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

Optimal. Leaf size=173 \[ -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}} \]

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Rubi [A]  time = 0.12, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1920, 1933, 843, 621, 206, 724} \begin {gather*} -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 1933

Int[((A_) + (B_.)*(x_)^(j_.))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[
(x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))])/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)], Int[(A + B*x^(n - q))/(
x^(q/2)*Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; FreeQ[{a, b, c, A, B, n, q}, x] && EqQ[j, n - q] &
& EqQ[r, 2*n - q] && PosQ[n - q] && EqQ[n, 3] && EqQ[q, 2]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 131, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {a+x (b+c x)} \left (b x \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {a} \left (\sqrt {a+x (b+c x)}-\sqrt {c} x \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )}{2 \sqrt {a} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[a + x*(b + c*x)]*(b*x*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[a]*(Sqrt[a +
x*(b + c*x)] - Sqrt[c]*x*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/(Sqrt[a]*Sqrt[x^2*(a + x*(b
+ c*x))])

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IntegrateAlgebraic [A]  time = 0.41, size = 128, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\sqrt {c} \log \left (-2 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}+b x+2 c x^2\right )+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {c} x^2-\sqrt {a x^2+b x^3+c x^4}}\right )}{\sqrt {a}}+\sqrt {c} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.43, size = 653, normalized size = 3.77 \begin {gather*} \left [\frac {2 \, a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + \sqrt {a} b x^{2} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, -\frac {4 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - \sqrt {a} b x^{2} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \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 ) + a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \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 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*a*sqrt(c)*x^2*log(-(8*c^2*x^3 + 8*b*c*x^2 + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(c) + (b^2 +
4*a*c)*x)/x) + sqrt(a)*b*x^2*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*sqrt(c*x^4 + b*x^3 + a*x^2)*a)/(a*x^2), -1/4*(4*a*sqrt(-c)*x^2*arctan(1/2*sqrt(c*x
^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(-c)/(c^2*x^3 + b*c*x^2 + a*c*x)) - sqrt(a)*b*x^2*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*sqrt(c*x^4 + b*x^3 + a*x^2
)*a)/(a*x^2), 1/2*(sqrt(-a)*b*x^2*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)) + a*sqrt(c)*x^2*log(-(8*c^2*x^3 + 8*b*c*x^2 + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(c) +
(b^2 + 4*a*c)*x)/x) - 2*sqrt(c*x^4 + b*x^3 + a*x^2)*a)/(a*x^2), 1/2*(sqrt(-a)*b*x^2*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*a*sqrt(-c)*x^2*arctan(1/2*sqrt(c*x^4 + b*x
^3 + a*x^2)*(2*c*x + b)*sqrt(-c)/(c^2*x^3 + b*c*x^2 + a*c*x)) - 2*sqrt(c*x^4 + b*x^3 + a*x^2)*a)/(a*x^2)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, choosing root of [1,0,%%%{-2,[1,0,0,2]%%%}+%%%{-2,[0,1,0,1]%%%}+%%%{-4,[0,0,1,0]%%%},0,%%%{1,[2,
0,0,4]%%%}+%%%{2,[1,1,0,3]%%%}+%%%{1,[0,2,0,2]%%%}] at parameters values [-97,-82,63.4443001123,-27]Warning, c
hoosing root of [1,0,%%%{-2,[1,0,0,2]%%%}+%%%{-2,[0,1,0,1]%%%}+%%%{-4,[0,0,1,0]%%%},0,%%%{1,[2,0,0,4]%%%}+%%%{
2,[1,1,0,3]%%%}+%%%{1,[0,2,0,2]%%%}] at parameters values [63,-49,35.2935628123,-64]Warning, choosing root of
[1,0,%%%{-2,[2,1,0,0]%%%}+%%%{-2,[1,0,1,0]%%%}+%%%{-4,[0,0,0,1]%%%},0,%%%{1,[4,2,0,0]%%%}+%%%{2,[3,1,1,0]%%%}+
%%%{1,[2,0,2,0]%%%}] at parameters values [22,42,56,43.9628838282]Sign error (%%%{b-2*sqrt(a)*sqrt(c),0%%%}+%%
%{-(-2*a*c+b*sqrt(a)*sqrt(c))/a,1%%%}+%%%{-(4*a*c*sqrt(a)*sqrt(c)-b^2*sqrt(a)*sqrt(c))/(4*a^2),2%%%}+%%%{undef
,3%%%})Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

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maple [A]  time = 0.01, size = 173, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (-2 a \,c^{2} x \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+\sqrt {a}\, b \,c^{\frac {3}{2}} x \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-2 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}} x^{2}-2 \sqrt {c \,x^{2}+b x +a}\, b \,c^{\frac {3}{2}} x +2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {3}{2}}\right )}{2 \sqrt {c \,x^{2}+b x +a}\, a \,c^{\frac {3}{2}} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2*(a + b*x + c*x**2))/x**3, x)

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