\(\int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx\) [2372]

Optimal result

Integrand size = 18, antiderivative size = 202 \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}} \]

[Out]

-1/256*b*(240*a^2*c^2-280*a*b^2*c+63*b^4)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)+1/240*(-
64*a*c+63*b^2)*x^2*(c*x^2+b*x+a)^(1/2)/c^3-9/40*b*x^3*(c*x^2+b*x+a)^(1/2)/c^2+1/5*x^4*(c*x^2+b*x+a)^(1/2)/c+1/
1920*(945*b^4-2940*a*b^2*c+1024*a^2*c^2-14*b*c*(-92*a*c+45*b^2)*x)*(c*x^2+b*x+a)^(1/2)/c^5

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {756, 846, 793, 635, 212} \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=-\frac {b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac {\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt {a+b x+c x^2}}{1920 c^5}+\frac {x^2 \left (63 b^2-64 a c\right ) \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c} \]

[In]

Int[x^5/Sqrt[a + b*x + c*x^2],x]

[Out]

((63*b^2 - 64*a*c)*x^2*Sqrt[a + b*x + c*x^2])/(240*c^3) - (9*b*x^3*Sqrt[a + b*x + c*x^2])/(40*c^2) + (x^4*Sqrt
[a + b*x + c*x^2])/(5*c) + ((945*b^4 - 2940*a*b^2*c + 1024*a^2*c^2 - 14*b*c*(45*b^2 - 92*a*c)*x)*Sqrt[a + b*x
+ c*x^2])/(1920*c^5) - (b*(63*b^4 - 280*a*b^2*c + 240*a^2*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c
*x^2])])/(256*c^(11/2))

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 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 793

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

Rule 846

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^3 \left (-4 a-\frac {9 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c} \\ & = -\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^2 \left (\frac {27 a b}{2}+\frac {1}{4} \left (63 b^2-64 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{20 c^2} \\ & = \frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x \left (-\frac {1}{2} a \left (63 b^2-64 a c\right )-\frac {7}{8} b \left (45 b^2-92 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{60 c^3} \\ & = \frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^5} \\ & = \frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right )\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 )}{128 c^5} \\ & = \frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.76 \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^4-630 b^3 c x+8 b c^2 x \left (161 a-54 c x^2\right )+84 b^2 c \left (-35 a+6 c x^2\right )+128 c^2 \left (8 a^2-4 a c x^2+3 c^2 x^4\right )\right )+15 \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{3840 c^{11/2}} \]

[In]

Integrate[x^5/Sqrt[a + b*x + c*x^2],x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^4 - 630*b^3*c*x + 8*b*c^2*x*(161*a - 54*c*x^2) + 84*b^2*c*(-35*a + 6*c
*x^2) + 128*c^2*(8*a^2 - 4*a*c*x^2 + 3*c^2*x^4)) + 15*(63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*Log[b + 2*c*x - 2
*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(3840*c^(11/2))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {\left (384 c^{4} x^{4}-432 b \,c^{3} x^{3}-512 a \,c^{3} x^{2}+504 b^{2} c^{2} x^{2}+1288 a b \,c^{2} x -630 b^{3} c x +1024 a^{2} c^{2}-2940 a \,b^{2} c +945 b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{5}}-\frac {b \left (240 a^{2} c^{2}-280 a \,b^{2} c +63 b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}\)	\(144\)
default	\(\frac {x^{4} \sqrt {c \,x^{2}+b x +a}}{5 c}-\frac {9 b \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{10 c}-\frac {4 a \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{5 c}\)	\(541\)

[In]

int(x^5/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/1920*(384*c^4*x^4-432*b*c^3*x^3-512*a*c^3*x^2+504*b^2*c^2*x^2+1288*a*b*c^2*x-630*b^3*c*x+1024*a^2*c^2-2940*a
*b^2*c+945*b^4)*(c*x^2+b*x+a)^(1/2)/c^5-1/256*b*(240*a^2*c^2-280*a*b^2*c+63*b^4)/c^(11/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.72 \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=\left [\frac {15 \, {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \, {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{6}}, \frac {15 \, {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \, {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{6}}\right ] \]

[In]

integrate(x^5/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/7680*(15*(63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x
+ a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(384*c^5*x^4 - 432*b*c^4*x^3 + 945*b^4*c - 2940*a*b^2*c^2 + 1024*a^2*c^
3 + 8*(63*b^2*c^3 - 64*a*c^4)*x^2 - 14*(45*b^3*c^2 - 92*a*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/3840*(15*(63
*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 +
b*c*x + a*c)) + 2*(384*c^5*x^4 - 432*b*c^4*x^3 + 945*b^4*c - 2940*a*b^2*c^2 + 1024*a^2*c^3 + 8*(63*b^2*c^3 - 6
4*a*c^4)*x^2 - 14*(45*b^3*c^2 - 92*a*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (199) = 398\).

