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

[Out]

-1/3*a*(d*x^2+e*x+c)^(3/2)*((b*x^2+a)^2)^(1/2)/c/x^3/(b*x^2+a)-1/16*e*(8*b*c^2-a*(4*c*d-e^2))*arctanh(1/2*(e*x
+2*c)/c^(1/2)/(d*x^2+e*x+c)^(1/2))*((b*x^2+a)^2)^(1/2)/c^(5/2)/(b*x^2+a)+b*arctanh(1/2*(2*d*x+e)/d^(1/2)/(d*x^
2+e*x+c)^(1/2))*d^(1/2)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+1/8*(2*a*c*e-(-a*e^2+8*b*c^2)*x)*(d*x^2+e*x+c)^(1/2)*((b
*x^2+a)^2)^(1/2)/c^2/x^2/(b*x^2+a)

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Rubi [A]  time = 0.79, antiderivative size = 294, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.29, size = 189, normalized size = 0.64 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (-3 e x^3 \left (a \left (e^2-4 c d\right )+8 b c^2\right ) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+x (d x+e)}}\right )-2 \sqrt {c} \sqrt {c+x (d x+e)} \left (a \left (8 c^2+2 c x (4 d x+e)-3 e^2 x^2\right )+24 b c^2 x^2\right )+48 b c^{5/2} \sqrt {d} x^3 \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )\right )}{48 c^{5/2} x^3 \left (a+b x^2\right )} \]

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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fricas [A]  time = 2.05, size = 791, normalized size = 2.69 \[ \left [\frac {48 \, b c^{3} \sqrt {d} x^{3} \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 3 \, {\left (a e^{3} + 4 \, {\left (2 \, b c^{2} - a c d\right )} e\right )} \sqrt {c} x^{3} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - 4 \, {\left (2 \, a c^{2} e x + 8 \, a c^{3} + {\left (24 \, b c^{3} + 8 \, a c^{2} d - 3 \, a c e^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + e x + c}}{96 \, c^{3} x^{3}}, -\frac {96 \, b c^{3} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) - 3 \, {\left (a e^{3} + 4 \, {\left (2 \, b c^{2} - a c d\right )} e\right )} \sqrt {c} x^{3} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) + 4 \, {\left (2 \, a c^{2} e x + 8 \, a c^{3} + {\left (24 \, b c^{3} + 8 \, a c^{2} d - 3 \, a c e^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + e x + c}}{96 \, c^{3} x^{3}}, \frac {24 \, b c^{3} \sqrt {d} x^{3} \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 3 \, {\left (a e^{3} + 4 \, {\left (2 \, b c^{2} - a c d\right )} e\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - 2 \, {\left (2 \, a c^{2} e x + 8 \, a c^{3} + {\left (24 \, b c^{3} + 8 \, a c^{2} d - 3 \, a c e^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + e x + c}}{48 \, c^{3} x^{3}}, -\frac {48 \, b c^{3} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) - 3 \, {\left (a e^{3} + 4 \, {\left (2 \, b c^{2} - a c d\right )} e\right )} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) + 2 \, {\left (2 \, a c^{2} e x + 8 \, a c^{3} + {\left (24 \, b c^{3} + 8 \, a c^{2} d - 3 \, a c e^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + e x + c}}{48 \, c^{3} x^{3}}\right ] \]

