3.341 \(\int \frac{1}{x^3 (d+e x) (a+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=276 \[ -\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^3}+\frac{2 c e x}{a^2 d^2 \sqrt{a+c x^2}}-\frac{3 c}{2 a^2 d \sqrt{a+c x^2}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2} d}-\frac{e^3 (a e+c d x)}{a d^3 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{e^2}{a d^3 \sqrt{a+c x^2}}+\frac{e^5 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}+\frac{e}{a d^2 x \sqrt{a+c x^2}}-\frac{1}{2 a d x^2 \sqrt{a+c x^2}} \]

[Out]

(-3*c)/(2*a^2*d*Sqrt[a + c*x^2]) + e^2/(a*d^3*Sqrt[a + c*x^2]) - 1/(2*a*d*x^2*Sqrt[a + c*x^2]) + e/(a*d^2*x*Sq
rt[a + c*x^2]) + (2*c*e*x)/(a^2*d^2*Sqrt[a + c*x^2]) - (e^3*(a*e + c*d*x))/(a*d^3*(c*d^2 + a*e^2)*Sqrt[a + c*x
^2]) + (e^5*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^3*(c*d^2 + a*e^2)^(3/2)) + (3*c*A
rcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(5/2)*d) - (e^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(a^(3/2)*d^3)

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Rubi [A]  time = 0.235688, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {961, 266, 51, 63, 208, 271, 191, 741, 12, 725, 206} \[ -\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{3/2} d^3}+\frac{2 c e x}{a^2 d^2 \sqrt{a+c x^2}}-\frac{3 \sqrt{a+c x^2}}{2 a^2 d x^2}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{5/2} d}-\frac{e^3 (a e+c d x)}{a d^3 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{e^2}{a d^3 \sqrt{a+c x^2}}+\frac{e^5 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \left (a e^2+c d^2\right )^{3/2}}+\frac{e}{a d^2 x \sqrt{a+c x^2}}+\frac{1}{a d x^2 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(d + e*x)*(a + c*x^2)^(3/2)),x]

[Out]

e^2/(a*d^3*Sqrt[a + c*x^2]) + 1/(a*d*x^2*Sqrt[a + c*x^2]) + e/(a*d^2*x*Sqrt[a + c*x^2]) + (2*c*e*x)/(a^2*d^2*S
qrt[a + c*x^2]) - (e^3*(a*e + c*d*x))/(a*d^3*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (3*Sqrt[a + c*x^2])/(2*a^2*d*x
^2) + (e^5*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(d^3*(c*d^2 + a*e^2)^(3/2)) + (3*c*Ar
cTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*a^(5/2)*d) - (e^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(a^(3/2)*d^3)

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 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 271

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

Rule 191

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

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*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] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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])

Rubi steps

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

Mathematica [C]  time = 0.382267, size = 203, normalized size = 0.74 \[ \frac{-\frac{c d^2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{c x^2}{a}+1\right )}{a^2 \sqrt{a+c x^2}}+\frac{d e \left (a+2 c x^2\right )}{a^2 x \sqrt{a+c x^2}}-\frac{e^3 (a e+c d x)}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{e^5 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{e^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{a}+1\right )}{a \sqrt{a+c x^2}}}{d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(d + e*x)*(a + c*x^2)^(3/2)),x]

[Out]

(-((e^3*(a*e + c*d*x))/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])) + (d*e*(a + 2*c*x^2))/(a^2*x*Sqrt[a + c*x^2]) + (e
^5*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(c*d^2 + a*e^2)^(3/2) + (e^2*Hypergeometric2F
1[-1/2, 1, 1/2, 1 + (c*x^2)/a])/(a*Sqrt[a + c*x^2]) - (c*d^2*Hypergeometric2F1[-1/2, 2, 1/2, 1 + (c*x^2)/a])/(
a^2*Sqrt[a + c*x^2]))/d^3

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Maple [A]  time = 0.247, size = 439, normalized size = 1.6 \begin{align*}{\frac{{e}^{2}}{a{d}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{{e}^{2}}{{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{e}^{4}}{{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{e}^{3}xc}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{e}^{4}}{{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{1}{2\,ad{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,c}{2\,{a}^{2}d}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,c}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{e}{a{d}^{2}x}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{cex}{{a}^{2}{d}^{2}\sqrt{c{x}^{2}+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(e*x+d)/(c*x^2+a)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)*x^3), x)

