\(\int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx\) [2371]

Optimal result

Integrand size = 18, antiderivative size = 261 \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=-\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}-\frac {\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}} \]

[Out]

1/1024*(-320*a^3*c^3+1680*a^2*b^2*c^2-1260*a*b^4*c+231*b^6)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))
/c^(13/2)-1/320*b*(-156*a*c+77*b^2)*x^2*(c*x^2+b*x+a)^(1/2)/c^4+1/480*(-100*a*c+99*b^2)*x^3*(c*x^2+b*x+a)^(1/2
)/c^3-11/60*b*x^4*(c*x^2+b*x+a)^(1/2)/c^2+1/6*x^5*(c*x^2+b*x+a)^(1/2)/c-1/2560*(7*b*(528*a^2*c^2-680*a*b^2*c+1
65*b^4)-2*c*(400*a^2*c^2-1176*a*b^2*c+385*b^4)*x)*(c*x^2+b*x+a)^(1/2)/c^6

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {756, 846, 793, 635, 212} \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=-\frac {\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (-320 a^3 c^3+1680 a^2 b^2 c^2-1260 a b^4 c+231 b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}}-\frac {b x^2 \left (77 b^2-156 a c\right ) \sqrt {a+b x+c x^2}}{320 c^4}+\frac {x^3 \left (99 b^2-100 a c\right ) \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c} \]

[In]

Int[x^6/Sqrt[a + b*x + c*x^2],x]

[Out]

-1/320*(b*(77*b^2 - 156*a*c)*x^2*Sqrt[a + b*x + c*x^2])/c^4 + ((99*b^2 - 100*a*c)*x^3*Sqrt[a + b*x + c*x^2])/(
480*c^3) - (11*b*x^4*Sqrt[a + b*x + c*x^2])/(60*c^2) + (x^5*Sqrt[a + b*x + c*x^2])/(6*c) - ((7*b*(165*b^4 - 68
0*a*b^2*c + 528*a^2*c^2) - 2*c*(385*b^4 - 1176*a*b^2*c + 400*a^2*c^2)*x)*Sqrt[a + b*x + c*x^2])/(2560*c^6) + (
(231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2]
)])/(1024*c^(13/2))

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^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x^4 \left (-5 a-\frac {11 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx}{6 c} \\ & = -\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x^3 \left (22 a b+\frac {1}{4} \left (99 b^2-100 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{30 c^2} \\ & = \frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x^2 \left (-\frac {3}{4} a \left (99 b^2-100 a c\right )-\frac {9}{8} b \left (77 b^2-156 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{120 c^3} \\ & = -\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}+\frac {\int \frac {x \left (\frac {9}{4} a b \left (77 b^2-156 a c\right )+\frac {9}{16} \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{360 c^4} \\ & = -\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}-\frac {\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^6} \\ & = -\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}-\frac {\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\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 )}{512 c^6} \\ & = -\frac {b \left (77 b^2-156 a c\right ) x^2 \sqrt {a+b x+c x^2}}{320 c^4}+\frac {\left (99 b^2-100 a c\right ) x^3 \sqrt {a+b x+c x^2}}{480 c^3}-\frac {11 b x^4 \sqrt {a+b x+c x^2}}{60 c^2}+\frac {x^5 \sqrt {a+b x+c x^2}}{6 c}-\frac {\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{2560 c^6}+\frac {\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{13/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.74 \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^5+2310 b^4 c x+168 b^3 c \left (85 a-11 c x^2\right )+144 b^2 c^2 x \left (-49 a+11 c x^2\right )+160 c^3 x \left (15 a^2-10 a c x^2+8 c^2 x^4\right )-16 b c^2 \left (693 a^2-234 a c x^2+88 c^2 x^4\right )\right )+15 \left (-231 b^6+1260 a b^4 c-1680 a^2 b^2 c^2+320 a^3 c^3\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{13/2}} \]

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^5 + 2310*b^4*c*x + 168*b^3*c*(85*a - 11*c*x^2) + 144*b^2*c^2*x*(-49*
a + 11*c*x^2) + 160*c^3*x*(15*a^2 - 10*a*c*x^2 + 8*c^2*x^4) - 16*b*c^2*(693*a^2 - 234*a*c*x^2 + 88*c^2*x^4)) +
15*(-231*b^6 + 1260*a*b^4*c - 1680*a^2*b^2*c^2 + 320*a^3*c^3)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]
])/(15360*c^(13/2))

