3.83 \(\int \frac{x^5 (d+e x)^3}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=174 \[ \frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{13 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6} \]

[Out]

(d^4*(d + e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (23*d^3*(d + e*x)^2)/(15*e^6*(d^2 - e^2*x^2)^(3/2)) + (127*d
^2*(d + e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + (3*d*Sqrt[d^2 - e^2*x^2])/e^6 + (x*Sqrt[d^2 - e^2*x^2])/(2*e^5) -
(13*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^6)

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Rubi [A]  time = 0.404562, antiderivative size = 174, 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 = {1635, 1815, 641, 217, 203} \[ \frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{13 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d^4*(d + e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (23*d^3*(d + e*x)^2)/(15*e^6*(d^2 - e^2*x^2)^(3/2)) + (127*d
^2*(d + e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + (3*d*Sqrt[d^2 - e^2*x^2])/e^6 + (x*Sqrt[d^2 - e^2*x^2])/(2*e^5) -
(13*d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^6)

Rule 1635

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -Simp[(d*f*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*
a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q
+ f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p +
1/2, 0] && GtQ[m, 0]

Rule 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

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{x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{(d+e x)^2 \left (\frac{3 d^5}{e^5}+\frac{5 d^4 x}{e^4}+\frac{5 d^3 x^2}{e^3}+\frac{5 d^2 x^3}{e^2}+\frac{5 d x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{(d+e x) \left (\frac{37 d^5}{e^5}+\frac{45 d^4 x}{e^4}+\frac{30 d^3 x^2}{e^3}+\frac{15 d^2 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{\frac{90 d^5}{e^5}+\frac{45 d^4 x}{e^4}+\frac{15 d^3 x^2}{e^3}}{\sqrt{d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{\int \frac{-\frac{195 d^5}{e^3}-\frac{90 d^4 x}{e^2}}{\sqrt{d^2-e^2 x^2}} \, dx}{30 d^3 e^2}\\ &=\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{\left (13 d^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^5}\\ &=\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{\left (13 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^5}\\ &=\frac{d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{127 d^2 (d+e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}-\frac{13 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6}\\ \end{align*}

Mathematica [A]  time = 0.262394, size = 131, normalized size = 0.75 \[ \frac{(d+e x) \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (479 d^2 e^2 x^2-717 d^3 e x+304 d^4-45 d e^3 x^3-15 e^4 x^4\right )-195 d (d-e x)^3 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{30 e^6 (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(304*d^4 - 717*d^3*e*x + 479*d^2*e^2*x^2 - 45*d*e^3*x^3 - 15*e^4*x^4) - 19
5*d*(d - e*x)^3*ArcSin[(e*x)/d]))/(30*e^6*(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 = 0.137, size = 222, normalized size = 1.3 \begin{align*} -{\frac{e{x}^{7}}{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{13\,{d}^{2}{x}^{5}}{10\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{13\,{d}^{2}{x}^{3}}{6\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{13\,{d}^{2}x}{2\,{e}^{5}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{13\,{d}^{2}}{2\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-3\,{\frac{d{x}^{6}}{ \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{5/2}}}+19\,{\frac{{d}^{3}{x}^{4}}{{e}^{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{76\,{d}^{5}{x}^{2}}{3\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{152\,{d}^{7}}{15\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/2*e*x^7/(-e^2*x^2+d^2)^(5/2)+13/10/e*d^2*x^5/(-e^2*x^2+d^2)^(5/2)-13/6/e^3*d^2*x^3/(-e^2*x^2+d^2)^(3/2)+13/
2/e^5*d^2*x/(-e^2*x^2+d^2)^(1/2)-13/2/e^5*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-3*d*x^6/(
-e^2*x^2+d^2)^(5/2)+19*d^3/e^2*x^4/(-e^2*x^2+d^2)^(5/2)-76/3*d^5/e^4*x^2/(-e^2*x^2+d^2)^(5/2)+152/15*d^7/e^6/(
-e^2*x^2+d^2)^(5/2)

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Maxima [B]  time = 1.50447, size = 429, normalized size = 2.47 \begin{align*} -\frac{e x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{13}{30} \, d^{2} e 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{3 \, d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{13 \, d^{2} 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 )}}{6 \, e} + \frac{19 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{76 \, d^{5} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{152 \, d^{7}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{26 \, d^{4} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{5}} - \frac{91 \, d^{2} x}{30 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{5}} - \frac{13 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/2*e*x^7/(-e^2*x^2 + d^2)^(5/2) + 13/30*d^2*e*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)) - 3*d*x^6/(-e^2*x^2 + d^2)^(5/2) - 13/6*d^2*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))/e + 19*d^3*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2)
- 76/3*d^5*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 152/15*d^7/((-e^2*x^2 + d^2)^(5/2)*e^6) + 26/15*d^4*x/((-e^2*x^
2 + d^2)^(3/2)*e^5) - 91/30*d^2*x/(sqrt(-e^2*x^2 + d^2)*e^5) - 13/2*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)
*e^5)

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Fricas [A]  time = 1.78923, size = 409, normalized size = 2.35 \begin{align*} \frac{304 \, d^{2} e^{3} x^{3} - 912 \, d^{3} e^{2} x^{2} + 912 \, d^{4} e x - 304 \, d^{5} + 390 \,{\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 (15 \, e^{4} x^{4} + 45 \, d e^{3} x^{3} - 479 \, d^{2} e^{2} x^{2} + 717 \, d^{3} e x - 304 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{9} x^{3} - 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x - d^{3} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(304*d^2*e^3*x^3 - 912*d^3*e^2*x^2 + 912*d^4*e*x - 304*d^5 + 390*(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)) + (15*e^4*x^4 + 45*d*e^3*x^3 - 479*d^2*e^2*x^2 + 717*d^3*e*x
- 304*d^4)*sqrt(-e^2*x^2 + d^2))/(e^9*x^3 - 3*d*e^8*x^2 + 3*d^2*e^7*x - d^3*e^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \left (d + e x\right )^{3}}{\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(x**5*(e*x+d)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(x**5*(d + e*x)**3/(-(-d + e*x)*(d + e*x))**(7/2), x)

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Giac [A]  time = 1.20391, size = 159, normalized size = 0.91 \begin{align*} -\frac{13}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-6\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (304 \, d^{7} e^{\left (-6\right )} +{\left (195 \, d^{6} e^{\left (-5\right )} -{\left (760 \, d^{5} e^{\left (-4\right )} +{\left (455 \, d^{4} e^{\left (-3\right )} -{\left (570 \, d^{3} e^{\left (-2\right )} +{\left (299 \, d^{2} e^{\left (-1\right )} - 15 \,{\left (x e + 6 \, d\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(x^5*(e*x+d)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-13/2*d^2*arcsin(x*e/d)*e^(-6)*sgn(d) - 1/30*(304*d^7*e^(-6) + (195*d^6*e^(-5) - (760*d^5*e^(-4) + (455*d^4*e^
(-3) - (570*d^3*e^(-2) + (299*d^2*e^(-1) - 15*(x*e + 6*d)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^
2)^3