3.9.58 \(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=230 \[ -\frac {\left (b^2-4 a c\right )^2 \left (-4 a A c-12 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{9/2}}+\frac {\left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2} \left (-4 a A c-12 a b B+7 A b^2\right )}{512 a^4 x^2}-\frac {(2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-12 a b B+7 A b^2\right )}{192 a^3 x^4}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6} \]

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Rubi [A]  time = 0.20, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 6, 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) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-12 a b B+7 A b^2\right )}{192 a^3 x^4}+\frac {\left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2} \left (-4 a A c-12 a b B+7 A b^2\right )}{512 a^4 x^2}-\frac {\left (b^2-4 a c\right )^2 \left (-4 a A c-12 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{9/2}}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x]

[Out]

((b^2 - 4*a*c)*(7*A*b^2 - 12*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(512*a^4*x^2) - ((7*A*b^2 - 1
2*a*b*B - 4*a*A*c)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(192*a^3*x^4) - (A*(a + b*x + c*x^2)^(5/2))/(6*a*x^6)
+ ((7*A*b - 12*a*B)*(a + b*x + c*x^2)^(5/2))/(60*a^2*x^5) - ((b^2 - 4*a*c)^2*(7*A*b^2 - 12*a*b*B - 4*a*A*c)*Ar
cTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(1024*a^(9/2))

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) \left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}-\frac {\int \frac {\left (\frac {1}{2} (7 A b-12 a B)+A c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx}{6 a}\\ &=-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}+\frac {\left (7 A b^2-12 a b B-4 a A c\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx}{24 a^2}\\ &=-\frac {\left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{192 a^3 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac {\left (\left (b^2-4 a c\right ) \left (7 A b^2-12 a b B-4 a A c\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{128 a^3}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^4 x^2}-\frac {\left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{192 a^3 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 A b^2-12 a b B-4 a A c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{1024 a^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^4 x^2}-\frac {\left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{192 a^3 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (7 A b^2-12 a b B-4 a A c\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^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{512 a^4 x^2}-\frac {\left (7 A b^2-12 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{192 a^3 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{6 a x^6}+\frac {(7 A b-12 a B) \left (a+b x+c x^2\right )^{5/2}}{60 a^2 x^5}-\frac {\left (b^2-4 a c\right )^2 \left (7 A b^2-12 a b B-4 a A c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{1024 a^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 182, normalized size = 0.79 \begin {gather*} \frac {-\frac {5 \left (-4 a A c-12 a b B+7 A b^2\right ) \left (2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)} \left (8 a^2+4 a x (2 b+5 c x)-3 b^2 x^2\right )+3 x^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )}{256 a^{5/2} x^4}+\frac {(7 A b-12 a B) (a+x (b+c x))^{5/2}}{x^5}-\frac {10 a A (a+x (b+c x))^{5/2}}{x^6}}{60 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x]

[Out]

((-10*a*A*(a + x*(b + c*x))^(5/2))/x^6 + ((7*A*b - 12*a*B)*(a + x*(b + c*x))^(5/2))/x^5 - (5*(7*A*b^2 - 12*a*b
*B - 4*a*A*c)*(2*Sqrt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)]*(8*a^2 - 3*b^2*x^2 + 4*a*x*(2*b + 5*c*x)) + 3*(b^2
- 4*a*c)^2*x^4*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))/(256*a^(5/2)*x^4))/(60*a^2)

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IntegrateAlgebraic [A]  time = 3.65, size = 388, normalized size = 1.69 \begin {gather*} \frac {7 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^{9/2}}+\frac {\left (16 a^2 A c^3+48 a^2 b B c^2-36 a A b^2 c^2-24 a b^3 B c+15 A b^4 c+3 b^5 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{128 a^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (-1280 a^5 A-1536 a^5 B x-1664 a^4 A b x-2240 a^4 A c x^2-2112 a^4 b B x^2-3072 a^4 B c x^3-48 a^3 A b^2 x^2-288 a^3 A b c x^3-480 a^3 A c^2 x^4-96 a^3 b^2 B x^3-672 a^3 b B c x^4-1536 a^3 B c^2 x^5+56 a^2 A b^3 x^3+432 a^2 A b^2 c x^4+1296 a^2 A b c^2 x^5+120 a^2 b^3 B x^4+1200 a^2 b^2 B c x^5-70 a A b^4 x^4-760 a A b^3 c x^5-180 a b^4 B x^5+105 A b^5 x^5\right )}{7680 a^4 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^7,x]

