\(\int \frac {(a x^2+b x^3+c x^4)^{3/2}}{x^7} \, dx\) [46]

Optimal result

Integrand size = 24, antiderivative size = 257 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=\frac {\left (b^2-8 a c+2 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac {b \left (b^2-12 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {c^{3/2} x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}} \]

[Out]

-1/3*(c*x^4+b*x^3+a*x^2)^(3/2)/x^6-1/4*b*(c*x^4+b*x^3+a*x^2)^(3/2)/a/x^5+1/16*b*(-12*a*c+b^2)*x*arctanh(1/2*(b
*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))*(c*x^2+b*x+a)^(1/2)/a^(3/2)/(c*x^4+b*x^3+a*x^2)^(1/2)+c^(3/2)*x*arctanh(1
/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*(c*x^2+b*x+a)^(1/2)/(c*x^4+b*x^3+a*x^2)^(1/2)+1/8*(2*b*c*x-8*a*c+b^2
)*(c*x^4+b*x^3+a*x^2)^(1/2)/a/x^2

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1934, 1965, 1955, 1947, 857, 635, 212, 738} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=\frac {b x \left (b^2-12 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {c^{3/2} x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}+\frac {\left (-8 a c+b^2+2 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5} \]

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^7,x]

[Out]

((b^2 - 8*a*c + 2*b*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(8*a*x^2) - (a*x^2 + b*x^3 + c*x^4)^(3/2)/(3*x^6) - (b*(
a*x^2 + b*x^3 + c*x^4)^(3/2))/(4*a*x^5) + (b*(b^2 - 12*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqr
t[a]*Sqrt[a + b*x + c*x^2])])/(16*a^(3/2)*Sqrt[a*x^2 + b*x^3 + c*x^4]) + (c^(3/2)*x*Sqrt[a + b*x + c*x^2]*ArcT
anh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[a*x^2 + b*x^3 + c*x^4]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 738

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 857

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 1934

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 1947

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]

Rule 1955

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

Rule 1965

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}+\frac {1}{2} \int \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx \\ & = -\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}-\frac {\int \frac {\left (\frac {1}{2} \left (b^2-8 a c\right )-b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx}{4 a} \\ & = \frac {\left (b^2-8 a c+2 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac {\int \frac {-\frac {1}{2} b \left (b^2-12 a c\right )+8 a c^2 x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 a} \\ & = \frac {\left (b^2-8 a c+2 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac {\left (x \sqrt {a+b x+c x^2}\right ) \int \frac {-\frac {1}{2} b \left (b^2-12 a c\right )+8 a c^2 x}{x \sqrt {a+b x+c x^2}} \, dx}{8 a \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {\left (b^2-8 a c+2 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac {\left (c^2 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 {\left (b \left (b^2-12 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{16 a \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {\left (b^2-8 a c+2 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac {\left (2 c^2 x \sqrt {a+b x+c x^2}\right ) \text {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 {\left (b \left (b^2-12 a c\right ) x \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{8 a \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {\left (b^2-8 a c+2 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{8 a x^2}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{3 x^6}-\frac {b \left (a x^2+b x^3+c x^4\right )^{3/2}}{4 a x^5}+\frac {b \left (b^2-12 a c\right ) 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 )}{16 a^{3/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {c^{3/2} 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*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {\sqrt {x^2 (a+x (b+c x))} \left (3 b \left (b^2-12 a c\right ) x^3 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+\sqrt {a} \left (\sqrt {a+x (b+c x)} \left (8 a^2+3 b^2 x^2+2 a x (7 b+16 c x)\right )+24 a c^{3/2} x^3 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )\right )}{24 a^{3/2} x^4 \sqrt {a+x (b+c x)}} \]

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x^7,x]

[Out]

-1/24*(Sqrt[x^2*(a + x*(b + c*x))]*(3*b*(b^2 - 12*a*c)*x^3*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]
] + Sqrt[a]*(Sqrt[a + x*(b + c*x)]*(8*a^2 + 3*b^2*x^2 + 2*a*x*(7*b + 16*c*x)) + 24*a*c^(3/2)*x^3*Log[b + 2*c*x
- 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])))/(a^(3/2)*x^4*Sqrt[a + x*(b + c*x)])

