### 3.2371 $$\int \frac{x^6}{\sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=261 $-\frac{\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (1680 a^2 b^2 c^2-320 a^3 c^3-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}+\frac{x^3 \left (99 b^2-100 a c\right ) \sqrt{a+b x+c x^2}}{480 c^3}-\frac{b x^2 \left (77 b^2-156 a c\right ) \sqrt{a+b x+c x^2}}{320 c^4}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}$

[Out]

-(b*(77*b^2 - 156*a*c)*x^2*Sqrt[a + b*x + c*x^2])/(320*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))

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Rubi [A]  time = 0.38183, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.278, Rules used = {742, 832, 779, 621, 206} $-\frac{\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (1680 a^2 b^2 c^2-320 a^3 c^3-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}+\frac{x^3 \left (99 b^2-100 a c\right ) \sqrt{a+b x+c x^2}}{480 c^3}-\frac{b x^2 \left (77 b^2-156 a c\right ) \sqrt{a+b x+c x^2}}{320 c^4}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^6/Sqrt[a + b*x + c*x^2],x]

[Out]

-(b*(77*b^2 - 156*a*c)*x^2*Sqrt[a + b*x + c*x^2])/(320*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 742

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 832

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])

Rule 779

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 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 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{x^6}{\sqrt{a+b x+c x^2}} \, dx &=\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 ) \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 )}{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]  time = 0.322678, size = 263, normalized size = 1.01 $\frac{8 a^2 c \left (-2268 b^2 c x+1785 b^3-618 b c^2 x^2+100 c^3 x^3\right )+48 a^3 c^2 (50 c x-231 b)+a \left (5376 b^3 c^2 x^2-1728 b^2 c^3 x^3+16590 b^4 c x-3465 b^5+736 b c^4 x^4-320 c^5 x^5\right )+x \left (462 b^4 c^2 x^2-264 b^3 c^3 x^3+176 b^2 c^4 x^4-1155 b^5 c x-3465 b^6-128 b c^5 x^5+1280 c^6 x^6\right )}{7680 c^6 \sqrt{a+x (b+c x)}}+\frac{\left (1680 a^2 b^2 c^2-320 a^3 c^3-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{1024 c^{13/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(48*a^3*c^2*(-231*b + 50*c*x) + 8*a^2*c*(1785*b^3 - 2268*b^2*c*x - 618*b*c^2*x^2 + 100*c^3*x^3) + a*(-3465*b^5
+ 16590*b^4*c*x + 5376*b^3*c^2*x^2 - 1728*b^2*c^3*x^3 + 736*b*c^4*x^4 - 320*c^5*x^5) + x*(-3465*b^6 - 1155*b^
5*c*x + 462*b^4*c^2*x^2 - 264*b^3*c^3*x^3 + 176*b^2*c^4*x^4 - 128*b*c^5*x^5 + 1280*c^6*x^6))/(7680*c^6*Sqrt[a
+ x*(b + c*x)]) + ((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]*Sq
rt[a + x*(b + c*x)])])/(1024*c^(13/2))

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Maple [A]  time = 0.049, size = 394, normalized size = 1.5 \begin{align*}{\frac{{x}^{5}}{6\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{11\,b{x}^{4}}{60\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{33\,{b}^{2}{x}^{3}}{160\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{77\,{b}^{3}{x}^{2}}{320\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{77\,{b}^{4}x}{256\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{231\,{b}^{5}}{512\,{c}^{6}}\sqrt{c{x}^{2}+bx+a}}+{\frac{231\,{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{13}{2}}}}-{\frac{315\,{b}^{4}a}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{11}{2}}}}+{\frac{119\,a{b}^{3}}{64\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{147\,{b}^{2}ax}{160\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{105\,{b}^{2}{a}^{2}}{64}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}+{\frac{39\,ab{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{231\,b{a}^{2}}{160\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,a{x}^{3}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{a}^{2}x}{16\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{a}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^5*(c*x^2+b*x+a)^(1/2)/c-11/60*b*x^4*(c*x^2+b*x+a)^(1/2)/c^2+33/160*b^2/c^3*x^3*(c*x^2+b*x+a)^(1/2)-77/32
0*b^3/c^4*x^2*(c*x^2+b*x+a)^(1/2)+77/256*b^4/c^5*x*(c*x^2+b*x+a)^(1/2)-231/512*b^5/c^6*(c*x^2+b*x+a)^(1/2)+231
/1024*b^6/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-315/256*b^4/c^(11/2)*a*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))+119/64*b^3/c^5*a*(c*x^2+b*x+a)^(1/2)-147/160*b^2/c^4*a*x*(c*x^2+b*x+a)^(1/2)+105/64*b^2/c^
(9/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+39/80*b/c^3*a*x^2*(c*x^2+b*x+a)^(1/2)-231/160*b/c^4*a^2*
(c*x^2+b*x+a)^(1/2)-5/24*a/c^2*x^3*(c*x^2+b*x+a)^(1/2)+5/16*a^2/c^3*x*(c*x^2+b*x+a)^(1/2)-5/16*a^3/c^(7/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.9075, size = 1075, normalized size = 4.12 \begin{align*} \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 ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16743, size = 281, normalized size = 1.08 \begin{align*} \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}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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 -
sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)