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

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

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Rubi [A]  time = 0.40, antiderivative size = 306, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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])

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 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 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])

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*}

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Mathematica [A]  time = 0.30, size = 233, normalized size = 0.76 \begin {gather*} \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 (96 a^4 (4 A+5 B x)-16 a^3 x (A (27 b+32 c x)+5 B x (7 b+9 c x))+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 )-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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1920*(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) - 16*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))))/(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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IntegrateAlgebraic [A]  time = 1.82, size = 300, normalized size = 0.98 \begin {gather*} \frac {5 \left (24 a A b c^2+24 a b^2 B c-28 A b^3 c-7 b^4 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{64 a^{9/2}}+\frac {3 \left (32 a^3 B c^2-21 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{128 a^{11/2}}+\frac {\sqrt {a+b x+c x^2} \left (-384 a^4 A-480 a^4 B x+432 a^3 A b x+512 a^3 A c x^2+560 a^3 b B x^2+720 a^3 B c x^3-504 a^2 A b^2 x^2-1288 a^2 A b c x^3-1024 a^2 A c^2 x^4-700 a^2 b^2 B x^3-2200 a^2 b B c x^4+630 a A b^3 x^3+2940 a A b^2 c x^4+1050 a b^3 B x^4-945 A b^4 x^4\right )}{1920 a^5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(-384*a^4*A + 432*a^3*A*b*x - 480*a^4*B*x - 504*a^2*A*b^2*x^2 + 560*a^3*b*B*x^2 + 512*a
^3*A*c*x^2 + 630*a*A*b^3*x^3 - 700*a^2*b^2*B*x^3 - 1288*a^2*A*b*c*x^3 + 720*a^3*B*c*x^3 - 945*A*b^4*x^4 + 1050
*a*b^3*B*x^4 + 2940*a*A*b^2*c*x^4 - 2200*a^2*b*B*c*x^4 - 1024*a^2*A*c^2*x^4))/(1920*a^5*x^5) + (3*(-21*A*b^5 +
32*a^3*B*c^2)*ArcTanh[(Sqrt[c]*x - Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(128*a^(11/2)) + (5*(-7*b^4*B - 28*A*b^3*
c + 24*a*b^2*B*c + 24*a*A*b*c^2)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(64*a^(9/2))

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fricas [A]  time = 1.94, size = 557, normalized size = 1.82 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.27, size = 1266, normalized size = 4.14

result too large to display

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)

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maple [B]  time = 0.07, size = 578, normalized size = 1.89 \begin {gather*} \frac {15 A b \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}-\frac {35 A \,b^{3} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{32 a^{\frac {9}{2}}}+\frac {63 A \,b^{5} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{256 a^{\frac {11}{2}}}-\frac {3 B \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}+\frac {15 B \,b^{2} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}-\frac {35 B \,b^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {9}{2}}}-\frac {8 \sqrt {c \,x^{2}+b x +a}\, A \,c^{2}}{15 a^{3} x}+\frac {49 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c}{32 a^{4} x}-\frac {63 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{128 a^{5} x}-\frac {55 \sqrt {c \,x^{2}+b x +a}\, B b c}{48 a^{3} x}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{64 a^{4} x}-\frac {161 \sqrt {c \,x^{2}+b x +a}\, A b c}{240 a^{3} x^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 a^{4} x^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B c}{8 a^{2} x^{2}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2}}{96 a^{3} x^{2}}+\frac {4 \sqrt {c \,x^{2}+b x +a}\, A c}{15 a^{2} x^{3}}-\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{80 a^{3} x^{3}}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, B b}{24 a^{2} x^{3}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, A b}{40 a^{2} x^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, B}{4 a \,x^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A}{5 a \,x^{5}} \end {gather*}

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((b*x+2
*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-35/32*A/a^(9/2)*b^3*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*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((b*x+2*a
+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+4/15*A*c/a^2/x^3*(c*x^2+b*x+a)^(1/2)-8/15*A*c^2/a^3/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((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+15/16*B
/a^(7/2)*b^2*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-55/48*B/a^3*b*c/x*(c*x^2+b*x+a)^(1/2)+3/8*B*c/a^2
/x^2*(c*x^2+b*x+a)^(1/2)-3/8*B*c^2/a^(5/2)*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> 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 details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^6\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{6} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

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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