3.960 \(\int \frac{A+B x}{x^6 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=306 \[ \frac{\sqrt{a+b x+c x^2} \left (50 a b B \left (21 b^2-44 a c\right )-A \left (1024 a^2 c^2-2940 a b^2 c+945 b^4\right )\right )}{1920 a^5 x}-\frac{\left (2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )-A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{11/2}}+\frac{\sqrt{a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{960 a^4 x^2}-\frac{\sqrt{a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{240 a^3 x^3}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5} \]

[Out]

-(A*Sqrt[a + b*x + c*x^2])/(5*a*x^5) + ((9*A*b - 10*a*B)*Sqrt[a + b*x + c*x^2])/(40*a^2*x^4) - ((63*A*b^2 - 70
*a*b*B - 64*a*A*c)*Sqrt[a + b*x + c*x^2])/(240*a^3*x^3) + ((315*A*b^3 - 350*a*b^2*B - 644*a*A*b*c + 360*a^2*B*
c)*Sqrt[a + b*x + c*x^2])/(960*a^4*x^2) + ((50*a*b*B*(21*b^2 - 44*a*c) - A*(945*b^4 - 2940*a*b^2*c + 1024*a^2*
c^2))*Sqrt[a + b*x + c*x^2])/(1920*a^5*x) - ((2*a*B*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2) - A*(63*b^5 - 280*a*b^
3*c + 240*a^2*b*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(11/2))

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Rubi [A]  time = 0.397192, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {834, 806, 724, 206} \[ \frac{\sqrt{a+b x+c x^2} \left (50 a b B \left (21 b^2-44 a c\right )-A \left (1024 a^2 c^2-2940 a b^2 c+945 b^4\right )\right )}{1920 a^5 x}-\frac{\left (2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )-A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{11/2}}+\frac{\sqrt{a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{960 a^4 x^2}-\frac{\sqrt{a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{240 a^3 x^3}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^6*Sqrt[a + b*x + c*x^2]),x]

[Out]

-(A*Sqrt[a + b*x + c*x^2])/(5*a*x^5) + ((9*A*b - 10*a*B)*Sqrt[a + b*x + c*x^2])/(40*a^2*x^4) - ((63*A*b^2 - 70
*a*b*B - 64*a*A*c)*Sqrt[a + b*x + c*x^2])/(240*a^3*x^3) + ((315*A*b^3 - 350*a*b^2*B - 644*a*A*b*c + 360*a^2*B*
c)*Sqrt[a + b*x + c*x^2])/(960*a^4*x^2) + ((50*a*b*B*(21*b^2 - 44*a*c) - A*(945*b^4 - 2940*a*b^2*c + 1024*a^2*
c^2))*Sqrt[a + b*x + c*x^2])/(1920*a^5*x) - ((2*a*B*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2) - A*(63*b^5 - 280*a*b^
3*c + 240*a^2*b*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(11/2))

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 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 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{A+B x}{x^6 \sqrt{a+b x+c x^2}} \, dx &=-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5}-\frac{\int \frac{\frac{1}{2} (9 A b-10 a B)+4 A c x}{x^5 \sqrt{a+b x+c x^2}} \, dx}{5 a}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}+\frac{\int \frac{\frac{1}{4} \left (63 A b^2-70 a b B-64 a A c\right )+\frac{3}{2} (9 A b-10 a B) c x}{x^4 \sqrt{a+b x+c x^2}} \, dx}{20 a^2}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}-\frac{\int \frac{\frac{1}{8} \left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right )+\frac{1}{2} c \left (63 A b^2-70 a b B-64 a A c\right ) x}{x^3 \sqrt{a+b x+c x^2}} \, dx}{60 a^3}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}+\frac{\int \frac{\frac{1}{16} \left (945 A b^4-1050 a b^3 B-2940 a A b^2 c+2200 a^2 b B c+1024 a^2 A c^2\right )+\frac{1}{8} c \left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{120 a^4}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}+\frac{\left (50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a b^2 c+1024 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{1920 a^5 x}+\frac{\left (2 a B \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )-A \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right )\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{256 a^5}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}+\frac{\left (50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a b^2 c+1024 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{1920 a^5 x}-\frac{\left (2 a B \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )-A \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right )\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 )}{128 a^5}\\ &=-\frac{A \sqrt{a+b x+c x^2}}{5 a x^5}+\frac{(9 A b-10 a B) \sqrt{a+b x+c x^2}}{40 a^2 x^4}-\frac{\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt{a+b x+c x^2}}{240 a^3 x^3}+\frac{\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt{a+b x+c x^2}}{960 a^4 x^2}+\frac{\left (50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a b^2 c+1024 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{1920 a^5 x}-\frac{\left (2 a B \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )-A \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.303844, size = 233, normalized size = 0.76 \[ \frac{\left (A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )-2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{256 a^{11/2}}-\frac{\sqrt{a+x (b+c x)} \left (4 a^2 x^2 \left (2 A \left (63 b^2+161 b c x+128 c^2 x^2\right )+25 b B x (7 b+22 c x)\right )-16 a^3 x (A (27 b+32 c x)+5 B x (7 b+9 c x))+96 a^4 (4 A+5 B x)-210 a b^2 x^3 (3 A b+14 A c x+5 b B x)+945 A b^4 x^4\right )}{1920 a^5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^6*Sqrt[a + b*x + c*x^2]),x]

