3.325 \(\int \frac{\sqrt{a+c x^2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=274 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2} d}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}-\frac{\sqrt{a} e^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^5}+\frac{e^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^5}-\frac{c e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{\sqrt{a+c x^2}}{4 d x^4} \]

[Out]

-Sqrt[a + c*x^2]/(4*d*x^4) - (c*Sqrt[a + c*x^2])/(8*a*d*x^2) - (e^2*Sqrt[a + c*x^2])/(2*d^3*x^2) + (e^3*Sqrt[a
+ c*x^2])/(d^4*x) + (e*(a + c*x^2)^(3/2))/(3*a*d^2*x^3) + (e^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqr
t[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/d^5 + (c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*a^(3/2)*d) - (c*e^2*ArcTan
h[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqrt[a]*d^3) - (Sqrt[a]*e^4*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^5

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Rubi [A]  time = 0.29656, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {961, 266, 47, 51, 63, 208, 264, 277, 217, 206, 50, 735, 844, 725} \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2} d}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}-\frac{\sqrt{a} e^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^5}+\frac{e^3 \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^5}-\frac{c e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{\sqrt{a+c x^2}}{4 d x^4} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^2]/(x^5*(d + e*x)),x]

[Out]

-Sqrt[a + c*x^2]/(4*d*x^4) - (c*Sqrt[a + c*x^2])/(8*a*d*x^2) - (e^2*Sqrt[a + c*x^2])/(2*d^3*x^2) + (e^3*Sqrt[a
+ c*x^2])/(d^4*x) + (e*(a + c*x^2)^(3/2))/(3*a*d^2*x^3) + (e^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqr
t[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/d^5 + (c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*a^(3/2)*d) - (c*e^2*ArcTan
h[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqrt[a]*d^3) - (Sqrt[a]*e^4*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^5

Rule 961

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 735

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

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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+c x^2}}{x^5 (d+e x)} \, dx &=\int \left (\frac{\sqrt{a+c x^2}}{d x^5}-\frac{e \sqrt{a+c x^2}}{d^2 x^4}+\frac{e^2 \sqrt{a+c x^2}}{d^3 x^3}-\frac{e^3 \sqrt{a+c x^2}}{d^4 x^2}+\frac{e^4 \sqrt{a+c x^2}}{d^5 x}-\frac{e^5 \sqrt{a+c x^2}}{d^5 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\sqrt{a+c x^2}}{x^5} \, dx}{d}-\frac{e \int \frac{\sqrt{a+c x^2}}{x^4} \, dx}{d^2}+\frac{e^2 \int \frac{\sqrt{a+c x^2}}{x^3} \, dx}{d^3}-\frac{e^3 \int \frac{\sqrt{a+c x^2}}{x^2} \, dx}{d^4}+\frac{e^4 \int \frac{\sqrt{a+c x^2}}{x} \, dx}{d^5}-\frac{e^5 \int \frac{\sqrt{a+c x^2}}{d+e x} \, dx}{d^5}\\ &=-\frac{e^4 \sqrt{a+c x^2}}{d^5}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^3} \, dx,x,x^2\right )}{2 d}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )}{2 d^3}-\frac{\left (c e^3\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{d^4}+\frac{e^4 \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x} \, dx,x,x^2\right )}{2 d^5}-\frac{e^4 \int \frac{a e-c d x}{(d+e x) \sqrt{a+c x^2}} \, dx}{d^5}\\ &=-\frac{\sqrt{a+c x^2}}{4 d x^4}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac{c \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+c x}} \, dx,x,x^2\right )}{8 d}+\frac{\left (c e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{4 d^3}+\frac{\left (c e^3\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{d^4}-\frac{\left (c e^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{d^4}+\frac{\left (a e^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{2 d^5}-\frac{\left (e^3 \left (c d^2+a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{d^5}\\ &=-\frac{\sqrt{a+c x^2}}{4 d x^4}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}-\frac{\sqrt{c} e^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{d^4}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{16 a d}+\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{2 d^3}+\frac{\left (c e^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{d^4}+\frac{\left (a e^4\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c d^5}+\frac{\left (e^3 \left (c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{d^5}\\ &=-\frac{\sqrt{a+c x^2}}{4 d x^4}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac{e^3 \sqrt{c d^2+a e^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^5}-\frac{c e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d^3}-\frac{\sqrt{a} e^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^5}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{8 a d}\\ &=-\frac{\sqrt{a+c x^2}}{4 d x^4}-\frac{c \sqrt{a+c x^2}}{8 a d x^2}-\frac{e^2 \sqrt{a+c x^2}}{2 d^3 x^2}+\frac{e^3 \sqrt{a+c x^2}}{d^4 x}+\frac{e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac{e^3 \sqrt{c d^2+a e^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^5}+\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{3/2} d}-\frac{c e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d^3}-\frac{\sqrt{a} e^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^5}\\ \end{align*}

