3.10.45 \(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^6} \, dx\) [945]

Optimal. Leaf size=346 \[ \frac {\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 a^2 x}+\frac {\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac {\left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 a b^3 c+240 a^2 b c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{5/2}}+\frac {1}{2} c^{3/2} (5 b B+2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]

[Out]

1/192*(4*a*(-16*A*a*c+3*A*b^2-10*B*a*b)-3*(10*a*B*(4*a*c+b^2)-A*(-20*a*b*c+3*b^3))*x)*(c*x^2+b*x+a)^(3/2)/a^2/
x^3-1/40*(8*a*A+5*(A*b+2*B*a)*x)*(c*x^2+b*x+a)^(5/2)/a/x^5+1/256*(10*a*B*(-48*a^2*c^2-24*a*b^2*c+b^4)-A*(240*a
^2*b*c^2-40*a*b^3*c+3*b^5))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2)+1/2*c^(3/2)*(2*A*c+5*B*
b)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))+1/128*(10*a*b*B*(-20*a*c+b^2)-A*(128*a^2*c^2-28*a*b^2*c+
3*b^4)+2*c*(10*a*B*(12*a*c+b^2)-A*(-28*a*b*c+3*b^3))*x)*(c*x^2+b*x+a)^(1/2)/a^2/x

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Rubi [A]
time = 0.32, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {824, 826, 857, 635, 212, 738} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{192 a^2 x^3}+\frac {\sqrt {a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{128 a^2 x}+\frac {\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 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^{5/2}}+\frac {1}{2} c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]

[Out]

((10*a*b*B*(b^2 - 20*a*c) - A*(3*b^4 - 28*a*b^2*c + 128*a^2*c^2) + 2*c*(10*a*B*(b^2 + 12*a*c) - A*(3*b^3 - 28*
a*b*c))*x)*Sqrt[a + b*x + c*x^2])/(128*a^2*x) + ((4*a*(3*A*b^2 - 10*a*b*B - 16*a*A*c) - 3*(10*a*B*(b^2 + 4*a*c
) - A*(3*b^3 - 20*a*b*c))*x)*(a + b*x + c*x^2)^(3/2))/(192*a^2*x^3) - ((8*a*A + 5*(A*b + 2*a*B)*x)*(a + b*x +
c*x^2)^(5/2))/(40*a*x^5) + ((10*a*B*(b^4 - 24*a*b^2*c - 48*a^2*c^2) - A*(3*b^5 - 40*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^(5/2)) + (c^(3/2)*(5*b*B + 2*A*c)*ArcTanh[(b +
2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/2

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 738

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 826

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m +
2*p + 2))), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^6} \, dx &=-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}-\frac {\int \frac {\left (\frac {1}{2} \left (3 A b^2-10 a b B-16 a A c\right )-(A b+10 a B) c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx}{8 a}\\ &=\frac {\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac {\int \frac {\left (\frac {1}{4} \left (-10 a b B \left (b^2-20 a c\right )+4 A \left (\frac {3 b^4}{4}-7 a b^2 c+32 a^2 c^2\right )\right )+\frac {1}{2} c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{x^2} \, dx}{32 a^2}\\ &=\frac {\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 a^2 x}+\frac {\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}-\frac {\int \frac {\frac {1}{4} \left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-4 A \left (\frac {3 b^5}{4}-10 a b^3 c+60 a^2 b c^2\right )\right )-32 a^2 c^2 (5 b B+2 A c) x}{x \sqrt {a+b x+c x^2}} \, dx}{64 a^2}\\ &=\frac {\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 a^2 x}+\frac {\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac {1}{2} \left (c^2 (5 b B+2 A c)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx-\frac {\left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 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^2}\\ &=\frac {\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 a^2 x}+\frac {\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\left (c^2 (5 b B+2 A c)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )+\frac {\left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 a b^3 c+240 a^2 b c^2\right )\right ) \text {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^2}\\ &=\frac {\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 a^2 x}+\frac {\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac {(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac {\left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 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^{5/2}}+\frac {1}{2} c^{3/2} (5 b B+2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 3.48, size = 313, normalized size = 0.90 \begin {gather*} \frac {-\frac {\sqrt {a+x (b+c x)} \left (-45 A b^4 x^4+96 a^4 (4 A+5 B x)+30 a b^2 x^3 (5 b B x+A (b+18 c x))+16 a^3 x (5 B x (17 b+27 c x)+A (63 b+88 c x))+4 a^2 x^2 \left (5 B x \left (59 b^2+278 b c x-96 c^2 x^2\right )+2 A \left (93 b^2+311 b c x+368 c^2 x^2\right )\right )\right )}{a^2 x^5}+\frac {45 \left (A b^5+160 a^3 B c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {150 b \left (b^3 B+4 A b^2 c-24 a b B c-24 a A c^2\right ) \tanh ^{-1}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{3/2}}-960 c^{3/2} (5 b B+2 A c) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{1920} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]

