3.9.46 \(\int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^7} \, dx\)

Optimal. Leaf size=310 \[ -\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{160 a^3 x^4}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}+\frac {(2 a+b x) \sqrt {a+b x+c x^2} \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right )}{512 a^5 x^2}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{960 a^4 x^3}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6} \]

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Rubi [A]  time = 0.36, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {834, 806, 720, 724, 206} \begin {gather*} \frac {(2 a+b x) \sqrt {a+b x+c x^2} \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right )}{512 a^5 x^2}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (128 a^2 B c-196 a A b c-140 a b^2 B+105 A b^3\right )}{960 a^4 x^3}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-20 a A c-28 a b B+21 A b^2\right )}{160 a^3 x^4}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^7,x]

[Out]

((4*a*b*B*(7*b^2 - 12*a*c) - A*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2))*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(512*a^5
*x^2) - (A*(a + b*x + c*x^2)^(3/2))/(6*a*x^6) + ((3*A*b - 4*a*B)*(a + b*x + c*x^2)^(3/2))/(20*a^2*x^5) - ((21*
A*b^2 - 28*a*b*B - 20*a*A*c)*(a + b*x + c*x^2)^(3/2))/(160*a^3*x^4) + ((105*A*b^3 - 140*a*b^2*B - 196*a*A*b*c
+ 128*a^2*B*c)*(a + b*x + c*x^2)^(3/2))/(960*a^4*x^3) - ((b^2 - 4*a*c)*(4*a*b*B*(7*b^2 - 12*a*c) - A*(21*b^4 -
56*a*b^2*c + 16*a^2*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(1024*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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, 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] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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) \sqrt {a+b x+c x^2}}{x^7} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}-\frac {\int \frac {\left (\frac {3}{2} (3 A b-4 a B)+3 A c x\right ) \sqrt {a+b x+c x^2}}{x^6} \, dx}{6 a}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}+\frac {\int \frac {\left (\frac {3}{4} \left (21 A b^2-28 a b B-20 a A c\right )+3 (3 A b-4 a B) c x\right ) \sqrt {a+b x+c x^2}}{x^5} \, dx}{30 a^2}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}-\frac {\int \frac {\left (\frac {3}{8} \left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right )+\frac {3}{4} c \left (21 A b^2-28 a b B-20 a A c\right ) x\right ) \sqrt {a+b x+c x^2}}{x^4} \, dx}{120 a^3}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{128 a^4}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}+\frac {\left (\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{1024 a^5}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\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 )}{512 a^5}\\ &=\frac {\left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^5 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{6 a x^6}+\frac {(3 A b-4 a B) \left (a+b x+c x^2\right )^{3/2}}{20 a^2 x^5}-\frac {\left (21 A b^2-28 a b B-20 a A c\right ) \left (a+b x+c x^2\right )^{3/2}}{160 a^3 x^4}+\frac {\left (105 A b^3-140 a b^2 B-196 a A b c+128 a^2 B c\right ) \left (a+b x+c x^2\right )^{3/2}}{960 a^4 x^3}-\frac {\left (b^2-4 a c\right ) \left (4 a b B \left (7 b^2-12 a c\right )-A \left (21 b^4-56 a b^2 c+16 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.44, size = 262, normalized size = 0.85 \begin {gather*} \frac {\frac {2 x^3 (a+x (b+c x))^{3/2} \left (7 A \left (15 b^3-28 a b c\right )+4 a B \left (32 a c-35 b^2\right )\right )}{a^2}+\frac {15 x^4 \left (A \left (16 a^2 c^2-56 a b^2 c+21 b^4\right )+4 a b B \left (12 a c-7 b^2\right )\right ) \left (x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}\right )}{8 a^{7/2}}+\frac {12 x^2 (a+x (b+c x))^{3/2} \left (20 a A c+28 a b B-21 A b^2\right )}{a}+96 x (3 A b-4 a B) (a+x (b+c x))^{3/2}-320 a A (a+x (b+c x))^{3/2}}{1920 a^2 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^7,x]

[Out]

(-320*a*A*(a + x*(b + c*x))^(3/2) + 96*(3*A*b - 4*a*B)*x*(a + x*(b + c*x))^(3/2) + (12*(-21*A*b^2 + 28*a*b*B +
20*a*A*c)*x^2*(a + x*(b + c*x))^(3/2))/a + (2*(4*a*B*(-35*b^2 + 32*a*c) + 7*A*(15*b^3 - 28*a*b*c))*x^3*(a + x
*(b + c*x))^(3/2))/a^2 + (15*(4*a*b*B*(-7*b^2 + 12*a*c) + A*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2))*x^4*(-2*Sqrt[a
]*(2*a + b*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*x^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]
))/(8*a^(7/2)))/(1920*a^2*x^6)