Mathematica [C]  time = 1.16158, size = 344, normalized size = 1.26 \[ \frac{-\frac{2 c^2 d^4 \left (a+c x^2\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{c x^2}{a}+1\right )}{a^3}-\frac{3 d^2 e^2 \left (c x^2 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x^2}{a}+1}\right )+a+c x^2\right )}{x^2 \sqrt{a+c x^2}}+6 e^3 \left (\sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )+\sqrt{c} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )\right )+\frac{2 d^3 e \left (a+c x^2\right )^{3/2}}{a x^3}+\frac{6 d e^3 \left (-\sqrt{a} \sqrt{c} x \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+a+c x^2\right )}{x \sqrt{a+c x^2}}-6 e^4 \sqrt{a+c x^2}+6 e^4 \left (\sqrt{a+c x^2}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\right )}{6 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^2]/(x^5*(d + e*x)),x]

[Out]

(-6*e^4*Sqrt[a + c*x^2] + (2*d^3*e*(a + c*x^2)^(3/2))/(a*x^3) + (6*d*e^3*(a + c*x^2 - Sqrt[a]*Sqrt[c]*x*Sqrt[1
+ (c*x^2)/a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/(x*Sqrt[a + c*x^2]) + 6*e^3*(Sqrt[c]*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a
+ c*x^2]] + Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])]) + 6*e^4*(Sqrt[a
+ c*x^2] - Sqrt[a]*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]) - (3*d^2*e^2*(a + c*x^2 + c*x^2*Sqrt[1 + (c*x^2)/a]*ArcT
anh[Sqrt[1 + (c*x^2)/a]]))/(x^2*Sqrt[a + c*x^2]) - (2*c^2*d^4*(a + c*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2,
1 + (c*x^2)/a])/a^3)/(6*d^5)

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Maple [B]  time = 0.24, size = 703, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(1/2)/x^5/(e*x+d),x)

[Out]

-e^4/d^5*a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+e^4/d^5*(c*x^2+a)^(1/2)+1/3*e*(c*x^2+a)^(3/2)/a/d^2/x^3
-e^4/d^5*((d/e+x)^2*c-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+e^3/d^4*c^(1/2)*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(
(d/e+x)^2*c-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+e^4/d^5/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^
2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*
a+e^2/d^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((d/e+
x)^2*c-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*c-1/4/d/a/x^4*(c*x^2+a)^(3/2)+1/8/d*c/a^2/x^2*(c*x^2
+a)^(3/2)+1/8/d*c^2/a^(3/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-1/8/d*c^2/a^2*(c*x^2+a)^(1/2)-1/2*e^2/d^3/a/
x^2*(c*x^2+a)^(3/2)-1/2*e^2/d^3*c/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+1/2*e^2/d^3*c/a*(c*x^2+a)^(1/2
)+e^3/d^4/a/x*(c*x^2+a)^(3/2)-e^3/d^4*c/a*x*(c*x^2+a)^(1/2)-e^3/d^4*c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}}{{\left (e x + d\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/x^5/(e*x+d),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)/((e*x + d)*x^5), x)

