3.10.31 \(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{x^5} \, dx\) [931]

Optimal. Leaf size=217 \[ -\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]

[Out]

-1/24*(6*a*A+(3*A*b+8*B*a)*x)*(c*x^2+b*x+a)^(3/2)/a/x^4+1/128*(8*a*b*B*(-12*a*c+b^2)-3*A*(-4*a*c+b^2)^2)*arcta
nh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2)+B*c^(3/2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1
/2))-1/64*(2*a*(8*a*b*B-3*A*(-4*a*c+b^2))+(8*a*B*(8*a*c+b^2)-3*A*(-4*a*b*c+b^3))*x)*(c*x^2+b*x+a)^(1/2)/a^2/x^
2

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Rubi [A]
time = 0.17, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {824, 857, 635, 212, 738} \begin {gather*} \frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{64 a^2 x^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5,x]

[Out]

-1/64*((2*a*(8*a*b*B - 3*A*(b^2 - 4*a*c)) + (8*a*B*(b^2 + 8*a*c) - 3*A*(b^3 - 4*a*b*c))*x)*Sqrt[a + b*x + c*x^
2])/(a^2*x^2) - ((6*a*A + (3*A*b + 8*a*B)*x)*(a + b*x + c*x^2)^(3/2))/(24*a*x^4) + ((8*a*b*B*(b^2 - 12*a*c) -
3*A*(b^2 - 4*a*c)^2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(5/2)) + B*c^(3/2)*ArcTanh
[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^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 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 )^{3/2}}{x^5} \, dx &=-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}-\frac {\int \frac {\left (\frac {1}{2} \left (-8 a b B+3 A \left (b^2-4 a c\right )\right )-8 a B c x\right ) \sqrt {a+b x+c x^2}}{x^3} \, dx}{8 a}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\int \frac {\frac {1}{4} \left (-8 a b B \left (b^2-12 a c\right )+3 A \left (b^2-4 a c\right )^2\right )+32 a^2 B c^2 x}{x \sqrt {a+b x+c x^2}} \, dx}{32 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\left (B c^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx-\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{128 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\left (2 B c^2\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 (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\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^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}+B c^{3/2} \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 = 1.64, size = 240, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {a+x (b+c x)} \left (-9 A b^3 x^3+16 a^3 (3 A+4 B x)+6 a b x^2 (4 b B x+A (b+10 c x))+8 a^2 x (3 A (3 b+5 c x)+2 B x (7 b+16 c x))\right )}{192 a^2 x^4}+\frac {3 A b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{64 a^{5/2}}+\frac {\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 a^{3/2}}-B c^{3/2} \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)^(3/2))/x^5,x]

[Out]

-1/192*(Sqrt[a + x*(b + c*x)]*(-9*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) + 6*a*b*x^2*(4*b*B*x + A*(b + 10*c*x)) + 8*
a^2*x*(3*A*(3*b + 5*c*x) + 2*B*x*(7*b + 16*c*x))))/(a^2*x^4) + (3*A*b^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c
*x)])/Sqrt[a]])/(64*a^(5/2)) + ((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*a^(3/2)) - B*c^(3/2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2179\) vs. \(2(191)=382\).
time = 0.88, size = 2180, normalized size = 10.05