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IntegrateAlgebraic [A]  time = 3.05, size = 388, normalized size = 1.25 \begin {gather*} -\frac {21 A b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{512 a^{11/2}}+\frac {\left (-16 a^2 A c^3-48 a^2 b B c^2+60 a A b^2 c^2+40 a b^3 B c-35 A b^4 c-7 b^5 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{128 a^{9/2}}+\frac {\sqrt {a+b x+c x^2} \left (-1280 a^5 A-1536 a^5 B x-128 a^4 A b x-320 a^4 A c x^2-192 a^4 b B x^2-512 a^4 B c x^3+144 a^3 A b^2 x^2+544 a^3 A b c x^3+480 a^3 A c^2 x^4+224 a^3 b^2 B x^3+928 a^3 b B c x^4+1024 a^3 B c^2 x^5-168 a^2 A b^3 x^3-896 a^2 A b^2 c x^4-1808 a^2 A b c^2 x^5-280 a^2 b^3 B x^4-1840 a^2 b^2 B c x^5+210 a A b^4 x^4+1680 a A b^3 c x^5+420 a b^4 B x^5-315 A b^5 x^5\right )}{7680 a^5 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^7,x]

[Out]

(Sqrt[a + b*x + c*x^2]*(-1280*a^5*A - 128*a^4*A*b*x - 1536*a^5*B*x + 144*a^3*A*b^2*x^2 - 192*a^4*b*B*x^2 - 320
*a^4*A*c*x^2 - 168*a^2*A*b^3*x^3 + 224*a^3*b^2*B*x^3 + 544*a^3*A*b*c*x^3 - 512*a^4*B*c*x^3 + 210*a*A*b^4*x^4 -
280*a^2*b^3*B*x^4 - 896*a^2*A*b^2*c*x^4 + 928*a^3*b*B*c*x^4 + 480*a^3*A*c^2*x^4 - 315*A*b^5*x^5 + 420*a*b^4*B
*x^5 + 1680*a*A*b^3*c*x^5 - 1840*a^2*b^2*B*c*x^5 - 1808*a^2*A*b*c^2*x^5 + 1024*a^3*B*c^2*x^5))/(7680*a^5*x^6)
+ ((-7*b^5*B - 35*A*b^4*c + 40*a*b^3*B*c + 60*a*A*b^2*c^2 - 48*a^2*b*B*c^2 - 16*a^2*A*c^3)*ArcTanh[(-(Sqrt[c]*
x) + Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(128*a^(9/2)) - (21*A*b^6*ArcTanh[(Sqrt[c]*x)/Sqrt[a] - Sqrt[a + b*x + c
*x^2]/Sqrt[a]])/(512*a^(11/2))

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fricas [A]  time = 3.40, size = 709, normalized size = 2.29 \begin {gather*} \left [\frac {15 \, {\left (28 \, B a b^{5} - 21 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} c^{2} - 20 \, {\left (8 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} c\right )} \sqrt {a} x^{6} \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 (1280 \, A a^{6} - {\left (420 \, B a^{2} b^{4} - 315 \, A a b^{5} + 16 \, {\left (64 \, B a^{4} - 113 \, A a^{3} b\right )} c^{2} - 80 \, {\left (23 \, B a^{3} b^{2} - 21 \, A a^{2} b^{3}\right )} c\right )} x^{5} + 2 \, {\left (140 \, B a^{3} b^{3} - 105 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 16 \, {\left (29 \, B a^{4} b - 28 \, A a^{3} b^{2}\right )} c\right )} x^{4} - 8 \, {\left (28 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3} - 4 \, {\left (16 \, B a^{5} - 17 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (12 \, B a^{5} b - 9 \, A a^{4} b^{2} + 20 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, a^{6} x^{6}}, \frac {15 \, {\left (28 \, B a b^{5} - 21 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} c^{2} - 20 \, {\left (8 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} c\right )} \sqrt {-a} x^{6} \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 (1280 \, A a^{6} - {\left (420 \, B a^{2} b^{4} - 315 \, A a b^{5} + 16 \, {\left (64 \, B a^{4} - 113 \, A a^{3} b\right )} c^{2} - 80 \, {\left (23 \, B a^{3} b^{2} - 21 \, A a^{2} b^{3}\right )} c\right )} x^{5} + 2 \, {\left (140 \, B a^{3} b^{3} - 105 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 16 \, {\left (29 \, B a^{4} b - 28 \, A a^{3} b^{2}\right )} c\right )} x^{4} - 8 \, {\left (28 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3} - 4 \, {\left (16 \, B a^{5} - 17 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (12 \, B a^{5} b - 9 \, A a^{4} b^{2} + 20 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, a^{6} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^7,x, algorithm="fricas")