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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giac [B]  time = 1.34, size = 720, normalized size = 2.45 \[ -b \sqrt {d} \log \left ({\left | 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} \sqrt {d} + e \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {{\left (8 \, b c^{2} e \mathrm {sgn}\left (b x^{2} + a\right ) - 4 \, a c d e \mathrm {sgn}\left (b x^{2} + a\right ) + a e^{3} \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 )}{8 \, \sqrt {-c} c^{2}} + \frac {24 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{5} b c^{2} \sqrt {d} e \mathrm {sgn}\left (b x^{2} + a\right ) + 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{5} a c d^{\frac {3}{2}} e \mathrm {sgn}\left (b x^{2} + a\right ) + 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{4} b c^{3} d \mathrm {sgn}\left (b x^{2} + a\right ) + 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{4} a c^{2} d^{2} \mathrm {sgn}\left (b x^{2} + a\right ) - 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} b c^{3} \sqrt {d} e \mathrm {sgn}\left (b x^{2} + a\right ) + 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} a c^{2} d^{\frac {3}{2}} e \mathrm {sgn}\left (b x^{2} + a\right ) - 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} b c^{4} d \mathrm {sgn}\left (b x^{2} + a\right ) - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{5} a \sqrt {d} e^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 24 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} b c^{4} \sqrt {d} e \mathrm {sgn}\left (b x^{2} + a\right ) + 36 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} a c^{3} d^{\frac {3}{2}} e \mathrm {sgn}\left (b x^{2} + a\right ) + 48 \, b c^{5} d \mathrm {sgn}\left (b x^{2} + a\right ) + 16 \, a c^{4} d^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} a c^{2} d e^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} a c \sqrt {d} e^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} a c^{2} \sqrt {d} e^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{24 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} - c\right )}^{3} c^{2} \sqrt {d}} \]

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))

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maple [A]  time = 0.01, size = 412, normalized size = 1.40 \[ -\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-12 a \,c^{\frac {3}{2}} d^{\frac {5}{2}} e \,x^{3} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )+3 a \sqrt {c}\, d^{\frac {3}{2}} e^{3} x^{3} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )-48 b \,c^{3} d^{2} x^{3} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+24 b \,c^{\frac {5}{2}} d^{\frac {3}{2}} e \,x^{3} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )-6 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}} e^{2} x^{4}-48 \sqrt {d \,x^{2}+e x +c}\, b \,c^{2} d^{\frac {5}{2}} x^{4}+12 \sqrt {d \,x^{2}+e x +c}\, a c \,d^{\frac {5}{2}} e \,x^{3}-6 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {3}{2}} e^{3} x^{3}-48 \sqrt {d \,x^{2}+e x +c}\, b \,c^{2} d^{\frac {3}{2}} e \,x^{3}+6 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {3}{2}} e^{2} x^{2}+48 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,c^{2} d^{\frac {3}{2}} x^{2}-12 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a c \,d^{\frac {3}{2}} e x +16 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,c^{2} d^{\frac {3}{2}}\right )}{48 \left (b \,x^{2}+a \right ) c^{3} d^{\frac {3}{2}} x^{3}} \]

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((e*x+2*c+2*(d*x^2+e*x+c)^(1/2)*c^(1/2))/x)*x^3*a*e+24*d^(3/2
)*c^(5/2)*ln((e*x+2*c+2*(d*x^2+e*x+c)^(1/2)*c^(1/2))/x)*x^3*b*e-6*(d*x^2+e*x+c)^(1/2)*d^(5/2)*x^4*a*e^2-48*(d*
x^2+e*x+c)^(1/2)*d^(5/2)*x^4*b*c^2+12*(d*x^2+e*x+c)^(1/2)*d^(5/2)*x^3*a*c*e+3*d^(3/2)*c^(1/2)*ln((e*x+2*c+2*(d
*x^2+e*x+c)^(1/2)*c^(1/2))/x)*x^3*a*e^3+6*(d*x^2+e*x+c)^(3/2)*d^(3/2)*x^2*a*e^2+48*(d*x^2+e*x+c)^(3/2)*d^(3/2)
*x^2*b*c^2-6*(d*x^2+e*x+c)^(1/2)*d^(3/2)*x^3*a*e^3-48*(d*x^2+e*x+c)^(1/2)*d^(3/2)*x^3*b*c^2*e-12*(d*x^2+e*x+c)
^(3/2)*d^(3/2)*x*a*c*e-48*ln(1/2*(2*d*x+e+2*(d*x^2+e*x+c)^(1/2)*d^(1/2))/d^(1/2))*x^3*b*c^3*d^2+16*(d*x^2+e*x+
c)^(3/2)*d^(3/2)*a*c^2)/d^(3/2)/x^3/c^3/(b*x^2+a)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e^2-4*c*d>0)', see `assume?` f
or more details)Is e^2-4*c*d positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((a + b*x^2)^2)^(1/2)*(c + e*x + d*x^2)^(1/2))/x^4,x)

[Out]

int((((a + b*x^2)^2)^(1/2)*(c + e*x + d*x^2)^(1/2))/x^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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