3.37 \(\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^6} \, dx\)

Optimal. Leaf size=205 \[ -\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5} \]

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(4*x^5) - (b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*a*x^4) + ((5*b^2 - 12*a*c)*Sqrt[a*x
^2 + b*x^3 + c*x^4])/(96*a^2*x^3) - (b*(15*b^2 - 52*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(192*a^3*x^2) + ((b^2 -
4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(7/2))

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Rubi [A]  time = 0.385848, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1920, 1951, 12, 1904, 206} \[ -\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^6,x]

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(4*x^5) - (b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*a*x^4) + ((5*b^2 - 12*a*c)*Sqrt[a*x
^2 + b*x^3 + c*x^4])/(96*a^2*x^3) - (b*(15*b^2 - 52*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(192*a^3*x^2) + ((b^2 -
4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(7/2))

Rule 1920

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a*
x^q + b*x^n + c*x^(2*n - q))^p)/(m + p*q + 1), x] - Dist[((n - q)*p)/(m + p*q + 1), Int[x^(m + n)*(b + 2*c*x^(
n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n -
q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && LeQ[m + p*q + 1, -
(n - q) + 1] && NeQ[m + p*q + 1, 0]

Rule 1951

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_.)*((A_) + (B_.)*(x_)^(r_.)), x_Sym
bol] :> Simp[(A*x^(m - q + 1)*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1))/(a*(m + p*q + 1)), x] + Dist[1/(a*(m +
p*q + 1)), Int[x^(m + n - q)*Simp[a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(p + 1) + 1) - A*c*(m + p*q + 2*(
n - q)*(p + 1) + 1)*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c, A, B}, x] && Eq
Q[r, n - q] && EqQ[j, 2*n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && RationalQ[m, p, q] &&
((GeQ[p, -1] && LtQ[p, 0]) || EqQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*
q + 1, 0]

Rule 12

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

Rule 1904

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^6} \, dx &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}+\frac{1}{8} \int \frac{b+2 c x}{x^3 \sqrt{a x^2+b x^3+c x^4}} \, dx\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\int \frac{\frac{1}{2} \left (5 b^2-12 a c\right )+2 b c x}{x^2 \sqrt{a x^2+b x^3+c x^4}} \, dx}{24 a}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}+\frac{\int \frac{\frac{1}{4} b \left (15 b^2-52 a c\right )+\frac{1}{2} c \left (5 b^2-12 a c\right ) x}{x \sqrt{a x^2+b x^3+c x^4}} \, dx}{48 a^2}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}-\frac{\int \frac{3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )}{8 \sqrt{a x^2+b x^3+c x^4}} \, dx}{48 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{128 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x (2 a+b x)}{\sqrt{a x^2+b x^3+c x^4}}\right )}{64 a^3}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.225157, size = 160, normalized size = 0.78 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (3 x^4 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (8 a^2 x (b+3 c x)+48 a^3-2 a b x^2 (5 b+26 c x)+15 b^3 x^3\right )\right )}{384 a^{7/2} x^5 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^6,x]

[Out]

(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(48*a^3 + 15*b^3*x^3 + 8*a^2*x*(b + 3*c*x) - 2*
a*b*x^2*(5*b + 26*c*x)) + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^4*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b
+ c*x)])]))/(384*a^(7/2)*x^5*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.006, size = 387, normalized size = 1.9 \begin{align*}{\frac{1}{384\,{x}^{5}{a}^{4}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 48\,{c}^{2}{a}^{5/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}+24\,{c}^{2}\sqrt{c{x}^{2}+bx+a}{x}^{5}ab-72\,c{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{2}-48\,{c}^{2}\sqrt{c{x}^{2}+bx+a}{x}^{4}{a}^{2}-30\,c\sqrt{c{x}^{2}+bx+a}{x}^{5}{b}^{3}-24\,c \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}ab+84\,c\sqrt{c{x}^{2}+bx+a}{x}^{4}a{b}^{2}+15\,\sqrt{a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{4}+48\,c \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}{a}^{2}+30\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}{b}^{3}-30\,\sqrt{c{x}^{2}+bx+a}{x}^{4}{b}^{4}-60\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}a{b}^{2}+80\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}x{a}^{2}b-96\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x)

[Out]

1/384*(c*x^4+b*x^3+a*x^2)^(1/2)*(48*c^2*a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^4+24*c^2*(c*x^
2+b*x+a)^(1/2)*x^5*a*b-72*c*a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^4*b^2-48*c^2*(c*x^2+b*x+a)
^(1/2)*x^4*a^2-30*c*(c*x^2+b*x+a)^(1/2)*x^5*b^3-24*c*(c*x^2+b*x+a)^(3/2)*x^3*a*b+84*c*(c*x^2+b*x+a)^(1/2)*x^4*
a*b^2+15*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^4*b^4+48*c*(c*x^2+b*x+a)^(3/2)*x^2*a^2+30*(c*
x^2+b*x+a)^(3/2)*x^3*b^3-30*(c*x^2+b*x+a)^(1/2)*x^4*b^4-60*(c*x^2+b*x+a)^(3/2)*x^2*a*b^2+80*(c*x^2+b*x+a)^(3/2
)*x*a^2*b-96*(c*x^2+b*x+a)^(3/2)*a^3)/x^5/(c*x^2+b*x+a)^(1/2)/a^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + b*x^3 + a*x^2)/x^6, x)

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Fricas [A]  time = 2.12879, size = 757, normalized size = 3.69 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{a} x^{5} \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 4 \,{\left (8 \, a^{3} b x + 48 \, a^{4} +{\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{768 \, a^{4} x^{5}}, -\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-a} x^{5} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \,{\left (8 \, a^{3} b x + 48 \, a^{4} +{\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{384 \, a^{4} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/768*(3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*sqrt(a)*x^5*log(-(8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x + 4*sqrt
(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(a))/x^3) - 4*(8*a^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^3 - 2*(5
*a^2*b^2 - 12*a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^4*x^5), -1/384*(3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*
sqrt(-a)*x^5*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(-a)/(a*c*x^3 + a*b*x^2 + a^2*x)) + 2*(8*a
^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^3 - 2*(5*a^2*b^2 - 12*a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^
4*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x**6,x)

[Out]

Integral(sqrt(x**2*(a + b*x + c*x**2))/x**6, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x, algorithm="giac")

[Out]

Exception raised: TypeError