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

Optimal. Leaf size=214 \[ -\frac{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{1}{2} d^2 e^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{85}{16} d^2 e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

-(d*e^5*(8*d - 85*e*x)*Sqrt[d^2 - e^2*x^2])/(16*x) + (d*e^3*(8*d + 85*e*x)*(d^2 - e^2*x^2)^(3/2))/(48*x^3) - (
e^2*(85*d + 12*e*x)*(d^2 - e^2*x^2)^(5/2))/(120*x^4) - (d*(d^2 - e^2*x^2)^(7/2))/(6*x^6) - (3*e*(d^2 - e^2*x^2
)^(7/2))/(5*x^5) - (d^2*e^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - (85*d^2*e^6*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])
/16

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Rubi [A]  time = 0.311736, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1807, 813, 811, 844, 217, 203, 266, 63, 208} \[ -\frac{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{1}{2} d^2 e^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{85}{16} d^2 e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*e^5*(8*d - 85*e*x)*Sqrt[d^2 - e^2*x^2])/(16*x) + (d*e^3*(8*d + 85*e*x)*(d^2 - e^2*x^2)^(3/2))/(48*x^3) - (
e^2*(85*d + 12*e*x)*(d^2 - e^2*x^2)^(5/2))/(120*x^4) - (d*(d^2 - e^2*x^2)^(7/2))/(6*x^6) - (3*e*(d^2 - e^2*x^2
)^(7/2))/(5*x^5) - (d^2*e^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - (85*d^2*e^6*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])
/16

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 811

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*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{5/2} \left (-18 d^4 e-17 d^3 e^2 x-6 d^2 e^3 x^2\right )}{x^6} \, dx}{6 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac{\int \frac{\left (85 d^5 e^2-6 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx}{30 d^4}\\ &=-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{\int \frac{\left (48 d^6 e^3+340 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx}{96 d^4}\\ &=\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac{\int \frac{\left (192 d^8 e^5+2040 d^7 e^6 x\right ) \sqrt{d^2-e^2 x^2}}{x^2} \, dx}{384 d^6}\\ &=-\frac{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{\int \frac{-4080 d^9 e^6+384 d^8 e^7 x}{x \sqrt{d^2-e^2 x^2}} \, dx}{768 d^6}\\ &=-\frac{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac{1}{16} \left (85 d^3 e^6\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx-\frac{1}{2} \left (d^2 e^7\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}+\frac{1}{32} \left (85 d^3 e^6\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )-\frac{1}{2} \left (d^2 e^7\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{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{1}{2} d^2 e^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{1}{16} \left (85 d^3 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )\\ &=-\frac{d e^5 (8 d-85 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{d e^3 (8 d+85 e x) \left (d^2-e^2 x^2\right )^{3/2}}{48 x^3}-\frac{e^2 (85 d+12 e x) \left (d^2-e^2 x^2\right )^{5/2}}{120 x^4}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{6 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac{1}{2} d^2 e^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{85}{16} d^2 e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}

Mathematica [C]  time = 0.242407, size = 286, normalized size = 1.34 \[ -\frac{3 d^6 e \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{5 x^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{d^4 e^3 \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 e^6 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )}{7 d^5}+\frac{34 d^7 e^2 x^2-59 d^5 e^4 x^4+33 d^3 e^6 x^6+15 d^3 e^6 x^6 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )-8 d^9}{48 x^6 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*d^9 + 34*d^7*e^2*x^2 - 59*d^5*e^4*x^4 + 33*d^3*e^6*x^6 + 15*d^3*e^6*x^6*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sq
rt[1 - (e^2*x^2)/d^2]])/(48*x^6*Sqrt[d^2 - e^2*x^2]) - (3*d^6*e*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, -5
/2, -3/2, (e^2*x^2)/d^2])/(5*x^5*Sqrt[1 - (e^2*x^2)/d^2]) - (d^4*e^3*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/
2, -3/2, -1/2, (e^2*x^2)/d^2])/(3*x^3*Sqrt[1 - (e^2*x^2)/d^2]) - (3*e^6*(d^2 - e^2*x^2)^(7/2)*Hypergeometric2F
1[3, 7/2, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^5)

