3.41 \(\int \frac{\sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx\)

Optimal. Leaf size=288 \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{8 c^{3/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (2 x (a d+2 b c)+a e)}{4 c x \left (a+b x^2\right )}+\frac{b e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{d} \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.784178, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {6744, 1650, 812, 843, 621, 206, 724} \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{8 c^{3/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} (2 x (a d+2 b c)+a e)}{4 c x \left (a+b x^2\right )}+\frac{b e \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{d} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^3,x]

[Out]

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

Rule 6744

Int[(u_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[Sqrt[a + b*x^n + c*x^(2*n)]/((4
*c)^(p - 1/2)*(b + 2*c*x^n)), Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] &
& EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 724

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

Rubi steps

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

Mathematica [A]  time = 0.358819, size = 177, normalized size = 0.61 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (4 b c^{3/2} e x^2 \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right )-\sqrt{d} \left (x^2 \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right )+2 \sqrt{c} \sqrt{c+x (d x+e)} \left (2 a c+a e x-4 b c x^2\right )\right )\right )}{8 c^{3/2} \sqrt{d} x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/x^3,x]

[Out]

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

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Maple [A]  time = 0.01, size = 329, normalized size = 1.1 \begin{align*}{\frac{1}{8\,{x}^{2}{c}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( -4\,{d}^{5/2}{c}^{3/2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{{x}^{2}d+ex+c}}{x}} \right ){x}^{2}a-8\,{d}^{3/2}{c}^{5/2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{{x}^{2}d+ex+c}}{x}} \right ){x}^{2}b-2\,{d}^{5/2}\sqrt{{x}^{2}d+ex+c}{x}^{3}ae+4\,{d}^{5/2}\sqrt{{x}^{2}d+ex+c}{x}^{2}ac+{d}^{{\frac{3}{2}}}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+ex+2\,\sqrt{c}\sqrt{{x}^{2}d+ex+c} \right ) } \right ){x}^{2}a{e}^{2}+2\,{d}^{3/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}xae-2\,{d}^{3/2}\sqrt{{x}^{2}d+ex+c}{x}^{2}a{e}^{2}+8\,{d}^{3/2}\sqrt{{x}^{2}d+ex+c}{x}^{2}b{c}^{2}+4\,\ln \left ( 1/2\,{\frac{2\,\sqrt{{x}^{2}d+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) d{x}^{2}b{c}^{2}e-4\,{d}^{3/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}ac \right ){d}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 5.94203, size = 1766, normalized size = 6.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(4*b*c^2*sqrt(d)*e*x^2*log(8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(d) + 4*c*d + e
^2) - (8*b*c^2*d + 4*a*c*d^2 - a*d*e^2)*sqrt(c)*x^2*log((8*c*e*x + (4*c*d + e^2)*x^2 + 4*sqrt(d*x^2 + e*x + c)
*(e*x + 2*c)*sqrt(c) + 8*c^2)/x^2) + 4*(4*b*c^2*d*x^2 - a*c*d*e*x - 2*a*c^2*d)*sqrt(d*x^2 + e*x + c))/(c^2*d*x
^2), -1/16*(8*b*c^2*sqrt(-d)*e*x^2*arctan(1/2*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(-d)/(d^2*x^2 + d*e*x + c*
d)) + (8*b*c^2*d + 4*a*c*d^2 - a*d*e^2)*sqrt(c)*x^2*log((8*c*e*x + (4*c*d + e^2)*x^2 + 4*sqrt(d*x^2 + e*x + c)
*(e*x + 2*c)*sqrt(c) + 8*c^2)/x^2) - 4*(4*b*c^2*d*x^2 - a*c*d*e*x - 2*a*c^2*d)*sqrt(d*x^2 + e*x + c))/(c^2*d*x
^2), 1/8*(2*b*c^2*sqrt(d)*e*x^2*log(8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(d) + 4*c*d
+ e^2) + (8*b*c^2*d + 4*a*c*d^2 - a*d*e^2)*sqrt(-c)*x^2*arctan(1/2*sqrt(d*x^2 + e*x + c)*(e*x + 2*c)*sqrt(-c)/
(c*d*x^2 + c*e*x + c^2)) + 2*(4*b*c^2*d*x^2 - a*c*d*e*x - 2*a*c^2*d)*sqrt(d*x^2 + e*x + c))/(c^2*d*x^2), -1/8*
(4*b*c^2*sqrt(-d)*e*x^2*arctan(1/2*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(-d)/(d^2*x^2 + d*e*x + c*d)) - (8*b*
c^2*d + 4*a*c*d^2 - a*d*e^2)*sqrt(-c)*x^2*arctan(1/2*sqrt(d*x^2 + e*x + c)*(e*x + 2*c)*sqrt(-c)/(c*d*x^2 + c*e
*x + c^2)) - 2*(4*b*c^2*d*x^2 - a*c*d*e*x - 2*a*c^2*d)*sqrt(d*x^2 + e*x + c))/(c^2*d*x^2)]

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

[Out]

Timed out

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Giac [A]  time = 1.23649, size = 500, normalized size = 1.74 \begin{align*} -\frac{b e \log \left ({\left | -2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} d - \sqrt{d} e \right |}\right ) \mathrm{sgn}\left (b x^{2} + a\right )}{2 \, \sqrt{d}} + \sqrt{d x^{2} + x e + c} b \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{{\left (8 \, b c^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 4 \, a c d \mathrm{sgn}\left (b x^{2} + a\right ) - a e^{2} \mathrm{sgn}\left (b x^{2} + a\right )\right )} \arctan \left (-\frac{\sqrt{d} x - \sqrt{d x^{2} + x e + c}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c} c} + \frac{4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )}^{3} a c d \mathrm{sgn}\left (b x^{2} + a\right ) + 8 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )}^{2} a c \sqrt{d} e \mathrm{sgn}\left (b x^{2} + a\right ) + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} a c^{2} d \mathrm{sgn}\left (b x^{2} + a\right ) +{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )}^{3} a e^{2} \mathrm{sgn}\left (b x^{2} + a\right ) +{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} a c e^{2} \mathrm{sgn}\left (b x^{2} + a\right )}{4 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )}^{2} - c\right )}^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*e*log(abs(-2*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))*d - sqrt(d)*e))*sgn(b*x^2 + a)/sqrt(d) + sqrt(d*x^2 +
x*e + c)*b*sgn(b*x^2 + a) + 1/4*(8*b*c^2*sgn(b*x^2 + a) + 4*a*c*d*sgn(b*x^2 + a) - a*e^2*sgn(b*x^2 + a))*arcta
n(-(sqrt(d)*x - sqrt(d*x^2 + x*e + c))/sqrt(-c))/(sqrt(-c)*c) + 1/4*(4*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^3*a
*c*d*sgn(b*x^2 + a) + 8*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^2*a*c*sqrt(d)*e*sgn(b*x^2 + a) + 4*(sqrt(d)*x - sq
rt(d*x^2 + x*e + c))*a*c^2*d*sgn(b*x^2 + a) + (sqrt(d)*x - sqrt(d*x^2 + x*e + c))^3*a*e^2*sgn(b*x^2 + a) + (sq
rt(d)*x - sqrt(d*x^2 + x*e + c))*a*c*e^2*sgn(b*x^2 + a))/(((sqrt(d)*x - sqrt(d*x^2 + x*e + c))^2 - c)^2*c)