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {\left (-1280 c^{5} x^{5}+1408 b \,c^{4} x^{4}+1600 a \,c^{4} x^{3}-1584 b^{2} c^{3} x^{3}-3744 a b \,c^{3} x^{2}+1848 b^{3} c^{2} x^{2}-2400 a^{2} c^{3} x +7056 a \,b^{2} c^{2} x -2310 b^{4} c x +11088 a^{2} b \,c^{2}-14280 a \,b^{3} c +3465 b^{5}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{6}}-\frac {\left (320 a^{3} c^{3}-1680 a^{2} b^{2} c^{2}+1260 a \,b^{4} c -231 b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {13}{2}}}\)	\(187\)
default	\(\frac {x^{5} \sqrt {c \,x^{2}+b x +a}}{6 c}-\frac {11 b \left (\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}\right )}{12 c}-\frac {5 a \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 )}{6 c}\)	\(896\)

[In]

int(x^6/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/7680*(-1280*c^5*x^5+1408*b*c^4*x^4+1600*a*c^4*x^3-1584*b^2*c^3*x^3-3744*a*b*c^3*x^2+1848*b^3*c^2*x^2-2400*a
^2*c^3*x+7056*a*b^2*c^2*x-2310*b^4*c*x+11088*a^2*b*c^2-14280*a*b^3*c+3465*b^5)*(c*x^2+b*x+a)^(1/2)/c^6-1/1024*
(320*a^3*c^3-1680*a^2*b^2*c^2+1260*a*b^4*c-231*b^6)/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.65 \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=\left [-\frac {15 \, {\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (1280 \, c^{6} x^{5} - 1408 \, b c^{5} x^{4} - 3465 \, b^{5} c + 14280 \, a b^{3} c^{2} - 11088 \, a^{2} b c^{3} + 16 \, {\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} x^{3} - 24 \, {\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} x^{2} + 6 \, {\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{7}}, -\frac {15 \, {\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (1280 \, c^{6} x^{5} - 1408 \, b c^{5} x^{4} - 3465 \, b^{5} c + 14280 \, a b^{3} c^{2} - 11088 \, a^{2} b c^{3} + 16 \, {\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} x^{3} - 24 \, {\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} x^{2} + 6 \, {\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{7}}\right ] \]

[In]

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

[Out]

[-1/30720*(15*(231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*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*(1280*c^6*x^5 - 1408*b*c^5*x^4 - 3465*b^5*c + 1428
0*a*b^3*c^2 - 11088*a^2*b*c^3 + 16*(99*b^2*c^4 - 100*a*c^5)*x^3 - 24*(77*b^3*c^3 - 156*a*b*c^4)*x^2 + 6*(385*b
^4*c^2 - 1176*a*b^2*c^3 + 400*a^2*c^4)*x)*sqrt(c*x^2 + b*x + a))/c^7, -1/15360*(15*(231*b^6 - 1260*a*b^4*c + 1
680*a^2*b^2*c^2 - 320*a^3*c^3)*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*(1280*c^6*x^5 - 1408*b*c^5*x^4 - 3465*b^5*c + 14280*a*b^3*c^2 - 11088*a^2*b*c^3 + 16*(99*b^2*c^4
- 100*a*c^5)*x^3 - 24*(77*b^3*c^3 - 156*a*b*c^4)*x^2 + 6*(385*b^4*c^2 - 1176*a*b^2*c^3 + 400*a^2*c^4)*x)*sqrt(
c*x^2 + b*x + a))/c^7]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (257) = 514\).