[Out]

(-((Sqrt[a + x*(b + c*x)]*(-45*A*b^4*x^4 + 96*a^4*(4*A + 5*B*x) + 30*a*b^2*x^3*(5*b*B*x + A*(b + 18*c*x)) + 16
*a^3*x*(5*B*x*(17*b + 27*c*x) + A*(63*b + 88*c*x)) + 4*a^2*x^2*(5*B*x*(59*b^2 + 278*b*c*x - 96*c^2*x^2) + 2*A*
(93*b^2 + 311*b*c*x + 368*c^2*x^2))))/(a^2*x^5)) + (45*(A*b^5 + 160*a^3*B*c^2)*ArcTanh[(Sqrt[c]*x - Sqrt[a + x
*(b + c*x)])/Sqrt[a]])/a^(5/2) + (150*b*(b^3*B + 4*A*b^2*c - 24*a*b*B*c - 24*a*A*c^2)*ArcTanh[(-(Sqrt[c]*x) +
Sqrt[a + x*(b + c*x)])/Sqrt[a]])/a^(3/2) - 960*c^(3/2)*(5*b*B + 2*A*c)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b
+ c*x)]])/1920

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5473\) vs. \(2(314)=628\).
time = 0.78, size = 5474, normalized size = 15.82

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (2944 a^{2} A \,c^{2} x^{4}+540 A \,b^{2} c \,x^{4} a -45 A \,b^{4} x^{4}+5560 B \,a^{2} b c \,x^{4}+150 x^{4} B a \,b^{3}+2488 A \,a^{2} b c \,x^{3}+30 a A \,b^{3} x^{3}+2160 a^{3} B c \,x^{3}+1180 x^{3} B \,a^{2} b^{2}+1408 a^{3} A c \,x^{2}+744 a^{2} A \,b^{2} x^{2}+1360 x^{2} B \,a^{3} b +1008 A \,a^{3} b x +480 B \,a^{4} x +384 a^{4} A \right )}{1920 x^{5} a^{2}}+B \,c^{2} \sqrt {c \,x^{2}+b x +a}+\frac {5 B b \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+A \,c^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {15 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A b \,c^{2}}{16 \sqrt {a}}+\frac {5 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{3} c}{32 a^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{5} A}{256 a^{\frac {5}{2}}}-\frac {15 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B \,c^{2}}{8}-\frac {15 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B \,b^{2} c}{16 \sqrt {a}}+\frac {5 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{4} B}{128 a^{\frac {3}{2}}}\)	\(482\)
default	\(\text {Expression too large to display}\)	\(5474\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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)^(5/2)/x^6,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 deta