[Out]

(Sqrt[a + b*x + c*x^2]*(-1280*a^5*A - 1664*a^4*A*b*x - 1536*a^5*B*x - 48*a^3*A*b^2*x^2 - 2112*a^4*b*B*x^2 - 22
40*a^4*A*c*x^2 + 56*a^2*A*b^3*x^3 - 96*a^3*b^2*B*x^3 - 288*a^3*A*b*c*x^3 - 3072*a^4*B*c*x^3 - 70*a*A*b^4*x^4 +
120*a^2*b^3*B*x^4 + 432*a^2*A*b^2*c*x^4 - 672*a^3*b*B*c*x^4 - 480*a^3*A*c^2*x^4 + 105*A*b^5*x^5 - 180*a*b^4*B
*x^5 - 760*a*A*b^3*c*x^5 + 1200*a^2*b^2*B*c*x^5 + 1296*a^2*A*b*c^2*x^5 - 1536*a^3*B*c^2*x^5))/(7680*a^4*x^6) +
((3*b^5*B + 15*A*b^4*c - 24*a*b^3*B*c - 36*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^(7/2)) + (7*A*b^6*ArcTanh[(Sqrt[c]*x)/Sqrt[a] - Sqrt[a + b*x + c*x^
2]/Sqrt[a]])/(512*a^(9/2))

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fricas [A]  time = 3.58, size = 709, normalized size = 3.08 \begin {gather*} \left [\frac {15 \, {\left (12 \, B a b^{5} - 7 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} c^{2} - 12 \, {\left (8 \, B a^{2} b^{3} - 5 \, 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 (180 \, B a^{2} b^{4} - 105 \, A a b^{5} + 48 \, {\left (32 \, B a^{4} - 27 \, A a^{3} b\right )} c^{2} - 40 \, {\left (30 \, B a^{3} b^{2} - 19 \, A a^{2} b^{3}\right )} c\right )} x^{5} - 2 \, {\left (60 \, B a^{3} b^{3} - 35 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 24 \, {\left (14 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} c\right )} x^{4} + 8 \, {\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3} + 12 \, {\left (32 \, B a^{5} + 3 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (132 \, B a^{5} b + 3 \, A a^{4} b^{2} + 140 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, a^{5} x^{6}}, -\frac {15 \, {\left (12 \, B a b^{5} - 7 \, A b^{6} + 64 \, A a^{3} c^{3} + 48 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} c^{2} - 12 \, {\left (8 \, B a^{2} b^{3} - 5 \, 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 (180 \, B a^{2} b^{4} - 105 \, A a b^{5} + 48 \, {\left (32 \, B a^{4} - 27 \, A a^{3} b\right )} c^{2} - 40 \, {\left (30 \, B a^{3} b^{2} - 19 \, A a^{2} b^{3}\right )} c\right )} x^{5} - 2 \, {\left (60 \, B a^{3} b^{3} - 35 \, A a^{2} b^{4} - 240 \, A a^{4} c^{2} - 24 \, {\left (14 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} c\right )} x^{4} + 8 \, {\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3} + 12 \, {\left (32 \, B a^{5} + 3 \, A a^{4} b\right )} c\right )} x^{3} + 16 \, {\left (132 \, B a^{5} b + 3 \, A a^{4} b^{2} + 140 \, A a^{5} c\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, a^{5} 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)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/30720*(15*(12*B*a*b^5 - 7*A*b^6 + 64*A*a^3*c^3 + 48*(4*B*a^3*b - 3*A*a^2*b^2)*c^2 - 12*(8*B*a^2*b^3 - 5*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 + (180*B*a^2*b^4 - 105*A*a*b^5 + 48*(32*B*a^4 - 27*A*a^3*b)*c^2 - 40*(30*B*a^3*b^2 - 19*A
*a^2*b^3)*c)*x^5 - 2*(60*B*a^3*b^3 - 35*A*a^2*b^4 - 240*A*a^4*c^2 - 24*(14*B*a^4*b - 9*A*a^3*b^2)*c)*x^4 + 8*(
12*B*a^4*b^2 - 7*A*a^3*b^3 + 12*(32*B*a^5 + 3*A*a^4*b)*c)*x^3 + 16*(132*B*a^5*b + 3*A*a^4*b^2 + 140*A*a^5*c)*x
^2 + 128*(12*B*a^6 + 13*A*a^5*b)*x)*sqrt(c*x^2 + b*x + a))/(a^5*x^6), -1/15360*(15*(12*B*a*b^5 - 7*A*b^6 + 64*
A*a^3*c^3 + 48*(4*B*a^3*b - 3*A*a^2*b^2)*c^2 - 12*(8*B*a^2*b^3 - 5*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 + (180*B*a^2*b^4 - 105*A*a*b^5 +
48*(32*B*a^4 - 27*A*a^3*b)*c^2 - 40*(30*B*a^3*b^2 - 19*A*a^2*b^3)*c)*x^5 - 2*(60*B*a^3*b^3 - 35*A*a^2*b^4 - 24
0*A*a^4*c^2 - 24*(14*B*a^4*b - 9*A*a^3*b^2)*c)*x^4 + 8*(12*B*a^4*b^2 - 7*A*a^3*b^3 + 12*(32*B*a^5 + 3*A*a^4*b)
*c)*x^3 + 16*(132*B*a^5*b + 3*A*a^4*b^2 + 140*A*a^5*c)*x^2 + 128*(12*B*a^6 + 13*A*a^5*b)*x)*sqrt(c*x^2 + b*x +
a))/(a^5*x^6)]