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(-\frac {3 \left (b \,x^{3} \left (a c -\frac {b^{2}}{12}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )-\frac {4 c^{\frac {3}{2}} a^{\frac {3}{2}} \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) x^{3}}{3}+\left (\frac {7 x \left (\frac {16 c x}{7}+b \right ) a^{\frac {3}{2}}}{9}+\frac {\sqrt {a}\, b^{2} x^{2}}{6}+\frac {4 a^{\frac {5}{2}}}{9}\right ) \sqrt {c \,x^{2}+b x +a}-\ln \left (2\right ) \left (a c -\frac {b^{2}}{12}\right ) x^{3} b \right )}{4 a^{\frac {3}{2}} x^{3}}\)	\(150\)
risch	\(-\frac {\left (32 a c \,x^{2}+3 b^{2} x^{2}+14 a b x +8 a^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{24 x^{4} a}+\frac {\left (16 a \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {b \left (12 a c -b^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {a}}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{16 a x \sqrt {c \,x^{2}+b x +a}}\)	\(165\)
default	\(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (32 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,x^{4}-36 c^{\frac {5}{2}} a^{\frac {5}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b \,x^{3}+48 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} x^{4}-2 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} x^{4}-32 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,x^{2}+28 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,x^{3}-6 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x^{4}+3 c^{\frac {3}{2}} a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{3} x^{3}+60 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,x^{3}+2 c^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} x^{2}-2 c^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} x^{3}+4 c^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a b x -6 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x^{3}-16 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} c^{\frac {3}{2}}+48 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{3} c^{3} x^{3}\right )}{48 x^{6} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{3} c^{\frac {3}{2}}}\)	\(435\)

[In]

int((c*x^4+b*x^3+a*x^2)^(3/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

-3/4*(b*x^3*(a*c-1/12*b^2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x/a^(1/2))-4/3*c^(3/2)*a^(3/2)*ln(2*(c*x
^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)*x^3+(7/9*x*(16/7*c*x+b)*a^(3/2)+1/6*a^(1/2)*b^2*x^2+4/9*a^(5/2))*(c*x^2+b*x+a
)^(1/2)-ln(2)*(a*c-1/12*b^2)*x^3*b)/a^(3/2)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 815, normalized size of antiderivative = 3.17 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=\left [\frac {48 \, a^{2} c^{\frac {3}{2}} x^{4} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{4} \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}} {\left (14 \, a^{2} b x + 8 \, a^{3} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{2}\right )}}{96 \, a^{2} x^{4}}, -\frac {96 \, a^{2} \sqrt {-c} c x^{4} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{4} \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}} {\left (14 \, a^{2} b x + 8 \, a^{3} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{2}\right )}}{96 \, a^{2} x^{4}}, \frac {24 \, a^{2} c^{\frac {3}{2}} x^{4} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{4} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (14 \, a^{2} b x + 8 \, a^{3} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{2}\right )}}{48 \, a^{2} x^{4}}, -\frac {48 \, a^{2} \sqrt {-c} c x^{4} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{4} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (14 \, a^{2} b x + 8 \, a^{3} + {\left (3 \, a b^{2} + 32 \, a^{2} c\right )} x^{2}\right )}}{48 \, a^{2} x^{4}}\right ] \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/96*(48*a^2*c^(3/2)*x^4*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) - 3*(b^3 - 12*a*b*c)*sqrt(a)*x^4*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)*(14*a^2*b*x + 8*a^3 + (3*a*b^2 +
32*a^2*c)*x^2))/(a^2*x^4), -1/96*(96*a^2*sqrt(-c)*c*x^4*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqr
t(-c)/(c^2*x^3 + b*c*x^2 + a*c*x)) + 3*(b^3 - 12*a*b*c)*sqrt(a)*x^4*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)*(14*a^2*b*x + 8*
a^3 + (3*a*b^2 + 32*a^2*c)*x^2))/(a^2*x^4), 1/48*(24*a^2*c^(3/2)*x^4*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) - 3*(b^3 - 12*a*b*c)*sqrt(-a)*x^4*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*sqrt(c*x^4 + b*x^3 + a*x^2)*(14*
a^2*b*x + 8*a^3 + (3*a*b^2 + 32*a^2*c)*x^2))/(a^2*x^4), -1/48*(48*a^2*sqrt(-c)*c*x^4*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)) + 3*(b^3 - 12*a*b*c)*sqrt(-a)*x^4*arctan(1/2*s
qrt(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*sqrt(c*x^4 + b*x^3 + a*x^2)*(
14*a^2*b*x + 8*a^3 + (3*a*b^2 + 32*a^2*c)*x^2))/(a^2*x^4)]

Sympy [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{7}}\, dx \]

[In]

integrate((c*x**4+b*x**3+a*x**2)**(3/2)/x**7,x)

[Out]

Integral((x**2*(a + b*x + c*x**2))**(3/2)/x**7, x)

Maxima [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{7}} \,d x } \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x^7,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^3 + a*x^2)^(3/2)/x^7, x)

Giac [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{7}} \,d x } \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x^7,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x^7} \,d x \]

[In]

int((a*x^2 + b*x^3 + c*x^4)^(3/2)/x^7,x)

[Out]

int((a*x^2 + b*x^3 + c*x^4)^(3/2)/x^7, x)