3.10.44 \(\int \frac {(A+B x) (a+b x+c x^2)^{5/2}}{x^5} \, dx\) [944]

Optimal. Leaf size=284 \[ -\frac {5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a x}-\frac {5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}-\frac {5 \left (8 a b B \left (b^2+12 a c\right )-A \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{3/2}}+\frac {5}{8} \sqrt {c} \left (3 b^2 B+4 A b c+4 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]

[Out]

-5/96*(4*a*(A*b+4*B*a)+3*(8*a*b*B+A*(4*a*c+b^2))*x)*(c*x^2+b*x+a)^(3/2)/a/x^3-1/4*(-2*B*x+A)*(c*x^2+b*x+a)^(5/
2)/x^4-5/128*(8*a*b*B*(12*a*c+b^2)-A*(-48*a^2*c^2-24*a*b^2*c+b^4))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)
^(1/2))/a^(3/2)+5/8*(4*A*b*c+4*B*a*c+3*B*b^2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/2)-5/64*
(8*a*B*(4*a*c+b^2)-A*(-20*a*b*c+b^3)-2*c*(16*a*b*B+A*(12*a*c+b^2))*x)*(c*x^2+b*x+a)^(1/2)/a/x

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Rubi [A]
time = 0.26, antiderivative size = 284, 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 = {826, 824, 857, 635, 212, 738} \begin {gather*} -\frac {5 \left (8 a b B \left (12 a c+b^2\right )-A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{3/2}}+\frac {5}{8} \sqrt {c} \left (4 a B c+4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {5 \left (a+b x+c x^2\right )^{3/2} \left (3 x \left (A \left (4 a c+b^2\right )+8 a b B\right )+4 a (4 a B+A b)\right )}{96 a x^3}-\frac {5 \sqrt {a+b x+c x^2} \left (-A \left (b^3-20 a b c\right )-2 c x \left (A \left (12 a c+b^2\right )+16 a b B\right )+8 a B \left (4 a c+b^2\right )\right )}{64 a x}-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(8*a*B*(b^2 + 4*a*c) - A*(b^3 - 20*a*b*c) - 2*c*(16*a*b*B + A*(b^2 + 12*a*c))*x)*Sqrt[a + b*x + c*x^2])/(6
4*a*x) - (5*(4*a*(A*b + 4*a*B) + 3*(8*a*b*B + A*(b^2 + 4*a*c))*x)*(a + b*x + c*x^2)^(3/2))/(96*a*x^3) - ((A -
2*B*x)*(a + b*x + c*x^2)^(5/2))/(4*x^4) - (5*(8*a*b*B*(b^2 + 12*a*c) - A*(b^4 - 24*a*b^2*c - 48*a^2*c^2))*ArcT
anh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(3/2)) + (5*Sqrt[c]*(3*b^2*B + 4*A*b*c + 4*a*B*c)*A
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/8

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^5} \, dx &=-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}-\frac {5}{16} \int \frac {(-2 (A b+4 a B)-4 (b B+A c) x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}+\frac {5 \int \frac {\left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )+2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{x^2} \, dx}{64 a}\\ &=-\frac {5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a x}-\frac {5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}-\frac {5 \int \frac {-8 a b B \left (b^2+12 a c\right )+A \left (b^4-24 a b^2 c-48 a^2 c^2\right )-16 a c \left (3 b^2 B+4 A b c+4 a B c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{128 a}\\ &=-\frac {5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a x}-\frac {5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}+\frac {1}{8} \left (5 c \left (3 b^2 B+4 A b c+4 a B c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx+\frac {\left (5 \left (8 a b B \left (b^2+12 a c\right )-A \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{128 a}\\ &=-\frac {5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a x}-\frac {5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}+\frac {1}{4} \left (5 c \left (3 b^2 B+4 A b c+4 a B c\right )\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 (5 \left (8 a b B \left (b^2+12 a c\right )-A \left (b^4-24 a b^2 c-48 a^2 c^2\right )\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 )}{64 a}\\ &=-\frac {5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a x}-\frac {5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac {(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}-\frac {5 \left (8 a b B \left (b^2+12 a c\right )-A \left (b^4-24 a b^2 c-48 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 )}{128 a^{3/2}}+\frac {5}{8} \sqrt {c} \left (3 b^2 B+4 A b c+4 a B c\right ) \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 = 2.38, size = 286, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {a+x (b+c x)} \left (15 A b^3 x^3+16 a^3 (3 A+4 B x)+8 a^2 x \left (17 A b+26 b B x+27 A c x+56 B c x^2\right )+2 a x^2 \left (A \left (59 b^2+278 b c x-96 c^2 x^2\right )-12 B x \left (-11 b^2+18 b c x+4 c^2 x^2\right )\right )\right )}{192 a x^4}-\frac {5 A b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{64 a^{3/2}}-\frac {5 \left (b^3 B+3 A b^2 c+12 a b B c+6 a A c^2\right ) \tanh ^{-1}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {5}{8} \sqrt {c} \left (3 b^2 B+4 A b c+4 a B c\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*(Sqrt[a + x*(b + c*x)]*(15*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) + 8*a^2*x*(17*A*b + 26*b*B*x + 27*A*c*x + 5
6*B*c*x^2) + 2*a*x^2*(A*(59*b^2 + 278*b*c*x - 96*c^2*x^2) - 12*B*x*(-11*b^2 + 18*b*c*x + 4*c^2*x^2))))/(a*x^4)
- (5*A*b^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(64*a^(3/2)) - (5*(b^3*B + 3*A*b^2*c + 12*a*
b*B*c + 6*a*A*c^2)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(8*Sqrt[a]) - (5*Sqrt[c]*(3*b^2*B
+ 4*A*b*c + 4*a*B*c)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3358\) vs. \(2(252)=504\).
time = 0.78, size = 3359, normalized size = 11.83

