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

Optimal. Leaf size=294 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{8 c^2 x^2 \left (a+b x^2\right )}-\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{16 c^{5/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{3 c x^3 \left (a+b x^2\right )}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{a+b x^2} \]

[Out]

((2*a*c*e - (8*b*c^2 - a*e^2)*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*c^2*x^2*(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])/(3*c*x^3*(a + b*x^2)) + (b*Sqrt[d]*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])])/(a + b*x^2) - (e*(8*b*c^2 - a*(4*
c*d - 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])])/(16*c^(5/2
)*(a + b*x^2))

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Rubi [A]  time = 0.786889, antiderivative size = 294, 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, 810, 843, 621, 206, 724} \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \sqrt{c+d x^2+e x} \left (2 a c e-x \left (8 b c^2-a e^2\right )\right )}{8 c^2 x^2 \left (a+b x^2\right )}-\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (8 b c^2-a \left (4 c d-e^2\right )\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{16 c^{5/2} \left (a+b x^2\right )}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{3 c x^3 \left (a+b x^2\right )}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{a+b x^2} \]

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^4,x]

[Out]

((2*a*c*e - (8*b*c^2 - a*e^2)*x)*Sqrt[c + e*x + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*c^2*x^2*(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])/(3*c*x^3*(a + b*x^2)) + (b*Sqrt[d]*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])])/(a + b*x^2) - (e*(8*b*c^2 - a*(4*
c*d - 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])])/(16*c^(5/2
)*(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 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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^4} \, 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^4} \, 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}}{3 c x^3 \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{\left (3 a b e-6 b^2 c x\right ) \sqrt{c+e x+d x^2}}{x^3} \, dx}{3 c \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{\left (2 a c e-\left (8 b c^2-a e^2\right ) x\right ) \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 c^2 x^2 \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}}{3 c x^3 \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{\frac{3}{2} b e \left (8 b c^2-a \left (4 c d-e^2\right )\right )+24 b^2 c^2 d x}{x \sqrt{c+e x+d x^2}} \, dx}{12 c^2 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{\left (2 a c e-\left (8 b c^2-a e^2\right ) x\right ) \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 c^2 x^2 \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}}{3 c x^3 \left (a+b x^2\right )}+\frac{\left (2 b^2 d \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 e \left (8 b c^2-a \left (4 c d-e^2\right )\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}{8 c^2 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{\left (2 a c e-\left (8 b c^2-a e^2\right ) x\right ) \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 c^2 x^2 \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}}{3 c x^3 \left (a+b x^2\right )}+\frac{\left (4 b^2 d \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 e \left (8 b c^2-a \left (4 c d-e^2\right )\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 )}{4 c^2 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{\left (2 a c e-\left (8 b c^2-a e^2\right ) x\right ) \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 c^2 x^2 \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}}{3 c x^3 \left (a+b x^2\right )}+\frac{b \sqrt{d} \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 )}{a+b x^2}-\frac{e \left (8 b c^2-a \left (4 c d-e^2\right )\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 )}{16 c^{5/2} \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.292935, size = 189, normalized size = 0.64 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (-2 \sqrt{c} \sqrt{c+x (d x+e)} \left (a \left (8 c^2+2 c x (4 d x+e)-3 e^2 x^2\right )+24 b c^2 x^2\right )-3 e x^3 \left (a \left (e^2-4 c d\right )+8 b c^2\right ) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+x (d x+e)}}\right )+48 b c^{5/2} \sqrt{d} x^3 \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right )\right )}{48 c^{5/2} x^3 \left (a+b x^2\right )} \]

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^4,x]

[Out]

(Sqrt[(a + b*x^2)^2]*(-2*Sqrt[c]*Sqrt[c + x*(e + d*x)]*(24*b*c^2*x^2 + a*(8*c^2 - 3*e^2*x^2 + 2*c*x*(e + 4*d*x
))) + 48*b*c^(5/2)*Sqrt[d]*x^3*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[c + x*(e + d*x)])] - 3*e*(8*b*c^2 + a*(-4*c
*d + e^2))*x^3*ArcTanh[(2*c + e*x)/(2*Sqrt[c]*Sqrt[c + x*(e + d*x)])]))/(48*c^(5/2)*x^3*(a + b*x^2))

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Maple [A]  time = 0.011, size = 412, normalized size = 1.4 \begin{align*}{\frac{1}{48\,{x}^{3}{c}^{3} \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 12\,{d}^{5/2}{c}^{3/2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{{x}^{2}d+ex+c}}{x}} \right ){x}^{3}ae-24\,{d}^{3/2}{c}^{5/2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{{x}^{2}d+ex+c}}{x}} \right ){x}^{3}be+6\,{d}^{5/2}\sqrt{{x}^{2}d+ex+c}{x}^{4}a{e}^{2}+48\,{d}^{5/2}\sqrt{{x}^{2}d+ex+c}{x}^{4}b{c}^{2}-12\,{d}^{5/2}\sqrt{{x}^{2}d+ex+c}{x}^{3}ace-3\,{d}^{3/2}\sqrt{c}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{{x}^{2}d+ex+c}}{x}} \right ){x}^{3}a{e}^{3}-6\,{d}^{3/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}{x}^{2}a{e}^{2}-48\,{d}^{3/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}{x}^{2}b{c}^{2}+6\,{d}^{3/2}\sqrt{{x}^{2}d+ex+c}{x}^{3}a{e}^{3}+48\,{d}^{3/2}\sqrt{{x}^{2}d+ex+c}{x}^{3}b{c}^{2}e+48\,\ln \left ( 1/2\,{\frac{2\,\sqrt{{x}^{2}d+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){x}^{3}b{c}^{3}{d}^{2}+12\,{d}^{3/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}xace-16\,{d}^{3/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}a{c}^{2} \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^4,x)