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giac [B]  time = 0.36, size = 2059, normalized size = 8.95

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)^(3/2)/x^7,x, algorithm="giac")

[Out]

-1/512*(12*B*a*b^5 - 7*A*b^6 - 96*B*a^2*b^3*c + 60*A*a*b^4*c + 192*B*a^3*b*c^2 - 144*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^4) + 1/7680*(180*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^11*B*a*b^5 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*b^6 - 1440*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^11*B*a^2*b^3*c + 900*(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 - 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^2*b^2*c^2 + 960*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^11*A*a^3*c^3 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^4*c^(5/2) - 1020*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^9*B*a^2*b^5 + 595*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b^6 + 8160*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^9*B*a^3*b^3*c - 5100*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b^4*c + 29760*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^9*B*a^4*b*c^2 + 12240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^3*b^2*c^2 + 15040*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^4*c^3 + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^4*b^2*c^(3/2
) - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^5*c^(5/2) + 76800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*
a^4*b*c^(5/2) + 2376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^5 - 1386*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^7*A*a^2*b^6 + 24000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*b^3*c + 11880*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^7*A*a^3*b^4*c + 13440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^5*b*c^2 + 97440*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^7*A*a^4*b^2*c^2 + 24960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^5*c^3 + 15360*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^6*B*a^4*b^4*sqrt(c) - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^5*b^2*c^(3/2) + 112640
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^4*b^3*c^(3/2) + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^6*c
^(5/2) + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^5*b*c^(5/2) - 696*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^5*B*a^4*b^5 + 1686*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*b^6 - 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^5*B*a^5*b^3*c + 42600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^4*b^4*c - 17280*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^5*B*a^6*b*c^2 + 128160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^5*b^2*c^2 + 24960*(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) + 15360*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^4*A*a^4*b^5*sqrt(c) + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^5*b^3*c^(3/2)
- 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^7*c^(5/2) + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^
6*b*c^(5/2) - 1020*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^5*b^5 + 595*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3
*A*a^4*b^6 - 22560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^6*b^3*c + 25620*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*A*a^5*b^4*c - 16320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^7*b*c^2 + 58320*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^3*A*a^6*b^2*c^2 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^7*c^3 - 30720*(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) + 3072*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^8*c^(5/2) + 12288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^7*b*c^(5/2)
+ 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^6*b^5 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*b^6 - 1440
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^7*b^3*c + 900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^6*b^4*c - 12480
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^8*b*c^2 + 13200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^7*b^2*c^2 + 9
60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^8*c^3 - 3072*B*a^9*c^(5/2) + 3072*A*a^8*b*c^(5/2))/(((sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^2 - a)^6*a^4)

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maple [B]  time = 0.08, size = 1264, normalized size = 5.50

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)^(3/2)/x^7,x)

[Out]