[Out]

-(Sqrt[a + x*(b + c*x)]*(945*A*b^4*x^4 + 96*a^4*(4*A + 5*B*x) - 210*a*b^2*x^3*(3*A*b + 5*b*B*x + 14*A*c*x) - 1
6*a^3*x*(5*B*x*(7*b + 9*c*x) + A*(27*b + 32*c*x)) + 4*a^2*x^2*(25*b*B*x*(7*b + 22*c*x) + 2*A*(63*b^2 + 161*b*c
*x + 128*c^2*x^2))))/(1920*a^5*x^5) + ((-2*a*B*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2) + A*(63*b^5 - 280*a*b^3*c +
240*a^2*b*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(256*a^(11/2))

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Maple [B]  time = 0.013, size = 578, normalized size = 1.9 \begin{align*} -{\frac{A}{5\,a{x}^{5}}\sqrt{c{x}^{2}+bx+a}}+{\frac{9\,Ab}{40\,{a}^{2}{x}^{4}}\sqrt{c{x}^{2}+bx+a}}-{\frac{21\,A{b}^{2}}{80\,{a}^{3}{x}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{21\,A{b}^{3}}{64\,{a}^{4}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{63\,A{b}^{4}}{128\,{a}^{5}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{63\,A{b}^{5}}{256}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{11}{2}}}}-{\frac{35\,A{b}^{3}c}{32}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{49\,A{b}^{2}c}{32\,{a}^{4}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{161\,Abc}{240\,{a}^{3}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{15\,Ab{c}^{2}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}+{\frac{4\,Ac}{15\,{a}^{2}{x}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{8\,A{c}^{2}}{15\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{B}{4\,a{x}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{7\,bB}{24\,{a}^{2}{x}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,{b}^{2}B}{96\,{a}^{3}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,{b}^{3}B}{64\,{a}^{4}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,{b}^{4}B}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{15\,{b}^{2}Bc}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{55\,Bcb}{48\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Bc}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,B{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*A*(c*x^2+b*x+a)^(1/2)/a/x^5+9/40*A/a^2*b/x^4*(c*x^2+b*x+a)^(1/2)-21/80*A/a^3*b^2/x^3*(c*x^2+b*x+a)^(1/2)+
21/64*A/a^4*b^3/x^2*(c*x^2+b*x+a)^(1/2)-63/128*A/a^5*b^4/x*(c*x^2+b*x+a)^(1/2)+63/256*A/a^(11/2)*b^5*ln((2*a+b
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-35/32*A/a^(9/2)*b^3*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+49/32
*A/a^4*b^2*c/x*(c*x^2+b*x+a)^(1/2)-161/240*A/a^3*b*c/x^2*(c*x^2+b*x+a)^(1/2)+15/16*A/a^(7/2)*b*c^2*ln((2*a+b*x
+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+4/15*A/a^2*c/x^3*(c*x^2+b*x+a)^(1/2)-8/15*A/a^3*c^2/x*(c*x^2+b*x+a)^(1/2)-1
/4*B/a/x^4*(c*x^2+b*x+a)^(1/2)+7/24*B/a^2*b/x^3*(c*x^2+b*x+a)^(1/2)-35/96*B/a^3*b^2/x^2*(c*x^2+b*x+a)^(1/2)+35
/64*B/a^4*b^3/x*(c*x^2+b*x+a)^(1/2)-35/128*B/a^(9/2)*b^4*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+15/16*B
/a^(7/2)*b^2*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-55/48*B/a^3*b*c/x*(c*x^2+b*x+a)^(1/2)+3/8*B/a^2*c
/x^2*(c*x^2+b*x+a)^(1/2)-3/8*B/a^(5/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.3809, size = 1327, normalized size = 4.34 \begin{align*} \left [-\frac{15 \,{\left (70 \, B a b^{4} - 63 \, A b^{5} + 48 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 40 \,{\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} c\right )} \sqrt{a} x^{5} \log \left (-\frac{8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \,{\left (384 \, A a^{5} -{\left (1050 \, B a^{2} b^{3} - 945 \, A a b^{4} - 1024 \, A a^{3} c^{2} - 20 \,{\left (110 \, B a^{3} b - 147 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \,{\left (350 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3} - 4 \,{\left (90 \, B a^{4} - 161 \, A a^{3} b\right )} c\right )} x^{3} - 8 \,{\left (70 \, B a^{4} b - 63 \, A a^{3} b^{2} + 64 \, A a^{4} c\right )} x^{2} + 48 \,{\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{c x^{2} + b x + a}}{7680 \, a^{6} x^{5}}, \frac{15 \,{\left (70 \, B a b^{4} - 63 \, A b^{5} + 48 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 40 \,{\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} c\right )} \sqrt{-a} x^{5} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \,{\left (384 \, A a^{5} -{\left (1050 \, B a^{2} b^{3} - 945 \, A a b^{4} - 1024 \, A a^{3} c^{2} - 20 \,{\left (110 \, B a^{3} b - 147 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \,{\left (350 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3} - 4 \,{\left (90 \, B a^{4} - 161 \, A a^{3} b\right )} c\right )} x^{3} - 8 \,{\left (70 \, B a^{4} b - 63 \, A a^{3} b^{2} + 64 \, A a^{4} c\right )} x^{2} + 48 \,{\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3840 \, a^{6} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(70*B*a*b^4 - 63*A*b^5 + 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - 7*A*a*b^3)*c)*sqrt(a)*x
