3.2.74 \(\int \frac {A+B x^2}{x^5 \sqrt {a+b x^2+c x^4}} \, dx\) [174]

Optimal. Leaf size=124 \[ -\frac {A \sqrt {a+b x^2+c x^4}}{4 a x^4}+\frac {(3 A b-4 a B) \sqrt {a+b x^2+c x^4}}{8 a^2 x^2}-\frac {\left (3 A b^2-4 a b B-4 a A c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{5/2}} \]

[Out]

-1/16*(-4*A*a*c+3*A*b^2-4*B*a*b)*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a)^(1/2))/a^(5/2)-1/4*A*(c*x^4+b
*x^2+a)^(1/2)/a/x^4+1/8*(3*A*b-4*B*a)*(c*x^4+b*x^2+a)^(1/2)/a^2/x^2

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Rubi [A]
time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1265, 848, 820, 738, 212} \begin {gather*} -\frac {\left (-4 a A c-4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{5/2}}+\frac {(3 A b-4 a B) \sqrt {a+b x^2+c x^4}}{8 a^2 x^2}-\frac {A \sqrt {a+b x^2+c x^4}}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^5*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

-1/4*(A*Sqrt[a + b*x^2 + c*x^4])/(a*x^4) + ((3*A*b - 4*a*B)*Sqrt[a + b*x^2 + c*x^4])/(8*a^2*x^2) - ((3*A*b^2 -
4*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(5/2))

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])

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{x^5 \sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{4 a x^4}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (3 A b-4 a B)+A c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{4 a x^4}+\frac {(3 A b-4 a B) \sqrt {a+b x^2+c x^4}}{8 a^2 x^2}+\frac {\left (3 A b^2-4 a b B-4 a A c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 a^2}\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{4 a x^4}+\frac {(3 A b-4 a B) \sqrt {a+b x^2+c x^4}}{8 a^2 x^2}-\frac {\left (3 A b^2-4 a b B-4 a A c\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )}{8 a^2}\\ &=-\frac {A \sqrt {a+b x^2+c x^4}}{4 a x^4}+\frac {(3 A b-4 a B) \sqrt {a+b x^2+c x^4}}{8 a^2 x^2}-\frac {\left (3 A b^2-4 a b B-4 a A c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 145, normalized size = 1.17 \begin {gather*} \frac {\sqrt {a} \sqrt {a+b x^2+c x^4} \left (3 A b x^2-2 a \left (A+2 B x^2\right )\right )+3 A b^2 x^4 \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )+4 a (b B+A c) x^4 \tanh ^{-1}\left (\frac {-\sqrt {c} x^2+\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{8 a^{5/2} x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^5*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

(Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]*(3*A*b*x^2 - 2*a*(A + 2*B*x^2)) + 3*A*b^2*x^4*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a +
b*x^2 + c*x^4])/Sqrt[a]] + 4*a*(b*B + A*c)*x^4*ArcTanh[(-(Sqrt[c]*x^2) + Sqrt[a + b*x^2 + c*x^4])/Sqrt[a]])/(
8*a^(5/2)*x^4)

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Maple [A]
time = 0.07, size = 194, normalized size = 1.56

method	result	size
risch	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-3 A b \,x^{2}+4 a B \,x^{2}+2 a A \right )}{8 a^{2} x^{4}}+\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) c A}{4 a^{\frac {3}{2}}}-\frac {3 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) A \,b^{2}}{16 a^{\frac {5}{2}}}+\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) b B}{4 a^{\frac {3}{2}}}\)	\(165\)
default	\(A \left (-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a \,x^{4}}+\frac {3 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2} x^{2}}-\frac {3 b^{2} \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{16 a^{\frac {5}{2}}}+\frac {c \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\right )+B \left (-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\right )\)	\(194\)
elliptic	\(-\frac {A \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 a \,x^{4}}+\frac {3 A b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 a^{2} x^{2}}-\frac {3 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) A \,b^{2}}{16 a^{\frac {5}{2}}}+\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) c A}{4 a^{\frac {3}{2}}}-\frac {B \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 a \,x^{2}}+\frac {\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) b B}{4 a^{\frac {3}{2}}}\)	\(194\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^5/(c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

A*(-1/4*(c*x^4+b*x^2+a)^(1/2)/a/x^4+3/8*b*(c*x^4+b*x^2+a)^(1/2)/a^2/x^2-3/16*b^2/a^(5/2)*ln((2*a+b*x^2+2*a^(1/
2)*(c*x^4+b*x^2+a)^(1/2))/x^2)+1/4*c/a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2))+B*(-1/2*(c*x
^4+b*x^2+a)^(1/2)/a/x^2+1/4*b/a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^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^2+A)/x^5/(c*x^4+b*x^2+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 = 0.44, size = 255, normalized size = 2.06 \begin {gather*} \left [\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \sqrt {a} x^{4} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )}}{32 \, a^{3} x^{4}}, -\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )}}{16 \, a^{3} x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^5/(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[1/32*((4*B*a*b - 3*A*b^2 + 4*A*a*c)*sqrt(a)*x^4*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 4*sqrt(c*x^4 + b*x^2 +
a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) - 4*sqrt(c*x^4 + b*x^2 + a)*(2*A*a^2 + (4*B*a^2 - 3*A*a*b)*x^2))/(a^3*x
^4), -1/16*((4*B*a*b - 3*A*b^2 + 4*A*a*c)*sqrt(-a)*x^4*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(-
a)/(a*c*x^4 + a*b*x^2 + a^2)) + 2*sqrt(c*x^4 + b*x^2 + a)*(2*A*a^2 + (4*B*a^2 - 3*A*a*b)*x^2))/(a^3*x^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x^{2}}{x^{5} \sqrt {a + b x^{2} + c x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**5/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral((A + B*x**2)/(x**5*sqrt(a + b*x**2 + c*x**4)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (106) = 212\).
time = 3.51, size = 339, normalized size = 2.73 \begin {gather*} -\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{2}} + \frac {4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} B a b - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} B a^{2} \sqrt {c} - 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} B a^{2} b + 5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} + 8 \, A a^{2} b \sqrt {c}}{8 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^5/(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

-1/8*(4*B*a*b - 3*A*b^2 + 4*A*a*c)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) +
1/8*(4*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*B*a*b - 3*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*A*b^2 + 4
*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*A*a*c + 8*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*B*a^2*sqrt(c) -
4*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*B*a^2*b + 5*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*A*a*b^2 + 4*(sq
rt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*A*a^2*c - 8*B*a^3*sqrt(c) + 8*A*a^2*b*sqrt(c))/(((sqrt(c)*x^2 - sqrt(c*x^
4 + b*x^2 + a))^2 - a)^2*a^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {B\,x^2+A}{x^5\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^5*(a + b*x^2 + c*x^4)^(1/2)),x)

[Out]

int((A + B*x^2)/(x^5*(a + b*x^2 + c*x^4)^(1/2)), x)

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