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (556 A a b c \,x^{3}+15 A \,b^{3} x^{3}+448 a^{2} B c \,x^{3}+264 B a \,b^{2} x^{3}+216 a^{2} A c \,x^{2}+118 A a \,b^{2} x^{2}+208 a^{2} b B \,x^{2}+136 A \,a^{2} b x +64 B \,a^{3} x +48 A \,a^{3}\right )}{192 x^{4} a}+\frac {B \,c^{2} x \sqrt {c \,x^{2}+b x +a}}{2}+\frac {9 B c b \sqrt {c \,x^{2}+b x +a}}{4}+\frac {15 b^{2} B \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8}+\frac {5 a B \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+A \,c^{2} \sqrt {c \,x^{2}+b x +a}+\frac {5 A b \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}-\frac {15 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,c^{2}}{8}-\frac {15 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{2} c}{16 \sqrt {a}}+\frac {5 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{4}}{128 a^{\frac {3}{2}}}-\frac {15 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b B c}{4}-\frac {5 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B \,b^{3}}{16 \sqrt {a}}\)	\(453\)
default	\(\text {Expression too large to display}\)	\(3359\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(-1/4/a/x^4*(c*x^2+b*x+a)^(7/2)-1/8*b/a*(-1/3/a/x^3*(c*x^2+b*x+a)^(7/2)+1/6*b/a*(-1/2/a/x^2*(c*x^2+b*x+a)^(7
/2)+3/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(7/2)+5/2*b/a*(1/5*(c*x^2+b*x+a)^(5/2)+1/2*b*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3
/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^
(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))+6*c/a*(1/12*(2*c*x+b)*(c*x^2+b*
x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+
b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))+5/2*c/a*(1/5*(c*x^2+b*x
+a)^(5/2)+1/2*b*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1
/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/4*(2*
c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+a*((c*x^2+b*
x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+
a)^(1/2))/x)))))+4/3*c/a*(-1/a/x*(c*x^2+b*x+a)^(7/2)+5/2*b/a*(1/5*(c*x^2+b*x+a)^(5/2)+1/2*b*(1/8*(2*c*x+b)*(c*
x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(
4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))+6*c/a*(1/12*(2*c*x
+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c
*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))))+3/4*c/a*(
-1/2/a/x^2*(c*x^2+b*x+a)^(7/2)+3/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(7/2)+5/2*b/a*(1/5*(c*x^2+b*x+a)^(5/2)+1/2*b*(1/8
*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/4*(2*c*x+b)*(c*x^2+b*x+a)
^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))+6*c/
a*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^
2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
)))+5/2*c/a*(1/5*(c*x^2+b*x+a)^(5/2)+1/2*b*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x
+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+
b*x+a)^(3/2)+1/2*b*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2)))+a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*
a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))))+B*(-1/3/a/x^3*(c*x^2+b*x+a)^(7/2)+1/6*b/a*(-1/2/a/x^2*(c*x^2+b*x+
a)^(7/2)+3/4*b/a*(-1/a/x*(c*x^2+b*x+a)^(7/2)+5/2*b/a*(1/5*(c*x^2+b*x+a)^(5/2)+1/2*b*(1/8*(2*c*x+b)*(c*x^2+b*x+
a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^
2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+a*((c*x^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))+6*c/a*(1/12*(2*c*x+b)*(c*x
^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c
*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))+5/2*c/a*(1/5*(c*x^
2+b*x+a)^(5/2)+1/2*b*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2
)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/
4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+a*((c*x
^2+b*x+a)^(1/2)+1/2*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2
+b*x+a)^(1/2))/x)))))+4/3*c/a*(-1/a/x*(c*x^2+b*x+a)^(7/2)+5/2*b/a*(1/5*(c*x^2+b*x+a)^(5/2)+1/2*b*(1/8*(2*c*x+b
)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+a*(1/3*(c*x^2+b*x+a)^(3/2)+1/2*b*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+
1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+...

