3.10.59 \(\int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx\) [959]

Optimal. Leaf size=231 \[ -\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}+\frac {\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{9/2}} \]

[Out]

1/128*(8*a*b*B*(-12*a*c+5*b^2)-A*(48*a^2*c^2-120*a*b^2*c+35*b^4))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^
(1/2))/a^(9/2)-1/4*A*(c*x^2+b*x+a)^(1/2)/a/x^4+1/24*(7*A*b-8*B*a)*(c*x^2+b*x+a)^(1/2)/a^2/x^3-1/96*(-36*A*a*c+
35*A*b^2-40*B*a*b)*(c*x^2+b*x+a)^(1/2)/a^3/x^2+1/192*(-220*A*a*b*c+105*A*b^3+128*B*a^2*c-120*B*a*b^2)*(c*x^2+b
*x+a)^(1/2)/a^4/x

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Rubi [A]
time = 0.19, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {848, 820, 738, 212} \begin {gather*} -\frac {\sqrt {a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{96 a^3 x^2}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}+\frac {\left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{9/2}}+\frac {\sqrt {a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{192 a^4 x}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(A*Sqrt[a + b*x + c*x^2])/(a*x^4) + ((7*A*b - 8*a*B)*Sqrt[a + b*x + c*x^2])/(24*a^2*x^3) - ((35*A*b^2 - 4
0*a*b*B - 36*a*A*c)*Sqrt[a + b*x + c*x^2])/(96*a^3*x^2) + ((105*A*b^3 - 120*a*b^2*B - 220*a*A*b*c + 128*a^2*B*
c)*Sqrt[a + b*x + c*x^2])/(192*a^4*x) + ((8*a*b*B*(5*b^2 - 12*a*c) - A*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2))*Ar
cTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(9/2))

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 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 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 848

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

Rubi steps

\begin {align*} \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx &=-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}-\frac {\int \frac {\frac {1}{2} (7 A b-8 a B)+3 A c x}{x^4 \sqrt {a+b x+c x^2}} \, dx}{4 a}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}+\frac {\int \frac {\frac {1}{4} \left (35 A b^2-40 a b B-36 a A c\right )+(7 A b-8 a B) c x}{x^3 \sqrt {a+b x+c x^2}} \, dx}{12 a^2}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}-\frac {\int \frac {\frac {1}{8} \left (-8 a B \left (15 b^2-16 a c\right )+5 A \left (21 b^3-44 a b c\right )\right )+\frac {1}{4} c \left (35 A b^2-40 a b B-36 a A c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{24 a^3}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}-\frac {\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{128 a^4}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}+\frac {\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 a b^2 c+48 a^2 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 )}{64 a^4}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}+\frac {\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 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^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.14, size = 201, normalized size = 0.87 \begin {gather*} \frac {\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (105 A b^3 x^3-16 a^3 (3 A+4 B x)-10 a b x^2 (7 A b+12 b B x+22 A c x)+8 a^2 x (2 B x (5 b+8 c x)+A (7 b+9 c x))\right )}{x^4}+105 A b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-24 a \left (-5 b^3 B-15 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 )}{192 a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[a]*Sqrt[a + x*(b + c*x)]*(105*A*b^3*x^3 - 16*a^3*(3*A + 4*B*x) - 10*a*b*x^2*(7*A*b + 12*b*B*x + 22*A*c*
x) + 8*a^2*x*(2*B*x*(5*b + 8*c*x) + A*(7*b + 9*c*x))))/x^4 + 105*A*b^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*
x)])/Sqrt[a]] - 24*a*(-5*b^3*B - 15*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]])/(192*a^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(570\) vs. \(2(205)=410\).
time = 0.76, size = 571, normalized size = 2.47