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Fricas [A]
time = 7.62, size = 1445, normalized size = 4.18 \begin {gather*} \left [\frac {1920 \, {\left (5 \, B a^{3} b c + 2 \, A a^{3} c^{2}\right )} \sqrt {c} x^{5} \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 ) - 15 \, {\left (10 \, B a b^{4} - 3 \, A b^{5} - 240 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 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 (1920 \, B a^{3} c^{2} x^{5} - 384 \, A a^{5} - {\left (150 \, B a^{2} b^{3} - 45 \, A a b^{4} + 2944 \, A a^{3} c^{2} + 20 \, {\left (278 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} c\right )} x^{4} - 2 \, {\left (590 \, B a^{3} b^{2} + 15 \, A a^{2} b^{3} + 4 \, {\left (270 \, B a^{4} + 311 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2} + 176 \, A a^{4} c\right )} x^{2} - 48 \, {\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, a^{3} x^{5}}, -\frac {3840 \, {\left (5 \, B a^{3} b c + 2 \, A a^{3} c^{2}\right )} \sqrt {-c} x^{5} \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 ) + 15 \, {\left (10 \, B a b^{4} - 3 \, A b^{5} - 240 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 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 (1920 \, B a^{3} c^{2} x^{5} - 384 \, A a^{5} - {\left (150 \, B a^{2} b^{3} - 45 \, A a b^{4} + 2944 \, A a^{3} c^{2} + 20 \, {\left (278 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} c\right )} x^{4} - 2 \, {\left (590 \, B a^{3} b^{2} + 15 \, A a^{2} b^{3} + 4 \, {\left (270 \, B a^{4} + 311 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2} + 176 \, A a^{4} c\right )} x^{2} - 48 \, {\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, a^{3} x^{5}}, -\frac {15 \, {\left (10 \, B a b^{4} - 3 \, A b^{5} - 240 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 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 ) - 960 \, {\left (5 \, B a^{3} b c + 2 \, A a^{3} c^{2}\right )} \sqrt {c} x^{5} \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 ) - 2 \, {\left (1920 \, B a^{3} c^{2} x^{5} - 384 \, A a^{5} - {\left (150 \, B a^{2} b^{3} - 45 \, A a b^{4} + 2944 \, A a^{3} c^{2} + 20 \, {\left (278 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} c\right )} x^{4} - 2 \, {\left (590 \, B a^{3} b^{2} + 15 \, A a^{2} b^{3} + 4 \, {\left (270 \, B a^{4} + 311 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2} + 176 \, A a^{4} c\right )} x^{2} - 48 \, {\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, a^{3} x^{5}}, -\frac {15 \, {\left (10 \, B a b^{4} - 3 \, A b^{5} - 240 \, {\left (2 \, B a^{3} + A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 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 ) + 1920 \, {\left (5 \, B a^{3} b c + 2 \, A a^{3} c^{2}\right )} \sqrt {-c} x^{5} \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 (1920 \, B a^{3} c^{2} x^{5} - 384 \, A a^{5} - {\left (150 \, B a^{2} b^{3} - 45 \, A a b^{4} + 2944 \, A a^{3} c^{2} + 20 \, {\left (278 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} c\right )} x^{4} - 2 \, {\left (590 \, B a^{3} b^{2} + 15 \, A a^{2} b^{3} + 4 \, {\left (270 \, B a^{4} + 311 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (170 \, B a^{4} b + 93 \, A a^{3} b^{2} + 176 \, A a^{4} c\right )} x^{2} - 48 \, {\left (10 \, B a^{5} + 21 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, a^{3} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(1920*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(c)*x^5*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) - 15*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - 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*(1920*B*a^3*c^2*x^5 - 384*A*a^5 - (150*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27
*A*a^2*b^2)*c)*x^4 - 2*(590*B*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^3*b)*c)*x^3 - 8*(170*B*a^4*b + 9
3*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B*a^5 + 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^5), -1/7680*(3840
*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(-c)*x^5*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)) + 15*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - 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*(1920*B*a^
3*c^2*x^5 - 384*A*a^5 - (150*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27*A*a^2*b^2)*c)*x^4
- 2*(590*B*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^3*b)*c)*x^3 - 8*(170*B*a^4*b + 93*A*a^3*b^2 + 176*A
*a^4*c)*x^2 - 48*(10*B*a^5 + 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^5), -1/3840*(15*(10*B*a*b^4 - 3*A*b^
5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - 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)) - 960*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(c)*x^5*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) - 2*(1920*B*a^3*c^2*x^5 - 384*A*a^5 - (150
*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27*A*a^2*b^2)*c)*x^4 - 2*(590*B*a^3*b^2 + 15*A*a^
2*b^3 + 4*(270*B*a^4 + 311*A*a^3*b)*c)*x^3 - 8*(170*B*a^4*b + 93*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B*a^5 +
21*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^5), -1/3840*(15*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*
c^2 - 40*(6*B*a^2*b^2 - 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)) + 1920*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(-c)*x^5*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*(1920*B*a^3*c^2*x^5 - 384*A*a^5 - (150*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*
a^3*c^2 + 20*(278*B*a^3*b + 27*A*a^2*b^2)*c)*x^4 - 2*(590*B*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^3*
b)*c)*x^3 - 8*(170*B*a^4*b + 93*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B*a^5 + 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x
+ a))/(a^3*x^5)]