Time = 0.46 (sec) , antiderivative size = 656, normalized size of antiderivative = 2.51 \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=\begin {cases} \left (- \frac {a \left (- \frac {3 a \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{4 c} - \frac {5 b \left (\frac {11 a b}{15 c^{2}} - \frac {7 b \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{8 c}\right )}{6 c}\right )}{2 c} - \frac {b \left (- \frac {2 a \left (\frac {11 a b}{15 c^{2}} - \frac {7 b \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{8 c}\right )}{3 c} - \frac {3 b \left (- \frac {3 a \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{4 c} - \frac {5 b \left (\frac {11 a b}{15 c^{2}} - \frac {7 b \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{8 c}\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 {11 b x^{4}}{60 c^{2}} + \frac {x^{5}}{6 c} + \frac {x^{3} \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{4 c} + \frac {x^{2} \cdot \left (\frac {11 a b}{15 c^{2}} - \frac {7 b \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{8 c}\right )}{3 c} + \frac {x \left (- \frac {3 a \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{4 c} - \frac {5 b \left (\frac {11 a b}{15 c^{2}} - \frac {7 b \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{8 c}\right )}{6 c}\right )}{2 c} + \frac {- \frac {2 a \left (\frac {11 a b}{15 c^{2}} - \frac {7 b \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{8 c}\right )}{3 c} - \frac {3 b \left (- \frac {3 a \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{4 c} - \frac {5 b \left (\frac {11 a b}{15 c^{2}} - \frac {7 b \left (- \frac {5 a}{6 c} + \frac {33 b^{2}}{40 c^{2}}\right )}{8 c}\right )}{6 c}\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (a^{6} \sqrt {a + b x} - 2 a^{5} \left (a + b x\right )^{\frac {3}{2}} + 3 a^{4} \left (a + b x\right )^{\frac {5}{2}} - \frac {20 a^{3} \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {5 a^{2} \left (a + b x\right )^{\frac {9}{2}}}{3} - \frac {6 a \left (a + b x\right )^{\frac {11}{2}}}{11} + \frac {\left (a + b x\right )^{\frac {13}{2}}}{13}\right )}{b^{7}} & \text {for}\: b \neq 0 \\\frac {x^{7}}{7 \sqrt {a}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((-a*(-3*a*(-5*a/(6*c) + 33*b**2/(40*c**2))/(4*c) - 5*b*(11*a*b/(15*c**2) - 7*b*(-5*a/(6*c) + 33*b**
2/(40*c**2))/(8*c))/(6*c))/(2*c) - b*(-2*a*(11*a*b/(15*c**2) - 7*b*(-5*a/(6*c) + 33*b**2/(40*c**2))/(8*c))/(3*
c) - 3*b*(-3*a*(-5*a/(6*c) + 33*b**2/(40*c**2))/(4*c) - 5*b*(11*a*b/(15*c**2) - 7*b*(-5*a/(6*c) + 33*b**2/(40*
c**2))/(8*c))/(6*c))/(4*c))/(2*c))*Piecewise((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)*(
-11*b*x**4/(60*c**2) + x**5/(6*c) + x**3*(-5*a/(6*c) + 33*b**2/(40*c**2))/(4*c) + x**2*(11*a*b/(15*c**2) - 7*b
*(-5*a/(6*c) + 33*b**2/(40*c**2))/(8*c))/(3*c) + x*(-3*a*(-5*a/(6*c) + 33*b**2/(40*c**2))/(4*c) - 5*b*(11*a*b/
(15*c**2) - 7*b*(-5*a/(6*c) + 33*b**2/(40*c**2))/(8*c))/(6*c))/(2*c) + (-2*a*(11*a*b/(15*c**2) - 7*b*(-5*a/(6*
c) + 33*b**2/(40*c**2))/(8*c))/(3*c) - 3*b*(-3*a*(-5*a/(6*c) + 33*b**2/(40*c**2))/(4*c) - 5*b*(11*a*b/(15*c**2
) - 7*b*(-5*a/(6*c) + 33*b**2/(40*c**2))/(8*c))/(6*c))/(4*c))/c), Ne(c, 0)), (2*(a**6*sqrt(a + b*x) - 2*a**5*(
a + b*x)**(3/2) + 3*a**4*(a + b*x)**(5/2) - 20*a**3*(a + b*x)**(7/2)/7 + 5*a**2*(a + b*x)**(9/2)/3 - 6*a*(a +
b*x)**(11/2)/11 + (a + b*x)**(13/2)/13)/b**7, Ne(b, 0)), (x**7/(7*sqrt(a)), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^6/(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 = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.79 \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, x {\left (\frac {10 \, x}{c} - \frac {11 \, b}{c^{2}}\right )} + \frac {99 \, b^{2} c^{3} - 100 \, a c^{4}}{c^{6}}\right )} x - \frac {3 \, {\left (77 \, b^{3} c^{2} - 156 \, a b c^{3}\right )}}{c^{6}}\right )} x + \frac {3 \, {\left (385 \, b^{4} c - 1176 \, a b^{2} c^{2} + 400 \, a^{2} c^{3}\right )}}{c^{6}}\right )} x - \frac {21 \, {\left (165 \, b^{5} - 680 \, a b^{3} c + 528 \, a^{2} b c^{2}\right )}}{c^{6}}\right )} - \frac {{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {13}{2}}} \]

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*x*(10*x/c - 11*b/c^2) + (99*b^2*c^3 - 100*a*c^4)/c^6)*x - 3*(77*b^3*c
^2 - 156*a*b*c^3)/c^6)*x + 3*(385*b^4*c - 1176*a*b^2*c^2 + 400*a^2*c^3)/c^6)*x - 21*(165*b^5 - 680*a*b^3*c + 5
28*a^2*b*c^2)/c^6) - 1/1024*(231*b^6 - 1260*a*b^4*c + 1680*a^2*b^2*c^2 - 320*a^3*c^3)*log(abs(2*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(13/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {x^6}{\sqrt {c\,x^2+b\,x+a}} \,d x \]

[In]

int(x^6/(a + b*x + c*x^2)^(1/2),x)

[Out]

int(x^6/(a + b*x + c*x^2)^(1/2), x)