Time = 0.41 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.17 \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \left (- \frac {a \left (\frac {27 a b}{40 c^{2}} - \frac {5 b \left (- \frac {4 a}{5 c} + \frac {63 b^{2}}{80 c^{2}}\right )}{6 c}\right )}{2 c} - \frac {b \left (- \frac {2 a \left (- \frac {4 a}{5 c} + \frac {63 b^{2}}{80 c^{2}}\right )}{3 c} - \frac {3 b \left (\frac {27 a b}{40 c^{2}} - \frac {5 b \left (- \frac {4 a}{5 c} + \frac {63 b^{2}}{80 c^{2}}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {a + b x + c x^{2}} \left (- \frac {9 b x^{3}}{40 c^{2}} + \frac {x^{4}}{5 c} + \frac {x^{2} \left (- \frac {4 a}{5 c} + \frac {63 b^{2}}{80 c^{2}}\right )}{3 c} + \frac {x \left (\frac {27 a b}{40 c^{2}} - \frac {5 b \left (- \frac {4 a}{5 c} + \frac {63 b^{2}}{80 c^{2}}\right )}{6 c}\right )}{2 c} + \frac {- \frac {2 a \left (- \frac {4 a}{5 c} + \frac {63 b^{2}}{80 c^{2}}\right )}{3 c} - \frac {3 b \left (\frac {27 a b}{40 c^{2}} - \frac {5 b \left (- \frac {4 a}{5 c} + \frac {63 b^{2}}{80 c^{2}}\right )}{6 c}\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (- a^{5} \sqrt {a + b x} + \frac {5 a^{4} \left (a + b x\right )^{\frac {3}{2}}}{3} - 2 a^{3} \left (a + b x\right )^{\frac {5}{2}} + \frac {10 a^{2} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {5 a \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a + b x\right )^{\frac {11}{2}}}{11}\right )}{b^{6}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 \sqrt {a}} & \text {otherwise} \end {cases} \]

[In]

integrate(x**5/(c*x**2+b*x+a)**(1/2),x)

[Out]

Piecewise(((-a*(27*a*b/(40*c**2) - 5*b*(-4*a/(5*c) + 63*b**2/(80*c**2))/(6*c))/(2*c) - b*(-2*a*(-4*a/(5*c) + 6
3*b**2/(80*c**2))/(3*c) - 3*b*(27*a*b/(40*c**2) - 5*b*(-4*a/(5*c) + 63*b**2/(80*c**2))/(6*c))/(4*c))/(2*c))*Pi
ecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log
(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)) + sqrt(a + b*x + c*x**2)*(-9*b*x**3/(40*c**2) + x**4/(5*c) + x*
*2*(-4*a/(5*c) + 63*b**2/(80*c**2))/(3*c) + x*(27*a*b/(40*c**2) - 5*b*(-4*a/(5*c) + 63*b**2/(80*c**2))/(6*c))/
(2*c) + (-2*a*(-4*a/(5*c) + 63*b**2/(80*c**2))/(3*c) - 3*b*(27*a*b/(40*c**2) - 5*b*(-4*a/(5*c) + 63*b**2/(80*c
**2))/(6*c))/(4*c))/c), Ne(c, 0)), (2*(-a**5*sqrt(a + b*x) + 5*a**4*(a + b*x)**(3/2)/3 - 2*a**3*(a + b*x)**(5/
2) + 10*a**2*(a + b*x)**(7/2)/7 - 5*a*(a + b*x)**(9/2)/9 + (a + b*x)**(11/2)/11)/b**6, Ne(b, 0)), (x**6/(6*sqr
t(a)), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^5/(c*x^2+b*x+a)^(1/2),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(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79 \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, x {\left (\frac {8 \, x}{c} - \frac {9 \, b}{c^{2}}\right )} + \frac {63 \, b^{2} c^{2} - 64 \, a c^{3}}{c^{5}}\right )} x - \frac {7 \, {\left (45 \, b^{3} c - 92 \, a b c^{2}\right )}}{c^{5}}\right )} x + \frac {945 \, b^{4} - 2940 \, a b^{2} c + 1024 \, a^{2} c^{2}}{c^{5}}\right )} + \frac {{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {11}{2}}} \]

[In]

integrate(x^5/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*x*(8*x/c - 9*b/c^2) + (63*b^2*c^2 - 64*a*c^3)/c^5)*x - 7*(45*b^3*c - 92*
a*b*c^2)/c^5)*x + (945*b^4 - 2940*a*b^2*c + 1024*a^2*c^2)/c^5) + 1/256*(63*b^5 - 280*a*b^3*c + 240*a^2*b*c^2)*
log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(11/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^5}{\sqrt {c\,x^2+b\,x+a}} \,d x \]

[In]

int(x^5/(a + b*x + c*x^2)^(1/2),x)

[Out]

int(x^5/(a + b*x + c*x^2)^(1/2), x)