[Out]

1/48*((b*x^2+a)^2)^(1/2)*(12*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^3*a*e-24*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^3*b*e+6*d^(5/2)*(d*x^2+e*x+c)^(1/2)*x^4*a*e^2+48*d^(5/
2)*(d*x^2+e*x+c)^(1/2)*x^4*b*c^2-12*d^(5/2)*(d*x^2+e*x+c)^(1/2)*x^3*a*c*e-3*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^3*a*e^3-6*d^(3/2)*(d*x^2+e*x+c)^(3/2)*x^2*a*e^2-48*d^(3/2)*(d*x^2+e*x+c)^(3/2)*x
^2*b*c^2+6*d^(3/2)*(d*x^2+e*x+c)^(1/2)*x^3*a*e^3+48*d^(3/2)*(d*x^2+e*x+c)^(1/2)*x^3*b*c^2*e+48*ln(1/2*(2*(d*x^
2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*x^3*b*c^3*d^2+12*d^(3/2)*(d*x^2+e*x+c)^(3/2)*x*a*c*e-16*d^(3/2)*(d*x^
2+e*x+c)^(3/2)*a*c^2)/d^(3/2)/x^3/c^3/(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^{4}}\,{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^4,x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + e*x + c)*sqrt((b*x^2 + a)^2)/x^4, x)

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Fricas [A]  time = 6.50311, size = 1887, normalized size = 6.42 \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^4,x, algorithm="fricas")

[Out]

[1/96*(48*b*c^3*sqrt(d)*x^3*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) + 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*sqrt(c)*x^3*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*(2*a*c^2*e*x + 8*a*c^3 + (24*b*c^3 + 8*a*c^2*d - 3*a*c*e^2)*x^2)*sqrt(d*
x^2 + e*x + c))/(c^3*x^3), -1/96*(96*b*c^3*sqrt(-d)*x^3*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)) - 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*sqrt(c)*x^3*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*(2*a*c^2*e*x + 8*a*c^3 + (24*b*c^3 + 8*a*c^2*d - 3
*a*c*e^2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^3*x^3), 1/48*(24*b*c^3*sqrt(d)*x^3*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) + 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*sqrt(-c)*x^3*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*(2*a*c^2*e*x + 8*a*c^3 + (24*b*c^3 +
8*a*c^2*d - 3*a*c*e^2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^3*x^3), -1/48*(48*b*c^3*sqrt(-d)*x^3*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)) - 3*(a*e^3 + 4*(2*b*c^2 - a*c*d)*e)*sqrt(-c)*x^3*a
rctan(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*(2*a*c^2*e*x + 8*a*c^3 + (24
*b*c^3 + 8*a*c^2*d - 3*a*c*e^2)*x^2)*sqrt(d*x^2 + e*x + c))/(c^3*x^3)]

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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**4,x)

[Out]

Timed out

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Giac [B]  time = 1.47226, size = 972, normalized size = 3.31 \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^4,x, algorithm="giac")

[Out]

-b*sqrt(d)*log(abs(2*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))*sqrt(d) + e))*sgn(b*x^2 + a) + 1/8*(8*b*c^2*e*sgn(b*x
^2 + a) - 4*a*c*d*e*sgn(b*x^2 + a) + a*e^3*sgn(b*x^2 + a))*arctan(-(sqrt(d)*x - sqrt(d*x^2 + x*e + c))/sqrt(-c
))/(sqrt(-c)*c^2) + 1/24*(24*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^5*b*c^2*sqrt(d)*e*sgn(b*x^2 + a) + 12*(sqrt(d
)*x - sqrt(d*x^2 + x*e + c))^5*a*c*d^(3/2)*e*sgn(b*x^2 + a) + 48*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^4*b*c^3*d
*sgn(b*x^2 + a) + 48*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^4*a*c^2*d^2*sgn(b*x^2 + a) - 48*(sqrt(d)*x - sqrt(d*x
^2 + x*e + c))^3*b*c^3*sqrt(d)*e*sgn(b*x^2 + a) + 48*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^3*a*c^2*d^(3/2)*e*sgn
(b*x^2 + a) - 96*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^2*b*c^4*d*sgn(b*x^2 + a) - 3*(sqrt(d)*x - sqrt(d*x^2 + x*
e + c))^5*a*sqrt(d)*e^3*sgn(b*x^2 + a) + 24*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))*b*c^4*sqrt(d)*e*sgn(b*x^2 + a)
+ 36*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))*a*c^3*d^(3/2)*e*sgn(b*x^2 + a) + 48*b*c^5*d*sgn(b*x^2 + a) + 16*a*c^
4*d^2*sgn(b*x^2 + a) + 48*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))^2*a*c^2*d*e^2*sgn(b*x^2 + a) + 8*(sqrt(d)*x - sq
rt(d*x^2 + x*e + c))^3*a*c*sqrt(d)*e^3*sgn(b*x^2 + a) + 3*(sqrt(d)*x - sqrt(d*x^2 + x*e + c))*a*c^2*sqrt(d)*e^
3*sgn(b*x^2 + a))/(((sqrt(d)*x - sqrt(d*x^2 + x*e + c))^2 - c)^3*c^2*sqrt(d))