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (220 A a b c \,x^{3}-105 A \,b^{3} x^{3}-128 a^{2} B c \,x^{3}+120 B a \,b^{2} x^{3}-72 a^{2} A c \,x^{2}+70 A a \,b^{2} x^{2}-80 a^{2} b B \,x^{2}-56 A \,a^{2} b x +64 B \,a^{3} x +48 A \,a^{3}\right )}{192 a^{4} x^{4}}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,c^{2}}{8 a^{\frac {5}{2}}}+\frac {15 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{2} c}{16 a^{\frac {7}{2}}}-\frac {35 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{4}}{128 a^{\frac {9}{2}}}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b B c}{4 a^{\frac {5}{2}}}+\frac {5 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B \,b^{3}}{16 a^{\frac {7}{2}}}\)	\(302\)
default	\(A \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{4 a \,x^{4}}-\frac {7 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )}{8 a}-\frac {3 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )+B \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )\)	\(571\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [A]
time = 4.27, size = 425, normalized size = 1.84 \begin {gather*} \left [-\frac {3 \, {\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \, {\left (4 \, B a^{2} b - 5 \, 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 (48 \, A a^{4} + {\left (120 \, B a^{2} b^{2} - 105 \, A a b^{3} - 4 \, {\left (32 \, B a^{3} - 55 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (40 \, B a^{3} b - 35 \, A a^{2} b^{2} + 36 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{5} x^{4}}, -\frac {3 \, {\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \, {\left (4 \, B a^{2} b - 5 \, 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 ) + 2 \, {\left (48 \, A a^{4} + {\left (120 \, B a^{2} b^{2} - 105 \, A a b^{3} - 4 \, {\left (32 \, B a^{3} - 55 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (40 \, B a^{3} b - 35 \, A a^{2} b^{2} + 36 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{5} x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(3*(40*B*a*b^3 - 35*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - 5*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*(48*A*a^4 + (120*B*a^2*b^2 -
105*A*a*b^3 - 4*(32*B*a^3 - 55*A*a^2*b)*c)*x^3 - 2*(40*B*a^3*b - 35*A*a^2*b^2 + 36*A*a^3*c)*x^2 + 8*(8*B*a^4 -
7*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^5*x^4), -1/384*(3*(40*B*a*b^3 - 35*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2
*b - 5*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))
+ 2*(48*A*a^4 + (120*B*a^2*b^2 - 105*A*a*b^3 - 4*(32*B*a^3 - 55*A*a^2*b)*c)*x^3 - 2*(40*B*a^3*b - 35*A*a^2*b^
2 + 36*A*a^3*c)*x^2 + 8*(8*B*a^4 - 7*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^5*x^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{5} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**5/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (205) = 410\).
time = 0.84, size = 884, normalized size = 3.83 \begin {gather*} -\frac {{\left (40 \, B a b^{3} - 35 \, A b^{4} - 96 \, B a^{2} b c + 120 \, 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^{4}} + \frac {120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a b^{3} - 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A b^{4} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{2} b c + 360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a b^{2} c - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a^{2} c^{2} - 440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} b^{3} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b^{4} + 1056 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{3} b c - 1320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{2} b^{2} c + 528 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{3} c^{2} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{4} c^{\frac {3}{2}} + 584 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{3} b^{3} - 511 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b^{4} - 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{4} b c + 1752 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{3} b^{2} c + 528 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{4} c^{2} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{4} b^{2} \sqrt {c} - 1024 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{5} c^{\frac {3}{2}} + 2048 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{4} b c^{\frac {3}{2}} - 264 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} b^{3} + 279 \, {\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 - 144 \, {\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} + 384 \, A a^{4} b^{3} \sqrt {c} + 256 \, B a^{6} c^{\frac {3}{2}} - 512 \, 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^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/64*(40*B*a*b^3 - 35*A*b^4 - 96*B*a^2*b*c + 120*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^4) + 1/192*(120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 - 105*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^7*A*b^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b*c + 360*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^7*A*a*b^2*c - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*c^2 - 440*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^5*B*a^2*b^3 + 385*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b^4 + 1056*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*B*a^3*b*c - 1320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^2*c + 528*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^5*A*a^3*c^2 + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^4*c^(3/2) + 584*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*B*a^3*b^3 - 511*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b^4 - 480*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^3*B*a^4*b*c + 1752*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^2*c + 528*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^3*A*a^4*c^2 + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^4*b^2*sqrt(c) - 1024*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*B*a^5*c^(3/2) + 2048*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^4*b*c^(3/2) - 264*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^4*b^3 + 279*(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 - 144*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*A*a^5*c^2 - 384*B*a^5*b^2*sqrt(c) + 384*A*a^4*b^3*sqrt(c) + 256*B*a^6*c^(3/2) - 512*A*a
^5*b*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^4*a^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^5\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^5*(a + b*x + c*x^2)^(1/2)),x)

[Out]

int((A + B*x)/(x^5*(a + b*x + c*x^2)^(1/2)), x)

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