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Fricas [A]  time = 2.70868, size = 2152, normalized size = 7.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/x^5/(e*x+d),x, algorithm="fricas")

[Out]

[1/48*(24*sqrt(c*d^2 + a*e^2)*a^2*e^3*x^4*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 +
2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2
- 8*a^2*e^4)*sqrt(a)*x^4*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(8*a^2*d^3*e*x - 6*a^2*d^4 +
8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2*d^2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^2*d^5*x^4), 1/48*(48*sq
rt(-c*d^2 - a*e^2)*a^2*e^3*x^4*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 +
(c^2*d^2 + a*c*e^2)*x^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*sqrt(a)*x^4*log(-(c*x^2 - 2*sqrt(c*x^2 + a
)*sqrt(a) + 2*a)/x^2) + 2*(8*a^2*d^3*e*x - 6*a^2*d^4 + 8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2*d^
2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^2*d^5*x^4), 1/24*(12*sqrt(c*d^2 + a*e^2)*a^2*e^3*x^4*log((2*a*c*d*e*x - a*c*d^
2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*
d*e*x + d^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*sqrt(-a)*x^4*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (8*a^2
*d^3*e*x - 6*a^2*d^4 + 8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2*d^2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^
2*d^5*x^4), 1/24*(24*sqrt(-c*d^2 - a*e^2)*a^2*e^3*x^4*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a
)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*sqrt(-a)*x^4*arctan
(sqrt(-a)/sqrt(c*x^2 + a)) + (8*a^2*d^3*e*x - 6*a^2*d^4 + 8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2
*d^2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^2*d^5*x^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(1/2)/x**5/(e*x+d),x)

[Out]

Integral(sqrt(a + c*x**2)/(x**5*(d + e*x)), x)

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Giac [B]  time = 1.23568, size = 805, normalized size = 2.94 \begin{align*} -\frac{2 \,{\left (c d^{2} e^{3} + a e^{5}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{\sqrt{-c d^{2} - a e^{2}} d^{5}} - \frac{{\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a d^{5}} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} c^{2} d^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} a c^{\frac{3}{2}} d^{2} e + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a c^{2} d^{3} + 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} a c d e^{2} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a^{2} c^{\frac{3}{2}} d^{2} e + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{2} c^{2} d^{3} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a^{2} c d e^{2} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} a^{2} \sqrt{c} e^{3} - 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{3} c^{\frac{3}{2}} d^{2} e + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{3} c^{2} d^{3} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{3} c d e^{2} + 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a^{3} \sqrt{c} e^{3} + 8 \, a^{4} c^{\frac{3}{2}} d^{2} e + 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{4} c d e^{2} - 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{4} \sqrt{c} e^{3} + 24 \, a^{5} \sqrt{c} e^{3}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4} a d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(1/2)/x^5/(e*x+d),x, algorithm="giac")

[Out]

-2*(c*d^2*e^3 + a*e^5)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/(sqrt(-c*d^
2 - a*e^2)*d^5) - 1/4*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(s
qrt(-a)*a*d^5) + 1/12*(3*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^2*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(3/2
)*d^2*e + 21*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c*d*e^2 + 24*(sq
rt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*d^2*e + 21*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d^3 - 12*(sqrt(c)*
x - sqrt(c*x^2 + a))^5*a^2*c*d*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*sqrt(c)*e^3 - 8*(sqrt(c)*x - sqrt(
c*x^2 + a))^2*a^3*c^(3/2)*d^2*e + 3*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + a
))^3*a^3*c*d*e^2 + 72*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*sqrt(c)*e^3 + 8*a^4*c^(3/2)*d^2*e + 12*(sqrt(c)*x -
sqrt(c*x^2 + a))*a^4*c*d*e^2 - 72*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*sqrt(c)*e^3 + 24*a^5*sqrt(c)*e^3)/(((sqr
t(c)*x - sqrt(c*x^2 + a))^2 - a)^4*a*d^4)