^5*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 + 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(384*A*a^
5 - (1050*B*a^2*b^3 - 945*A*a*b^4 - 1024*A*a^3*c^2 - 20*(110*B*a^3*b - 147*A*a^2*b^2)*c)*x^4 + 2*(350*B*a^3*b^
2 - 315*A*a^2*b^3 - 4*(90*B*a^4 - 161*A*a^3*b)*c)*x^3 - 8*(70*B*a^4*b - 63*A*a^3*b^2 + 64*A*a^4*c)*x^2 + 48*(1
0*B*a^5 - 9*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^5), 1/3840*(15*(70*B*a*b^4 - 63*A*b^5 + 48*(2*B*a^3 - 5*
A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - 7*A*a*b^3)*c)*sqrt(-a)*x^5*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(
-a)/(a*c*x^2 + a*b*x + a^2)) - 2*(384*A*a^5 - (1050*B*a^2*b^3 - 945*A*a*b^4 - 1024*A*a^3*c^2 - 20*(110*B*a^3*b
- 147*A*a^2*b^2)*c)*x^4 + 2*(350*B*a^3*b^2 - 315*A*a^2*b^3 - 4*(90*B*a^4 - 161*A*a^3*b)*c)*x^3 - 8*(70*B*a^4*
b - 63*A*a^3*b^2 + 64*A*a^4*c)*x^2 + 48*(10*B*a^5 - 9*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.3692, size = 1709, normalized size = 5.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(70*B*a*b^4 - 63*A*b^5 - 240*B*a^2*b^2*c + 280*A*a*b^3*c + 96*B*a^3*c^2 - 240*A*a^2*b*c^2)*arctan(-(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^5) - 1/1920*(1050*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B
*a*b^4 - 945*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*b^5 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^2*b^
2*c + 4200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b^3*c + 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*
c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b*c^2 - 4900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^
2*b^4 + 4410*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^5 + 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3
*b^2*c - 19600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^3*c - 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*
B*a^4*c^2 + 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b*c^2 + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^5*B*a^3*b^4 - 8064*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^5 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^5*B*a^4*b^2*c + 35840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*b^3*c - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^5*A*a^4*b*c^2 - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^5*b*c^(3/2) - 20480*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^4*A*a^5*c^(5/2) - 7900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^4*b^4 + 7110*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^3*A*a^3*b^5 + 13920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^5*b^2*c - 31600*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*A*a^4*b^3*c + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^6*c^2 - 16800*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*A*a^5*b*c^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^5*b^3*sqrt(c) + 25
600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^6*b*c^(3/2) - 38400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^5*
b^2*c^(3/2) + 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^6*c^(5/2) + 2790*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a^5*b^4 - 2895*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^4*b^5 + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*B*a^6*b^2*c - 4200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*b^3*c - 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*B*a^7*c^2 + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^6*b*c^2 + 3840*B*a^6*b^3*sqrt(c) - 3840*A*a^5*b^4*sq
rt(c) - 5120*B*a^7*b*c^(3/2) + 7680*A*a^6*b^2*c^(3/2) - 2048*A*a^7*c^(5/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2 - a)^5*a^5)