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (60 A a b c \,x^{3}-9 A \,b^{3} x^{3}+256 a^{2} B c \,x^{3}+24 B a \,b^{2} x^{3}+120 a^{2} A c \,x^{2}+6 A a \,b^{2} x^{2}+112 a^{2} b B \,x^{2}+72 A \,a^{2} b x +64 B \,a^{3} x +48 A \,a^{3}\right )}{192 x^{4} a^{2}}+B \,c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,c^{2}}{8 \sqrt {a}}+\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{2} c}{16 a^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) A \,b^{4}}{128 a^{\frac {5}{2}}}-\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b B c}{4 \sqrt {a}}+\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) B \,b^{3}}{16 a^{\frac {3}{2}}}\)	\(332\)
default	\(\text {Expression too large to display}\)	\(2180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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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^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 = 7.42, size = 1083, normalized size = 4.99 \begin {gather*} \left [\frac {384 \, B a^{3} c^{\frac {3}{2}} 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 ) - 3 \, {\left (8 \, B a b^{3} - 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 (48 \, A a^{4} + {\left (24 \, B a^{2} b^{2} - 9 \, A a b^{3} + 4 \, {\left (64 \, B a^{3} + 15 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2} + 60 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{3} x^{4}}, -\frac {768 \, B a^{3} \sqrt {-c} 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 ) + 3 \, {\left (8 \, B a b^{3} - 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 (48 \, A a^{4} + {\left (24 \, B a^{2} b^{2} - 9 \, A a b^{3} + 4 \, {\left (64 \, B a^{3} + 15 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2} + 60 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{3} x^{4}}, \frac {192 \, B a^{3} c^{\frac {3}{2}} 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 ) - 3 \, {\left (8 \, B a b^{3} - 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 ) - 2 \, {\left (48 \, A a^{4} + {\left (24 \, B a^{2} b^{2} - 9 \, A a b^{3} + 4 \, {\left (64 \, B a^{3} + 15 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2} + 60 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{3} x^{4}}, -\frac {384 \, B a^{3} \sqrt {-c} 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 ) + 3 \, {\left (8 \, B a b^{3} - 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 ) + 2 \, {\left (48 \, A a^{4} + {\left (24 \, B a^{2} b^{2} - 9 \, A a b^{3} + 4 \, {\left (64 \, B a^{3} + 15 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2} + 60 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{3} 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)^(3/2)/x^5,x, algorithm="fricas")

[Out]

[1/768*(384*B*a^3*c^(3/2)*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) - 3*(8*B*a*b^3 - 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*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*
b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*
b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), -1/768*(768*B*a^3*sqrt(-c)*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)) + 3*(8*B*a*b^3 - 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*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*
A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), 1/384*(192*B*a^3*c^(3/2)*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) - 3*(8*B*a*b^3 - 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)) - 2*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^2*b)*c)*x^3 + 2*(56*B*
a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^4), -1/384*(3
84*B*a^3*sqrt(-c)*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)) + 3*(8*
B*a*b^3 - 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)) + 2*(48*A*a^4 + (24*B*a^2*b^2 - 9*A*a*b^3 + 4*(64*B*a^3 + 15*A*a^
2*b)*c)*x^3 + 2*(56*B*a^3*b + 3*A*a^2*b^2 + 60*A*a^3*c)*x^2 + 8*(8*B*a^4 + 9*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a)
)/(a^3*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 {3}{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)**(3/2)/x**5,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (191) = 382\).
time = 1.80, size = 1019, normalized size = 4.70 \begin {gather*} -B c^{\frac {3}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c - b \sqrt {c} \right |}\right ) - \frac {{\left (8 \, B a b^{3} - 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^{2}} + \frac {24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a b^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A b^{4} + 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{2} b c + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a b^{2} c + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a^{2} c^{2} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{2} b^{2} \sqrt {c} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{3} c^{\frac {3}{2}} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} A a^{2} b c^{\frac {3}{2}} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} b^{3} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b^{4} - 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{3} b c + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{2} b^{2} c + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{3} c^{2} - 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{3} b^{2} \sqrt {c} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} b^{3} \sqrt {c} - 1536 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{4} c^{\frac {3}{2}} - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{3} b^{3} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b^{4} + 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{4} b c + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{3} b^{2} c + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{4} c^{2} + 1280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{5} c^{\frac {3}{2}} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{4} b c^{\frac {3}{2}} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} b^{3} - 9 \, {\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 + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{4} b^{2} c + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{5} c^{2} - 512 \, B a^{6} c^{\frac {3}{2}}}{192 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{4} 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)^(3/2)/x^5,x, algorithm="giac")

[Out]

-B*c^(3/2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c - b*sqrt(c))) - 1/64*(8*B*a*b^3 - 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^2) + 1
/192*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*b^4 + 480*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b*c + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^2*c + 240*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*c^2 + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^2*b^2*sqrt(c) + 76
8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3*c^(3/2) + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^2*b*c^(3
/2) + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*b^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b^4 -
480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b*c + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^2*c +
144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*c^2 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^3*b^2*sqrt
(c) + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^2*b^3*sqrt(c) - 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4
*B*a^4*c^(3/2) - 88*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^3*b^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3
*A*a^2*b^4 + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^4*b*c + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A
*a^3*b^2*c + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*c^2 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
B*a^5*c^(3/2) + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^4*b*c^(3/2) + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a^4*b^3 - 9*(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 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^4*b^2*c + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*c^
2 - 512*B*a^6*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^4*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 )}^{3/2}}{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)^(3/2))/x^5,x)

[Out]

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/x^5, x)

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