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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^5,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 = 4.46, size = 1305, normalized size = 4.60 \begin {gather*} \left [\frac {240 \, {\left (3 \, B a^{2} b^{2} + 4 \, {\left (B a^{3} + A a^{2} b\right )} c\right )} \sqrt {c} x^{4} \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 (8 \, B a b^{3} - A b^{4} + 48 \, A a^{2} c^{2} + 24 \, {\left (4 \, B a^{2} b + A a b^{2}\right )} c\right )} \sqrt {a} x^{4} \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 (96 \, B a^{2} c^{2} x^{5} - 48 \, A a^{4} + 48 \, {\left (9 \, B a^{2} b c + 4 \, A a^{2} c^{2}\right )} x^{4} - {\left (264 \, B a^{2} b^{2} + 15 \, A a b^{3} + 4 \, {\left (112 \, B a^{3} + 139 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2} + 108 \, A a^{3} c\right )} x^{2} - 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{2} x^{4}}, -\frac {480 \, {\left (3 \, B a^{2} b^{2} + 4 \, {\left (B a^{3} + A a^{2} b\right )} c\right )} \sqrt {-c} x^{4} \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 (8 \, B a b^{3} - A b^{4} + 48 \, A a^{2} c^{2} + 24 \, {\left (4 \, B a^{2} b + A a b^{2}\right )} c\right )} \sqrt {a} x^{4} \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 (96 \, B a^{2} c^{2} x^{5} - 48 \, A a^{4} + 48 \, {\left (9 \, B a^{2} b c + 4 \, A a^{2} c^{2}\right )} x^{4} - {\left (264 \, B a^{2} b^{2} + 15 \, A a b^{3} + 4 \, {\left (112 \, B a^{3} + 139 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2} + 108 \, A a^{3} c\right )} x^{2} - 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{2} x^{4}}, \frac {15 \, {\left (8 \, B a b^{3} - A b^{4} + 48 \, A a^{2} c^{2} + 24 \, {\left (4 \, B a^{2} b + A a b^{2}\right )} c\right )} \sqrt {-a} x^{4} \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 ) + 120 \, {\left (3 \, B a^{2} b^{2} + 4 \, {\left (B a^{3} + A a^{2} b\right )} c\right )} \sqrt {c} x^{4} \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 (96 \, B a^{2} c^{2} x^{5} - 48 \, A a^{4} + 48 \, {\left (9 \, B a^{2} b c + 4 \, A a^{2} c^{2}\right )} x^{4} - {\left (264 \, B a^{2} b^{2} + 15 \, A a b^{3} + 4 \, {\left (112 \, B a^{3} + 139 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2} + 108 \, A a^{3} c\right )} x^{2} - 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{2} x^{4}}, \frac {15 \, {\left (8 \, B a b^{3} - A b^{4} + 48 \, A a^{2} c^{2} + 24 \, {\left (4 \, B a^{2} b + A a b^{2}\right )} c\right )} \sqrt {-a} x^{4} \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 ) - 240 \, {\left (3 \, B a^{2} b^{2} + 4 \, {\left (B a^{3} + A a^{2} b\right )} c\right )} \sqrt {-c} x^{4} \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 (96 \, B a^{2} c^{2} x^{5} - 48 \, A a^{4} + 48 \, {\left (9 \, B a^{2} b c + 4 \, A a^{2} c^{2}\right )} x^{4} - {\left (264 \, B a^{2} b^{2} + 15 \, A a b^{3} + 4 \, {\left (112 \, B a^{3} + 139 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2} + 108 \, A a^{3} c\right )} x^{2} - 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{2} x^{4}}\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^5,x, algorithm="fricas")

[Out]

