3.136 \(\int \frac{x^7}{(d+e x) (d^2-e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=162 \[ \frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8} \]

[Out]

(x^6*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d - 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*
d - 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d - 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) + (7*d^2*ArcTan[(e*
x)/Sqrt[d^2 - e^2*x^2]])/(2*e^8)

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Rubi [A]  time = 0.160423, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {850, 819, 780, 217, 203} \[ \frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8} \]

Antiderivative was successfully verified.

[In]

Int[x^7/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(x^6*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d - 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*
d - 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d - 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) + (7*d^2*ArcTan[(e*
x)/Sqrt[d^2 - e^2*x^2]])/(2*e^8)

Rule 850

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

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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{x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^7 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x^5 \left (6 d^3-7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{x^3 \left (24 d^5-35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{x \left (48 d^7-105 d^6 e x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{\left (7 d^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=\frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{\left (7 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 )}{2 e^7}\\ &=\frac{x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d-35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d-35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}+\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8}\\ \end{align*}

Mathematica [A]  time = 0.261823, size = 128, normalized size = 0.79 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-249 d^4 e^2 x^2-4 d^3 e^3 x^3+176 d^2 e^4 x^4-9 d^5 e x+96 d^6+15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^2 (d+e x)^3}+105 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{30 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(96*d^6 - 9*d^5*e*x - 249*d^4*e^2*x^2 - 4*d^3*e^3*x^3 + 176*d^2*e^4*x^4 + 15*d*e^5*x^5 -
15*e^6*x^6))/((d - e*x)^2*(d + e*x)^3) + 105*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^8)

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Maple [B]  time = 0.087, size = 318, normalized size = 2. \begin{align*} -{\frac{{x}^{5}}{2\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{d}^{2}{x}^{3}}{6\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{19\,{d}^{2}x}{6\,{e}^{7}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+{\frac{7\,{d}^{2}}{2\,{e}^{7}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d{x}^{4}}{{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-5\,{\frac{{d}^{3}{x}^{2}}{{e}^{6} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{3/2}}}+3\,{\frac{{d}^{5}}{{e}^{8} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{3/2}}}+{\frac{2\,{d}^{4}x}{3\,{e}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{{d}^{6}}{5\,{e}^{9}} \left ({\frac{d}{e}}+x \right ) ^{-1} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{d}^{4}x}{15\,{e}^{7}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{d}^{2}x}{15\,{e}^{7}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/2/e^3*x^5/(-e^2*x^2+d^2)^(3/2)+7/6/e^5*d^2*x^3/(-e^2*x^2+d^2)^(3/2)-19/6/e^7*d^2*x/(-e^2*x^2+d^2)^(1/2)+7/2
/e^7*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+d/e^4*x^4/(-e^2*x^2+d^2)^(3/2)-5*d^3/e^6*x^2/(
-e^2*x^2+d^2)^(3/2)+3*d^5/e^8/(-e^2*x^2+d^2)^(3/2)+2/3*d^4/e^7*x/(-e^2*x^2+d^2)^(3/2)+1/5*d^6/e^9/(d/e+x)/(-(d
/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)-4/15*d^4/e^7/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x-8/15*d^2/e^7/(-(d/e+x)^2*
e^2+2*d*e*(d/e+x))^(1/2)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.00057, size = 575, normalized size = 3.55 \begin{align*} \frac{96 \, d^{2} e^{5} x^{5} + 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} - 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x + 96 \, d^{7} - 210 \,{\left (d^{2} e^{5} x^{5} + d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} - 2 \, d^{5} e^{2} x^{2} + d^{6} e x + d^{7}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{6} x^{6} - 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} + 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} + 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{13} x^{5} + d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} - 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x + d^{5} e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

1/30*(96*d^2*e^5*x^5 + 96*d^3*e^4*x^4 - 192*d^4*e^3*x^3 - 192*d^5*e^2*x^2 + 96*d^6*e*x + 96*d^7 - 210*(d^2*e^5
*x^5 + d^3*e^4*x^4 - 2*d^4*e^3*x^3 - 2*d^5*e^2*x^2 + d^6*e*x + d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x))
- (15*e^6*x^6 - 15*d*e^5*x^5 - 176*d^2*e^4*x^4 + 4*d^3*e^3*x^3 + 249*d^4*e^2*x^2 + 9*d^5*e*x - 96*d^6)*sqrt(-e
^2*x^2 + d^2))/(e^13*x^5 + d*e^12*x^4 - 2*d^2*e^11*x^3 - 2*d^3*e^10*x^2 + d^4*e^9*x + d^5*e^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(x**7/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]