 x^2-815 d^2 e^3 x^3-12843 d^4 e x+5446 d^5-105 d e^4 x^4-10 e^5 x^5\right )-3465 d^2 (d-e x)^3 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{30 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^9/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(5446*d^5 - 12843*d^4*e*x + 8711*d^3*e^2*x^2 - 815*d^2*e^3*x^3 - 105*d*e^4
*x^4 - 10*e^5*x^5) - 3465*d^2*(d - e*x)^3*ArcSin[(e*x)/d]))/(30*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 = 0.209, size = 309, normalized size = 1.5 \begin{align*} -{\frac{152\,{d}^{7}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{157\,{d}^{5}x}{15} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+63\,{\frac{{d}^{5}{e}^{2}{x}^{3}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{231\,{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{4093\,{d}^{3}x}{30}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{9\,d{e}^{6}{x}^{7}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{231\,{d}^{3}{e}^{4}{x}^{5}}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{77\,{d}^{3}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{116\,{e}^{5}{d}^{2}{x}^{6}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+358\,{\frac{{e}^{3}{d}^{4}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{1348\,e{d}^{6}{x}^{2}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{e}^{7}{x}^{8}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{2723\,{d}^{8}}{15\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^9/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-152/5*d^7*x/(-e^2*x^2+d^2)^(5/2)+157/15*d^5*x/(-e^2*x^2+d^2)^(3/2)+63*d^5*e^2*x^3/(-e^2*x^2+d^2)^(5/2)-231/2*
d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+4093/30*d^3*x/(-e^2*x^2+d^2)^(1/2)-9/2*d*e^6*x^7/(-
e^2*x^2+d^2)^(5/2)+231/10*d^3*e^4*x^5/(-e^2*x^2+d^2)^(5/2)-77/2*d^3*e^2*x^3/(-e^2*x^2+d^2)^(3/2)-116/3*e^5*d^2
*x^6/(-e^2*x^2+d^2)^(5/2)+358*e^3*d^4*x^4/(-e^2*x^2+d^2)^(5/2)-1348/3*e*d^6*x^2/(-e^2*x^2+d^2)^(5/2)-1/3*e^7*x
^8/(-e^2*x^2+d^2)^(5/2)+2723/15*d^8/e/(-e^2*x^2+d^2)^(5/2)

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Maxima [B]  time = 1.87807, size = 518, normalized size = 2.51 \begin{align*} -\frac{e^{7} x^{8}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{9 \, d e^{6} x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{77}{10} \, d^{3} 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{116 \, d^{2} e^{5} x^{6}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{77}{2} \, d^{3} 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{358 \, d^{4} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{63 \, d^{5} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{1348 \, d^{6} e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{152 \, d^{7} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2723 \, d^{8}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{619 \, d^{5} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{989 \, d^{3} x}{30 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{231 \, d^{3} \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)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/3*e^7*x^8/(-e^2*x^2 + d^2)^(5/2) - 9/2*d*e^6*x^7/(-e^2*x^2 + d^2)^(5/2) + 77/10*d^3*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)) - 116/3*d^
2*e^5*x^6/(-e^2*x^2 + d^2)^(5/2) - 77/2*d^3*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)) + 358*d^4*e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 63*d^5*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) - 1348/3*d^6*e*
x^2/(-e^2*x^2 + d^2)^(5/2) - 152/5*d^7*x/(-e^2*x^2 + d^2)^(5/2) + 2723/15*d^8/((-e^2*x^2 + d^2)^(5/2)*e) + 619
/15*d^5*x/(-e^2*x^2 + d^2)^(3/2) - 989/30*d^3*x/sqrt(-e^2*x^2 + d^2) - 231/2*d^3*arcsin(e^2*x/sqrt(d^2*e^2))/s
qrt(e^2)

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Fricas [A]  time = 2.54416, size = 447, normalized size = 2.17 \begin{align*} \frac{5446 \, d^{3} e^{3} x^{3} - 16338 \, d^{4} e^{2} x^{2} + 16338 \, d^{5} e x - 5446 \, d^{6} + 6930 \,{\left (d^{3} e^{3} x^{3} - 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x - d^{6}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (10 \, e^{5} x^{5} + 105 \, d e^{4} x^{4} + 815 \, d^{2} e^{3} x^{3} - 8711 \, d^{3} e^{2} x^{2} + 12843 \, d^{4} e x - 5446 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\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)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(5446*d^3*e^3*x^3 - 16338*d^4*e^2*x^2 + 16338*d^5*e*x - 5446*d^6 + 6930*(d^3*e^3*x^3 - 3*d^4*e^2*x^2 + 3*
d^5*e*x - d^6)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (10*e^5*x^5 + 105*d*e^4*x^4 + 815*d^2*e^3*x^3 - 871
1*d^3*e^2*x^2 + 12843*d^4*e*x - 5446*d^5)*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 )^{9}}{\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)**9/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**9/(-(-d + e*x)*(d + e*x))**(7/2), x)

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Giac [A]  time = 1.32293, size = 174, normalized size = 0.84 \begin{align*} -\frac{231}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (5446 \, d^{8} e^{\left (-1\right )} +{\left (3495 \, d^{7} -{\left (13480 \, d^{6} e +{\left (7765 \, d^{5} e^{2} -{\left (10740 \, d^{4} e^{3} +{\left (5941 \, d^{3} e^{4} - 5 \,{\left (232 \, d^{2} e^{5} +{\left (2 \, x e^{7} + 27 \, 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}}}{30 \,{\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)^9/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-231/2*d^3*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/30*(5446*d^8*e^(-1) + (3495*d^7 - (13480*d^6*e + (7765*d^5*e^2 - (1
0740*d^4*e^3 + (5941*d^3*e^4 - 5*(232*d^2*e^5 + (2*x*e^7 + 27*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)
/(x^2*e^2 - d^2)^3