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Fricas [A]  time = 6.96821, size = 3868, normalized size = 14.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(2*(a^3*c*e^5*x^4 + a^4*e^5*x^2)*sqrt(c*d^2 + a*e^2)*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^4*d^6
+ 4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4 - 2*a^3*c*e^6)*x^4 + (3*a*c^3*d^6 + 4*a^2*c^2*d^4*e^2 - a^3*c*d^2*e^4 - 2
*a^4*e^6)*x^2)*sqrt(a)*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(a^2*c^2*d^6 + 2*a^3*c*d^4*e^2
+ a^4*d^2*e^4 - 2*(2*a*c^3*d^5*e + 3*a^2*c^2*d^3*e^3 + a^3*c*d*e^5)*x^3 + (3*a*c^3*d^6 + 4*a^2*c^2*d^4*e^2 + a
^3*c*d^2*e^4)*x^2 - 2*(a^2*c^2*d^5*e + 2*a^3*c*d^3*e^3 + a^4*d*e^5)*x)*sqrt(c*x^2 + a))/((a^3*c^3*d^7 + 2*a^4*
c^2*d^5*e^2 + a^5*c*d^3*e^4)*x^4 + (a^4*c^2*d^7 + 2*a^5*c*d^5*e^2 + a^6*d^3*e^4)*x^2), 1/4*(4*(a^3*c*e^5*x^4 +
a^4*e^5*x^2)*sqrt(-c*d^2 - a*e^2)*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^4*d^6 + 4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4 - 2*a^3*c*e^6)*x^4 + (3*a*c^3*
d^6 + 4*a^2*c^2*d^4*e^2 - a^3*c*d^2*e^4 - 2*a^4*e^6)*x^2)*sqrt(a)*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*
a)/x^2) - 2*(a^2*c^2*d^6 + 2*a^3*c*d^4*e^2 + a^4*d^2*e^4 - 2*(2*a*c^3*d^5*e + 3*a^2*c^2*d^3*e^3 + a^3*c*d*e^5)
*x^3 + (3*a*c^3*d^6 + 4*a^2*c^2*d^4*e^2 + a^3*c*d^2*e^4)*x^2 - 2*(a^2*c^2*d^5*e + 2*a^3*c*d^3*e^3 + a^4*d*e^5)
*x)*sqrt(c*x^2 + a))/((a^3*c^3*d^7 + 2*a^4*c^2*d^5*e^2 + a^5*c*d^3*e^4)*x^4 + (a^4*c^2*d^7 + 2*a^5*c*d^5*e^2 +
a^6*d^3*e^4)*x^2), -1/2*(((3*c^4*d^6 + 4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4 - 2*a^3*c*e^6)*x^4 + (3*a*c^3*d^6 +
4*a^2*c^2*d^4*e^2 - a^3*c*d^2*e^4 - 2*a^4*e^6)*x^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (a^3*c*e^5*x^4
+ a^4*e^5*x^2)*sqrt(c*d^2 + a*e^2)*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*sqr
t(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)) + (a^2*c^2*d^6 + 2*a^3*c*d^4*e^2 +
a^4*d^2*e^4 - 2*(2*a*c^3*d^5*e + 3*a^2*c^2*d^3*e^3 + a^3*c*d*e^5)*x^3 + (3*a*c^3*d^6 + 4*a^2*c^2*d^4*e^2 + a^3
*c*d^2*e^4)*x^2 - 2*(a^2*c^2*d^5*e + 2*a^3*c*d^3*e^3 + a^4*d*e^5)*x)*sqrt(c*x^2 + a))/((a^3*c^3*d^7 + 2*a^4*c^
2*d^5*e^2 + a^5*c*d^3*e^4)*x^4 + (a^4*c^2*d^7 + 2*a^5*c*d^5*e^2 + a^6*d^3*e^4)*x^2), 1/2*(2*(a^3*c*e^5*x^4 + a
^4*e^5*x^2)*sqrt(-c*d^2 - a*e^2)*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^4*d^6 + 4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4 - 2*a^3*c*e^6)*x^4 + (3*a*c^3*d^
6 + 4*a^2*c^2*d^4*e^2 - a^3*c*d^2*e^4 - 2*a^4*e^6)*x^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (a^2*c^2*d
^6 + 2*a^3*c*d^4*e^2 + a^4*d^2*e^4 - 2*(2*a*c^3*d^5*e + 3*a^2*c^2*d^3*e^3 + a^3*c*d*e^5)*x^3 + (3*a*c^3*d^6 +
4*a^2*c^2*d^4*e^2 + a^3*c*d^2*e^4)*x^2 - 2*(a^2*c^2*d^5*e + 2*a^3*c*d^3*e^3 + a^4*d*e^5)*x)*sqrt(c*x^2 + a))/(
(a^3*c^3*d^7 + 2*a^4*c^2*d^5*e^2 + a^5*c*d^3*e^4)*x^4 + (a^4*c^2*d^7 + 2*a^5*c*d^5*e^2 + a^6*d^3*e^4)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

Integral(1/(x**3*(a + c*x**2)**(3/2)*(d + e*x)), x)

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Giac [A]  time = 1.25382, size = 483, normalized size = 1.75 \begin{align*} \frac{\frac{{\left (a^{2} c^{3} d^{2} e + a^{3} c^{2} e^{3}\right )} x}{a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4}} - \frac{a^{2} c^{3} d^{3} + a^{3} c^{2} d e^{2}}{a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4}}}{\sqrt{c x^{2} + a}} - \frac{2 \, \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 ) e^{5}}{{\left (c d^{5} + a d^{3} e^{2}\right )} \sqrt{-c d^{2} - a e^{2}}} - \frac{{\left (3 \, c d^{2} - 2 \, a e^{2}\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} d^{3}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c d - 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a \sqrt{c} e +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a c d + 2 \, a^{2} \sqrt{c} e}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

((a^2*c^3*d^2*e + a^3*c^2*e^3)*x/(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4) - (a^2*c^3*d^3 + a^3*c^2*d*e^2)/(a^
4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4))/sqrt(c*x^2 + a) - 2*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)
*d)/sqrt(-c*d^2 - a*e^2))*e^5/((c*d^5 + a*d^3*e^2)*sqrt(-c*d^2 - a*e^2)) - (3*c*d^2 - 2*a*e^2)*arctan(-(sqrt(c
)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2*d^3) + ((sqrt(c)*x - sqrt(c*x^2 + a))^3*c*d - 2*(sqrt(c)*x - sq
rt(c*x^2 + a))^2*a*sqrt(c)*e + (sqrt(c)*x - sqrt(c*x^2 + a))*a*c*d + 2*a^2*sqrt(c)*e)/(((sqrt(c)*x - sqrt(c*x^
2 + a))^2 - a)^2*a^2*d^2)