7/1536*A/a^6*b^5*c*(c*x^2+b*x+a)^(3/2)*x+7/512*A/a^5*b^5*c*(c*x^2+b*x+a)^(1/2)*x+11/192*A/a^5*b^3*c/x*(c*x^2+b
*x+a)^(5/2)-1/32*A/a^4*b^2*c/x^2*(c*x^2+b*x+a)^(5/2)-1/16*A/a^4*b^3*c^2*(c*x^2+b*x+a)^(1/2)*x-11/192*A/a^5*b^3
*c^2*(c*x^2+b*x+a)^(3/2)*x-1/32*A*c^2/a^4*b/x*(c*x^2+b*x+a)^(5/2)-1/48*A*c/a^3*b/x^3*(c*x^2+b*x+a)^(5/2)+1/32*
A*c^3/a^3*b*(c*x^2+b*x+a)^(1/2)*x+1/32*A*c^3/a^4*b*(c*x^2+b*x+a)^(3/2)*x+3/32*B/a^3*b^2*c^2*(c*x^2+b*x+a)^(1/2
)*x+3/32*B/a^4*b^2*c^2*(c*x^2+b*x+a)^(3/2)*x+1/16*B/a^3*b*c/x^2*(c*x^2+b*x+a)^(5/2)-3/32*B/a^4*b^2*c/x*(c*x^2+
b*x+a)^(5/2)-3/128*B/a^4*b^4*c*(c*x^2+b*x+a)^(1/2)*x-1/128*B/a^5*b^4*c*(c*x^2+b*x+a)^(3/2)*x-7/768*A/a^5*b^4/x
^2*(c*x^2+b*x+a)^(5/2)-7/1536*A/a^6*b^5/x*(c*x^2+b*x+a)^(5/2)-37/768*A/a^5*b^4*c*(c*x^2+b*x+a)^(3/2)-23/256*A/
a^4*b^4*c*(c*x^2+b*x+a)^(1/2)+3/256*B/a^(7/2)*b^5*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+7/1536*A/a^6*b
^6*(c*x^2+b*x+a)^(3/2)-3/32*B/a^(5/2)*b^3*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+1/8*B/a^2*b/x^4*(c*x
^2+b*x+a)^(5/2)+7/512*A/a^5*b^6*(c*x^2+b*x+a)^(1/2)-1/48*A*c^3/a^3*(c*x^2+b*x+a)^(3/2)-1/16*A*c^3/a^2*(c*x^2+b
*x+a)^(1/2)-1/5*B/a/x^5*(c*x^2+b*x+a)^(5/2)-1/128*B/a^5*b^5*(c*x^2+b*x+a)^(3/2)-7/1024*A/a^(9/2)*b^6*ln((b*x+2
*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+1/16*A*c^3/a^(3/2)*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-3/128*B/
a^4*b^5*(c*x^2+b*x+a)^(1/2)+1/24*A*c/a^2/x^4*(c*x^2+b*x+a)^(5/2)+1/48*A*c^2/a^3/x^2*(c*x^2+b*x+a)^(5/2)+15/256
*A/a^(7/2)*b^4*c*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)-9/64*A/a^(5/2)*b^2*c^2*ln((b*x+2*a+2*(c*x^2+b*x
+a)^(1/2)*a^(1/2))/x)+1/128*B/a^5*b^4/x*(c*x^2+b*x+a)^(5/2)+5/64*B/a^4*b^3*c*(c*x^2+b*x+a)^(3/2)+9/64*B/a^3*b^
3*c*(c*x^2+b*x+a)^(1/2)+7/60*A/a^2*b/x^5*(c*x^2+b*x+a)^(5/2)-7/96*A/a^3*b^2/x^4*(c*x^2+b*x+a)^(5/2)+1/16*A/a^4
*b^2*c^2*(c*x^2+b*x+a)^(3/2)+5/32*A/a^3*b^2*c^2*(c*x^2+b*x+a)^(1/2)+7/192*A/a^4*b^3/x^3*(c*x^2+b*x+a)^(5/2)-1/
16*B/a^3*b*c^2*(c*x^2+b*x+a)^(3/2)-3/16*B/a^2*b*c^2*(c*x^2+b*x+a)^(1/2)+3/16*B/a^(3/2)*b*c^2*ln((b*x+2*a+2*(c*
x^2+b*x+a)^(1/2)*a^(1/2))/x)-1/16*B/a^3*b^2/x^3*(c*x^2+b*x+a)^(5/2)+1/64*B/a^4*b^3/x^2*(c*x^2+b*x+a)^(5/2)-1/6
*A*(c*x^2+b*x+a)^(5/2)/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)^(3/2)/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 )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{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)^(3/2))/x^7,x)

[Out]

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/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 ) \left (a + b x + c x^{2}\right )^{\frac {3}{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)**(3/2)/x**7,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**7, x)

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