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

Optimal. Leaf size=607 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-2 a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left (-a b \left (e \left (2 d \sqrt{b^2-4 a c}-a e\right )+3 c d^2\right )+a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )\right )+b^2 d \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \sqrt{c} \left (-a b \left (3 c d^2-e \left (2 d \sqrt{b^2-4 a c}+a e\right )\right )-a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )\right )-b^2 d \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x} (b d-2 a e)}{a^2 x}-\frac{e (b d-2 a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^2 \sqrt{d}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 a \sqrt{d}}-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a x} \]

[Out]

-(d*Sqrt[d + e*x])/(2*a*x^2) + (3*e*Sqrt[d + e*x])/(4*a*x) + ((b*d - 2*a*e)*Sqrt[d + e*x])/(a^2*x) - (3*e^2*Ar
cTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*a*Sqrt[d]) - (e*(b*d - 2*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(a^2*Sqrt[d]) -
(2*(b^2*d^2 - 2*a*b*d*e - a*(c*d^2 - a*e^2))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(a^3*Sqrt[d]) + (Sqrt[2]*Sqrt[c]
*(b^3*d^2 + b^2*d*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + a*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*
e)) - a*b*(3*c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b
- Sqrt[b^2 - 4*a*c])*e]])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*
(b^3*d^2 - b^2*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b*(3*c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + a*e)) - a*(a*Sqrt[b
^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

________________________________________________________________________________________

Rubi [A]  time = 3.93064, antiderivative size = 607, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {897, 1287, 199, 206, 1166, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-2 a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left (-a b \left (e \left (2 d \sqrt{b^2-4 a c}-a e\right )+3 c d^2\right )+a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )\right )+b^2 d \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \sqrt{c} \left (-a b \left (3 c d^2-e \left (2 d \sqrt{b^2-4 a c}+a e\right )\right )-a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )\right )-b^2 d \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x} (b d-2 a e)}{a^2 x}-\frac{e (b d-2 a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^2 \sqrt{d}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 a \sqrt{d}}-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*Sqrt[d + e*x])/(2*a*x^2) + (3*e*Sqrt[d + e*x])/(4*a*x) + ((b*d - 2*a*e)*Sqrt[d + e*x])/(a^2*x) - (3*e^2*Ar
cTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*a*Sqrt[d]) - (e*(b*d - 2*a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(a^2*Sqrt[d]) -
(2*(b^2*d^2 - 2*a*b*d*e - a*(c*d^2 - a*e^2))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(a^3*Sqrt[d]) + (Sqrt[2]*Sqrt[c]
*(b^3*d^2 + b^2*d*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + a*(a*Sqrt[b^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*
e)) - a*b*(3*c*d^2 + e*(2*Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b
- Sqrt[b^2 - 4*a*c])*e]])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*
(b^3*d^2 - b^2*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b*(3*c*d^2 - e*(2*Sqrt[b^2 - 4*a*c]*d + a*e)) - a*(a*Sqrt[b
^2 - 4*a*c]*e^2 - c*d*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1287

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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

Mathematica [A]  time = 2.87339, size = 587, normalized size = 0.97 \[ \frac{-\frac{2 a^2 d \sqrt{d+e x}}{x^2}+3 a^2 e \left (\frac{\sqrt{d+e x}}{x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}\right )-\frac{8 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-2 a b d e+a \left (a e^2-c d^2\right )+b^2 d^2\right )}{\sqrt{d}}+\frac{4 \sqrt{2} \sqrt{c} \left (a b \left (e \left (a e-2 d \sqrt{b^2-4 a c}\right )-3 c d^2\right )+a \left (c d \left (4 a e-d \sqrt{b^2-4 a c}\right )+a e^2 \sqrt{b^2-4 a c}\right )+b^2 d \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{4 \sqrt{2} \sqrt{c} \left (a b \left (e \left (2 d \sqrt{b^2-4 a c}+a e\right )-3 c d^2\right )+a \left (c d \left (d \sqrt{b^2-4 a c}+4 a e\right )-a e^2 \sqrt{b^2-4 a c}\right )-b^2 d \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{4 a \sqrt{d+e x} (b d-2 a e)}{x}+\frac{4 a e (2 a e-b d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.304, size = 1880, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*e/a^2*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*b-3*e/a^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^
2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*
b*d^2+e/a^3*c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(
1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3*d^2-2*e^2/a^2*c/(-e^2*(4*a*c-b^2))^(1/2)*2^
(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*
c-b^2))^(1/2))*c)^(1/2))*b^2*d-3*e/a^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1
/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^2+e/a^3*c/(
-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2
)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3*d^2-2*e^2/a^2*c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-
2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*
c)^(1/2))*b^2*d+2/a^2*d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c-2/a^3*d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*b^
2+1/a^3*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d
+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d^2-1/a^3*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*ar
ctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d^2+2*e/a^2*c*2^(1/2)/((b*e-2
*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c
)^(1/2))*b*d+4*e^2/a*c^2/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arct
anh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d+e^3/a*c/(-e^2*(4*a*c-b^2))^(1/2
)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(
4*a*c-b^2))^(1/2))*c)^(1/2))*b+e^3/a*c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*
c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b+4*e^2/a*c^2/(-e^2*(4
*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2
*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d-2*e/a^2*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*a
rctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d-5/4/a/x^2*(e*x+d)^(3/2)+1/
a^2*c^2*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^
2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2-1/a^2*c^2*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(
(e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2+1/e/a^2/x^2*(e*x+d)^(3/2)*b*d-e^2
/a*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(
4*a*c-b^2))^(1/2))*c)^(1/2))+e^2/a*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(
1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/e/a^2/x^2*(e*x+d)^(1/2)*b*d^2-3/4*e^2*arctan
h((e*x+d)^(1/2)/d^(1/2))/a/d^(1/2)+3/4*d*(e*x+d)^(1/2)/a/x^2

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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)^(3/2)/x^3/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out