[Out]

[1/30720*(15*(28*B*a*b^5 - 21*A*b^6 + 64*A*a^3*c^3 + 48*(4*B*a^3*b - 5*A*a^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 7*A*
a*b^4)*c)*sqrt(a)*x^6*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*(1280*A*a^6 - (420*B*a^2*b^4 - 315*A*a*b^5 + 16*(64*B*a^4 - 113*A*a^3*b)*c^2 - 80*(23*B*a^3*b^2 - 21
*A*a^2*b^3)*c)*x^5 + 2*(140*B*a^3*b^3 - 105*A*a^2*b^4 - 240*A*a^4*c^2 - 16*(29*B*a^4*b - 28*A*a^3*b^2)*c)*x^4
- 8*(28*B*a^4*b^2 - 21*A*a^3*b^3 - 4*(16*B*a^5 - 17*A*a^4*b)*c)*x^3 + 16*(12*B*a^5*b - 9*A*a^4*b^2 + 20*A*a^5*
c)*x^2 + 128*(12*B*a^6 + A*a^5*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^6), 1/15360*(15*(28*B*a*b^5 - 21*A*b^6 + 64
*A*a^3*c^3 + 48*(4*B*a^3*b - 5*A*a^2*b^2)*c^2 - 20*(8*B*a^2*b^3 - 7*A*a*b^4)*c)*sqrt(-a)*x^6*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*(1280*A*a^6 - (420*B*a^2*b^4 - 315*A*a*b^5 +
16*(64*B*a^4 - 113*A*a^3*b)*c^2 - 80*(23*B*a^3*b^2 - 21*A*a^2*b^3)*c)*x^5 + 2*(140*B*a^3*b^3 - 105*A*a^2*b^4
- 240*A*a^4*c^2 - 16*(29*B*a^4*b - 28*A*a^3*b^2)*c)*x^4 - 8*(28*B*a^4*b^2 - 21*A*a^3*b^3 - 4*(16*B*a^5 - 17*A*
a^4*b)*c)*x^3 + 16*(12*B*a^5*b - 9*A*a^4*b^2 + 20*A*a^5*c)*x^2 + 128*(12*B*a^6 + A*a^5*b)*x)*sqrt(c*x^2 + b*x
+ a))/(a^6*x^6)]

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giac [B]  time = 0.32, size = 1955, normalized size = 6.31

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^7,x, algorithm="giac")

[Out]

1/512*(28*B*a*b^5 - 21*A*b^6 - 160*B*a^2*b^3*c + 140*A*a*b^4*c + 192*B*a^3*b*c^2 - 240*A*a^2*b^2*c^2 + 64*A*a^
3*c^3)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^5) - 1/7680*(420*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^11*B*a*b^5 - 315*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*b^6 - 2400*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^11*B*a^2*b^3*c + 2100*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a*b^4*c + 2880*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^11*B*a^3*b*c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^2*b^2*c^2 + 960*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^11*A*a^3*c^3 - 2380*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^2*b^5 + 1785*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^9*A*a*b^6 + 13600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*b^3*c - 11900*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^9*A*a^2*b^4*c - 16320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^4*b*c^2 + 20400*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^3*b^2*c^2 - 5440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^4*c^3 - 30720*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^5*c^(5/2) + 5544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^5 - 4
158*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^6 - 31680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*b^3*c
+ 27720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b^4*c - 48000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^5*
b*c^2 - 47520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^4*b^2*c^2 - 36480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
7*A*a^5*c^3 - 97280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^5*b^2*c^(3/2) + 20480*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^6*B*a^6*c^(5/2) - 163840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^5*b*c^(5/2) - 6744*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^5*B*a^4*b^5 + 5058*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*b^6 - 16320*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*B*a^5*b^3*c - 33720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^4*b^4*c + 13440*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^5*B*a^6*b*c^2 - 170400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^5*b^2*c^2 - 3648
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^6*c^3 - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^5*b^4*sqrt
(c) + 76800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^6*b^2*c^(3/2) - 168960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^4*A*a^5*b^3*c^(3/2) - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^6*b*c^(5/2) + 2740*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*B*a^5*b^5 - 3335*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*b^6 + 23840*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*B*a^6*b^3*c - 47740*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^5*b^4*c + 45120*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*B*a^7*b*c^2 - 102480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^6*b^2*c^2 - 5440*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^7*c^3 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^6*b^4*sqrt(c) -
15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^5*b^5*sqrt(c) + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B
*a^7*b^2*c^(3/2) - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^6*b^3*c^(3/2) + 12288*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*B*a^8*c^(5/2) - 24576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^7*b*c^(5/2) + 420*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*B*a^6*b^5 - 315*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*b^6 + 12960*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*B*a^7*b^3*c - 13260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^6*b^4*c + 2880*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*B*a^8*b*c^2 - 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^7*b^2*c^2 + 960*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*A*a^8*c^3 + 5120*B*a^8*b^2*c^(3/2) - 5120*A*a^7*b^3*c^(3/2) - 2048*B*a^9*c^(5/2) + 4096
*A*a^8*b*c^(5/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^6*a^5)