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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 {5}{2}}}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**6,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**6, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1526 vs. \(2 (314) = 628\).
time = 1.36, size = 1526, normalized size = 4.41 \begin {gather*} \sqrt {c x^{2} + b x + a} B c^{2} - \frac {{\left (5 \, B b c^{\frac {5}{2}} + 2 \, A c^{\frac {7}{2}}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c - b \sqrt {c} \right |}\right )}{2 \, c} - \frac {{\left (10 \, B a b^{4} - 3 \, A b^{5} - 240 \, B a^{2} b^{2} c + 40 \, A a b^{3} c - 480 \, B a^{3} c^{2} - 240 \, A a^{2} b c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{128 \, \sqrt {-a} a^{2}} + \frac {150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} B a b^{4} - 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} A b^{5} + 7920 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} B a^{2} b^{2} c + 600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} A a b^{3} c + 4320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} B a^{3} c^{2} + 7920 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{9} A a^{2} b c^{2} + 3840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} B a^{2} b^{3} \sqrt {c} + 23040 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} B a^{3} b c^{\frac {3}{2}} + 11520 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} A a^{2} b^{2} c^{\frac {3}{2}} + 11520 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} A a^{3} c^{\frac {5}{2}} + 580 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{2} b^{4} + 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a b^{5} - 13920 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{3} b^{2} c + 6160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a^{2} b^{3} c - 4800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{4} c^{2} - 2400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a^{3} b c^{2} - 3840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{3} b^{3} \sqrt {c} + 3840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} A a^{2} b^{4} \sqrt {c} - 57600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{4} b c^{\frac {3}{2}} - 23040 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} A a^{4} c^{\frac {5}{2}} - 1280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{3} b^{4} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{2} b^{5} + 19200 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{4} b^{2} c + 6400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{3} b^{3} c + 19200 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{4} b c^{2} + 70400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{5} b c^{\frac {3}{2}} + 19200 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{4} b^{2} c^{\frac {3}{2}} + 35840 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{5} c^{\frac {5}{2}} + 700 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{4} b^{4} - 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{3} b^{5} - 16800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{5} b^{2} c + 2800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{4} b^{3} c + 4800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{6} c^{2} + 2400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{5} b c^{2} - 44800 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{6} b c^{\frac {3}{2}} - 17920 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{6} c^{\frac {5}{2}} - 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{5} b^{4} + 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{4} b^{5} + 3600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{6} b^{2} c - 600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{5} b^{3} c - 4320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{7} c^{2} + 3600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{6} b c^{2} + 8960 \, B a^{7} b c^{\frac {3}{2}} + 5888 \, A a^{7} c^{\frac {5}{2}}}{1920 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{5} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^6,x, algorithm="giac")

[Out]

sqrt(c*x^2 + b*x + a)*B*c^2 - 1/2*(5*B*b*c^(5/2) + 2*A*c^(7/2))*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*c - b*sqrt(c)))/c - 1/128*(10*B*a*b^4 - 3*A*b^5 - 240*B*a^2*b^2*c + 40*A*a*b^3*c - 480*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^2) + 1/1920*(150*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^9*B*a*b^4 - 45*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*b^5 + 7920*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^9*B*a^2*b^2*c + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b^3*c + 4320*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^9*B*a^3*c^2 + 7920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b*c^2 + 3840*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^8*B*a^2*b^3*sqrt(c) + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^3*b*c^(3/2) + 11520*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^8*A*a^2*b^2*c^(3/2) + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^3*c^(5/2) + 5
80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b^4 + 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^5 - 13920
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^2*c + 6160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*b^3*c -
4800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*c^2 - 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b*c^2
- 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3*b^3*sqrt(c) + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*
a^2*b^4*sqrt(c) - 57600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^4*b*c^(3/2) - 23040*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^6*A*a^4*c^(5/2) - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b^4 + 384*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*A*a^2*b^5 + 19200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^4*b^2*c + 6400*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^5*A*a^3*b^3*c + 19200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^4*b*c^2 + 70400*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*B*a^5*b*c^(3/2) + 19200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^4*b^2*c^(3/2) + 35
840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^5*c^(5/2) + 700*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^4*b^4
- 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^5 - 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^5*b^2*
c + 2800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*b^3*c + 4800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^6*
c^2 + 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^5*b*c^2 - 44800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a
^6*b*c^(3/2) - 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^6*c^(5/2) - 150*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a^5*b^4 + 45*(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 - 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*b^3*c - 4320*(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 + 8960*B*a^7*b*c^(3/2) + 5888*A*a^7*c^(5/2))/((
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^5*a^2)

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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 )}^{5/2}}{x^6} \,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)^(5/2))/x^6,x)

[Out]

int(((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6, x)

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