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Maple [A]  time = 0.106, size = 352, normalized size = 1.6 \begin{align*}{\frac{{e}^{3}}{15\,{d}^{2}{x}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{e}^{5}}{15\,{d}^{4}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{e}^{7}x}{15\,{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{7}x}{3\,{d}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{7}{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{7}x}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,e}{5\,{x}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{17\,{e}^{6}}{16\,{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{85\,{e}^{6}}{48\,d} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{85\,d{e}^{6}}{16}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{d}{6\,{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{17\,{e}^{2}}{24\,d{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{17\,{e}^{4}}{16\,{d}^{3}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{85\,{d}^{3}{e}^{6}}{16}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*e^3/d^2/x^3*(-e^2*x^2+d^2)^(7/2)-4/15*e^5/d^4/x*(-e^2*x^2+d^2)^(7/2)-4/15*e^7/d^4*x*(-e^2*x^2+d^2)^(5/2)-
1/3*e^7/d^2*x*(-e^2*x^2+d^2)^(3/2)-1/2*e^7*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2*e^7*
x*(-e^2*x^2+d^2)^(1/2)-3/5*e*(-e^2*x^2+d^2)^(7/2)/x^5+17/16/d^3*e^6*(-e^2*x^2+d^2)^(5/2)+85/48/d*e^6*(-e^2*x^2
+d^2)^(3/2)+85/16*d*e^6*(-e^2*x^2+d^2)^(1/2)-1/6*d*(-e^2*x^2+d^2)^(7/2)/x^6-17/24/d*e^2/x^4*(-e^2*x^2+d^2)^(7/
2)+17/16/d^3*e^4/x^2*(-e^2*x^2+d^2)^(7/2)-85/16*d^3*e^6/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(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((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.04116, size = 392, normalized size = 1.83 \begin{align*} \frac{240 \, d^{2} e^{6} x^{6} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 1275 \, d^{2} e^{6} x^{6} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 720 \, d^{2} e^{6} x^{6} +{\left (120 \, e^{7} x^{7} + 720 \, d e^{6} x^{6} - 544 \, d^{2} e^{5} x^{5} + 645 \, d^{3} e^{4} x^{4} + 448 \, d^{4} e^{3} x^{3} - 50 \, d^{5} e^{2} x^{2} - 144 \, d^{6} e x - 40 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(240*d^2*e^6*x^6*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 1275*d^2*e^6*x^6*log(-(d - sqrt(-e^2*x^2 +
d^2))/x) + 720*d^2*e^6*x^6 + (120*e^7*x^7 + 720*d*e^6*x^6 - 544*d^2*e^5*x^5 + 645*d^3*e^4*x^4 + 448*d^4*e^3*x^
3 - 50*d^5*e^2*x^2 - 144*d^6*e*x - 40*d^7)*sqrt(-e^2*x^2 + d^2))/x^6

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Sympy [C]  time = 23.5367, size = 1420, normalized size = 6.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**7*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/
(48*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/
(16*d**5), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x*
*5*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-
d**2/(e**2*x**2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) + 3*d**6*e*Piecewise((3*I*d**3*sqrt(-1 + e**2*
x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2
*x**7) + 2*I*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e
**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2)/Abs(d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**
2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*
e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-
15*d**3*x**5 + 15*d*e**2*x**7), True)) + d**5*e**2*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*
e/(8*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d*
*3), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt
(-d**2/(e**2*x**2) + 1)) + I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True
)) - 5*d**4*e**3*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2),
Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x*
*2) + 1)/(3*d**2), True)) - 5*d**3*e**4*Piecewise((-d**2/(2*e*x**3*sqrt(d**2/(e**2*x**2) - 1)) + e/(2*x*sqrt(d
**2/(e**2*x**2) - 1)) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (-I*e*sqrt(-d**2/(e**
2*x**2) + 1)/(2*x) - I*e**2*asin(d/(e*x))/(2*d), True)) + d**2*e**5*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2
)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (-d/(x*sqrt(1 -
e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True)) + 3*d*e**6*Piecewise((d**2/(e*
x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2)/(Abs(e**2)*Abs(x*
*2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1),
True)) + e**7*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)/Abs(d**2) > 1), (d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d*
*2)/2, True))

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Giac [B]  time = 1.32402, size = 655, normalized size = 3.06 \begin{align*} -\frac{1}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{6} \mathrm{sgn}\left (d\right ) - \frac{85}{16} \, d^{2} e^{6} \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) + \frac{{\left (5 \, d^{2} e^{14} + \frac{36 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{12}}{x} + \frac{45 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{10}}{x^{2}} - \frac{340 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} e^{8}}{x^{3}} - \frac{1215 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{6}}{x^{4}} + \frac{1800 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{2} e^{4}}{x^{5}}\right )} x^{6} e^{4}}{1920 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6}} - \frac{1}{1920} \,{\left (\frac{1800 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{2} e^{52}}{x} - \frac{1215 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{50}}{x^{2}} - \frac{340 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} e^{48}}{x^{3}} + \frac{45 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} e^{46}}{x^{4}} + \frac{36 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{2} e^{44}}{x^{5}} + \frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{2} e^{42}}{x^{6}}\right )} e^{\left (-48\right )} + \frac{1}{2} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (x e^{7} + 6 \, d e^{6}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d^2*arcsin(x*e/d)*e^6*sgn(d) - 85/16*d^2*e^6*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x)
) + 1/1920*(5*d^2*e^14 + 36*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*e^12/x + 45*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^
2*e^10/x^2 - 340*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^2*e^8/x^3 - 1215*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^2*e^6/
x^4 + 1800*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d^2*e^4/x^5)*x^6*e^4/(d*e + sqrt(-x^2*e^2 + d^2)*e)^6 - 1/1920*(18
00*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*e^52/x - 1215*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*e^50/x^2 - 340*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^3*d^2*e^48/x^3 + 45*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^2*e^46/x^4 + 36*(d*e + sqrt(-x
^2*e^2 + d^2)*e)^5*d^2*e^44/x^5 + 5*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*d^2*e^42/x^6)*e^(-48) + 1/2*sqrt(-x^2*e^2
+ d^2)*(x*e^7 + 6*d*e^6)