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maple [B]  time = 0.07, size = 1014, normalized size = 3.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^7,x)

[Out]

7/128*B/a^5*b^4*c*(c*x^2+b*x+a)^(1/2)*x-3/32*B/a^4*b^2*c^2*(c*x^2+b*x+a)^(1/2)*x+3/32*B/a^4*b^2*c/x*(c*x^2+b*x
+a)^(3/2)-3/16*B/a^3*b*c/x^2*(c*x^2+b*x+a)^(3/2)+7/64*A/a^5*b^3*c^2*(c*x^2+b*x+a)^(1/2)*x-7/64*A/a^5*b^3*c/x*(
c*x^2+b*x+a)^(3/2)-49/240*A/a^3*b*c/x^3*(c*x^2+b*x+a)^(3/2)+1/32*A*c^2/a^4*b/x*(c*x^2+b*x+a)^(3/2)-1/32*A*c^3/
a^4*b*(c*x^2+b*x+a)^(1/2)*x-21/512*A/a^6*b^5*c*(c*x^2+b*x+a)^(1/2)*x+7/32*A/a^4*b^2*c/x^2*(c*x^2+b*x+a)^(3/2)+
21/1024*A/a^(11/2)*b^6*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-21/512*A/a^6*b^6*(c*x^2+b*x+a)^(1/2)-1/16
*A*c^3/a^(5/2)*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+1/16*A*c^3/a^3*(c*x^2+b*x+a)^(1/2)-1/5*B/a/x^5*(c
*x^2+b*x+a)^(3/2)+7/128*B/a^5*b^5*(c*x^2+b*x+a)^(1/2)-7/256*B/a^(9/2)*b^5*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^
(1/2))/x)-21/160*A/a^3*b^2/x^4*(c*x^2+b*x+a)^(3/2)+3/20*A/a^2*b/x^5*(c*x^2+b*x+a)^(3/2)+7/64*A/a^4*b^3/x^3*(c*
x^2+b*x+a)^(3/2)-21/256*A/a^5*b^4/x^2*(c*x^2+b*x+a)^(3/2)+21/512*A/a^6*b^5/x*(c*x^2+b*x+a)^(3/2)+49/256*A/a^5*
b^4*c*(c*x^2+b*x+a)^(1/2)-1/4*A/a^4*b^2*c^2*(c*x^2+b*x+a)^(1/2)+1/8*A*c/a^2/x^4*(c*x^2+b*x+a)^(3/2)-1/16*A*c^2
/a^3/x^2*(c*x^2+b*x+a)^(3/2)-35/256*A/a^(9/2)*b^4*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+15/64*A/a^(7
/2)*b^2*c^2*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+7/40*B/a^2*b/x^4*(c*x^2+b*x+a)^(3/2)-7/48*B/a^3*b^2/
x^3*(c*x^2+b*x+a)^(3/2)+7/64*B/a^4*b^3/x^2*(c*x^2+b*x+a)^(3/2)-7/128*B/a^5*b^4/x*(c*x^2+b*x+a)^(3/2)-13/64*B/a
^4*b^3*c*(c*x^2+b*x+a)^(1/2)+5/32*B/a^(7/2)*b^3*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+3/16*B/a^3*b*c
^2*(c*x^2+b*x+a)^(1/2)-3/16*B/a^(5/2)*b*c^2*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+2/15*B*c/a^2/x^3*(c*
x^2+b*x+a)^(3/2)-1/6*A*(c*x^2+b*x+a)^(3/2)/a/x^6

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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)*(c*x^2+b*x+a)^(1/2)/x^7,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 {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^7,x)

[Out]

int(((A + B*x)*(a + b*x + c*x^2)^(1/2))/x^7, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**7,x)

[Out]

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**7, x)

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