[1/768*(240*(3*B*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(c)*x^4*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*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*a*b^2)*c)*sqr
t(a)*x^4*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*(96
*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b^2 + 15*A*a*b^3 + 4*(112*B*a^3 +
139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8*B*a^4 + 17*A*a^3*b)*x)*sqrt(c*x^
2 + b*x + a))/(a^2*x^4), -1/768*(480*(3*B*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(-c)*x^4*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*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*
b + A*a*b^2)*c)*sqrt(a)*x^4*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*(96*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b^2 + 15*A*a*b^
3 + 4*(112*B*a^3 + 139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8*B*a^4 + 17*A*
a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^4), 1/384*(15*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*a*
b^2)*c)*sqrt(-a)*x^4*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)) + 120*(3*B
*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(c)*x^4*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*(96*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b^2 +
15*A*a*b^3 + 4*(112*B*a^3 + 139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8*B*a
^4 + 17*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^4), 1/384*(15*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^
2*b + A*a*b^2)*c)*sqrt(-a)*x^4*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))
- 240*(3*B*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(-c)*x^4*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*(96*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b
^2 + 15*A*a*b^3 + 4*(112*B*a^3 + 139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8
*B*a^4 + 17*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^4)]

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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^{5}}\, 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**5,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1163 vs. \(2 (254) = 508\).
time = 0.93, size = 1163, normalized size = 4.10 \begin {gather*} \frac {1}{4} \, {\left (2 \, B c^{2} x + \frac {9 \, B b c^{2} + 4 \, A c^{3}}{c}\right )} \sqrt {c x^{2} + b x + a} - \frac {5 \, {\left (3 \, B b^{2} c + 4 \, B a c^{2} + 4 \, A b c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, \sqrt {c}} + \frac {5 \, {\left (8 \, B a b^{3} - A b^{4} + 96 \, B a^{2} b c + 24 \, A a b^{2} c + 48 \, A a^{2} c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a} + \frac {264 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a b^{3} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A b^{4} + 864 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{2} b c + 792 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a b^{2} c + 432 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a^{2} c^{2} + 1152 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{2} b^{2} \sqrt {c} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} A a b^{3} \sqrt {c} + 1152 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{3} c^{\frac {3}{2}} + 2304 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} A a^{2} b c^{\frac {3}{2}} - 584 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} b^{3} + 73 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b^{4} - 1248 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{3} b c - 600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{2} b^{2} c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{3} c^{2} - 2304 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{3} b^{2} \sqrt {c} - 2688 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{4} c^{\frac {3}{2}} - 3456 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{3} b c^{\frac {3}{2}} + 440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{3} b^{3} - 55 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b^{4} + 672 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{4} b c + 1320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{3} b^{2} c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{4} c^{2} + 1536 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{4} b^{2} \sqrt {c} + 2432 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{5} c^{\frac {3}{2}} + 3584 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{4} b c^{\frac {3}{2}} - 120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} b^{3} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b^{4} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{5} b c - 360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{4} b^{2} c + 432 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{5} c^{2} - 384 \, B a^{5} b^{2} \sqrt {c} - 896 \, B a^{6} c^{\frac {3}{2}} - 896 \, A a^{5} b c^{\frac {3}{2}}}{192 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{4} a} \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^5,x, algorithm="giac")

[Out]

1/4*(2*B*c^2*x + (9*B*b*c^2 + 4*A*c^3)/c)*sqrt(c*x^2 + b*x + a) - 5/8*(3*B*b^2*c + 4*B*a*c^2 + 4*A*b*c^2)*log(
abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/sqrt(c) + 5/64*(8*B*a*b^3 - A*b^4 + 96*B*a^2*b*c + 24*
A*a*b^2*c + 48*A*a^2*c^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a) + 1/192*(264*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*b^4 + 864*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^7*B*a^2*b*c + 792*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^2*c + 432*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^7*A*a^2*c^2 + 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^2*b^2*sqrt(c) + 384*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^6*A*a*b^3*sqrt(c) + 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3*c^(3/2) + 230
4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^2*b*c^(3/2) - 584*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*b^3
+ 73*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b^4 - 1248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b*c - 60
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^2*c - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*c^2 - 230
4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^3*b^2*sqrt(c) - 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^4*c
^(3/2) - 3456*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^3*b*c^(3/2) + 440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*B*a^3*b^3 - 55*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b^4 + 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B
*a^4*b*c + 1320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^2*c - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A
*a^4*c^2 + 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^4*b^2*sqrt(c) + 2432*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*B*a^5*c^(3/2) + 3584*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^4*b*c^(3/2) - 120*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*B*a^4*b^3 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*B*a^5*b*c - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^4*b^2*c + 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*A*a^5*c^2 - 384*B*a^5*b^2*sqrt(c) - 896*B*a^6*c^(3/2) - 896*A*a^5*b*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^2 - a)^4*a)

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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^5} \,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^5,x)

[Out]

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

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