3.8.43 \(\int \frac {(a+b x^2+c x^4)^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=216 \[ -\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2048 a^{9/2}}+\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}} \]

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Rubi [A]  time = 0.22, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1114, 744, 806, 720, 724, 206} \begin {gather*} -\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}+\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2048 a^{9/2}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]

[Out]

((b^2 - 4*a*c)*(7*b^2 - 4*a*c)*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(1024*a^4*x^4) - ((7*b^2 - 4*a*c)*(2*a +
b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(384*a^3*x^8) - (a + b*x^2 + c*x^4)^(5/2)/(12*a*x^12) + (7*b*(a + b*x^2 + c
*x^4)^(5/2))/(120*a^2*x^10) - ((b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2
+ c*x^4])])/(2048*a^(9/2))

Rule 206

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

Rule 720

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

Rule 724

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 744

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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[Sim
plify[m + 2*p + 3], 0]

Rule 1114

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}-\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {7 b}{2}+c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{12 a}\\ &=-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}+\frac {\left (7 b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{48 a^2}\\ &=-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{256 a^3}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2048 a^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \operatorname {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 )}{1024 a^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{1024 a^4 x^4}-\frac {\left (7 b^2-4 a c\right ) \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{384 a^3 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{12 a x^{12}}+\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{120 a^2 x^{10}}-\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2048 a^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 206, normalized size = 0.95 \begin {gather*} -\frac {\frac {\left (\frac {7 b^2}{2}-2 a c\right ) \left (16 a^{3/2} \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}-3 x^4 \left (b^2-4 a c\right ) \left (2 \sqrt {a} \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}-x^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )\right )\right )}{256 a^{7/2} x^8}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{x^{12}}-\frac {7 b \left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}}{12 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]

[Out]

-1/12*((a + b*x^2 + c*x^4)^(5/2)/x^12 - (7*b*(a + b*x^2 + c*x^4)^(5/2))/(10*a*x^10) + (((7*b^2)/2 - 2*a*c)*(16
*a^(3/2)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2) - 3*(b^2 - 4*a*c)*x^4*(2*Sqrt[a]*(2*a + b*x^2)*Sqrt[a + b*x^2
+ c*x^4] - (b^2 - 4*a*c)*x^4*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])))/(256*a^(7/2)*x^8))
/a

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IntegrateAlgebraic [A]  time = 1.94, size = 221, normalized size = 1.02 \begin {gather*} \frac {\left (-64 a^3 c^3+144 a^2 b^2 c^2-60 a b^4 c+7 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{1024 a^{9/2}}+\frac {\sqrt {a+b x^2+c x^4} \left (-1280 a^5-1664 a^4 b x^2-2240 a^4 c x^4-48 a^3 b^2 x^4-288 a^3 b c x^6-480 a^3 c^2 x^8+56 a^2 b^3 x^6+432 a^2 b^2 c x^8+1296 a^2 b c^2 x^{10}-70 a b^4 x^8-760 a b^3 c x^{10}+105 b^5 x^{10}\right )}{15360 a^4 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^(3/2)/x^13,x]

[Out]

(Sqrt[a + b*x^2 + c*x^4]*(-1280*a^5 - 1664*a^4*b*x^2 - 48*a^3*b^2*x^4 - 2240*a^4*c*x^4 + 56*a^2*b^3*x^6 - 288*
a^3*b*c*x^6 - 70*a*b^4*x^8 + 432*a^2*b^2*c*x^8 - 480*a^3*c^2*x^8 + 105*b^5*x^10 - 760*a*b^3*c*x^10 + 1296*a^2*
b*c^2*x^10))/(15360*a^4*x^12) + ((7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*ArcTanh[(Sqrt[c]*x^2 - Sq
rt[a + b*x^2 + c*x^4])/Sqrt[a]])/(1024*a^(9/2))

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fricas [A]  time = 1.51, size = 473, normalized size = 2.19 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {a} x^{12} \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 \, {\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{61440 \, a^{5} x^{12}}, \frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-a} x^{12} \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 \, {\left ({\left (105 \, a b^{5} - 760 \, a^{2} b^{3} c + 1296 \, a^{3} b c^{2}\right )} x^{10} - 2 \, {\left (35 \, a^{2} b^{4} - 216 \, a^{3} b^{2} c + 240 \, a^{4} c^{2}\right )} x^{8} - 1664 \, a^{5} b x^{2} + 8 \, {\left (7 \, a^{3} b^{3} - 36 \, a^{4} b c\right )} x^{6} - 1280 \, a^{6} - 16 \, {\left (3 \, a^{4} b^{2} + 140 \, a^{5} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30720 \, a^{5} x^{12}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="fricas")

[Out]

[-1/61440*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(a)*x^12*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*((105*a*b^5 - 760*a^2*b^3*c + 1296*a^
3*b*c^2)*x^10 - 2*(35*a^2*b^4 - 216*a^3*b^2*c + 240*a^4*c^2)*x^8 - 1664*a^5*b*x^2 + 8*(7*a^3*b^3 - 36*a^4*b*c)
*x^6 - 1280*a^6 - 16*(3*a^4*b^2 + 140*a^5*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^12), 1/30720*(15*(7*b^6 - 60
*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-a)*x^12*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*((105*a*b^5 - 760*a^2*b^3*c + 1296*a^3*b*c^2)*x^10 - 2*(35*a^2*b^4 - 216*a^3
*b^2*c + 240*a^4*c^2)*x^8 - 1664*a^5*b*x^2 + 8*(7*a^3*b^3 - 36*a^4*b*c)*x^6 - 1280*a^6 - 16*(3*a^4*b^2 + 140*a
^5*c)*x^4)*sqrt(c*x^4 + b*x^2 + a))/(a^5*x^12)]

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giac [B]  time = 0.70, size = 1235, normalized size = 5.72

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^13,x, algorithm="giac")

[Out]

1/1024*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqr
t(-a))/(sqrt(-a)*a^4) - 1/15360*(105*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^11*b^6 - 900*(sqrt(c)*x^2 - sqrt(
c*x^4 + b*x^2 + a))^11*a*b^4*c + 2160*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^11*a^2*b^2*c^2 - 960*(sqrt(c)*x^
2 - sqrt(c*x^4 + b*x^2 + a))^11*a^3*c^3 - 595*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a*b^6 + 5100*(sqrt(c)*
x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a^2*b^4*c - 12240*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a^3*b^2*c^2 - 150
40*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^9*a^4*c^3 - 76800*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^8*a^4*b*c
^(5/2) + 1386*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a^2*b^6 - 11880*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a)
)^7*a^3*b^4*c - 97440*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^7*a^4*b^2*c^2 - 24960*(sqrt(c)*x^2 - sqrt(c*x^4
+ b*x^2 + a))^7*a^5*c^3 - 112640*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^6*a^4*b^3*c^(3/2) - 61440*(sqrt(c)*x^
2 - sqrt(c*x^4 + b*x^2 + a))^6*a^5*b*c^(5/2) - 1686*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^3*b^6 - 42600*
(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^4*b^4*c - 128160*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^5*b^2
*c^2 - 24960*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^5*a^6*c^3 - 15360*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))
^4*a^4*b^5*sqrt(c) - 61440*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^4*a^5*b^3*c^(3/2) - 92160*(sqrt(c)*x^2 - sq
rt(c*x^4 + b*x^2 + a))^4*a^6*b*c^(5/2) - 595*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^4*b^6 - 25620*(sqrt(c
)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^5*b^4*c - 58320*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^6*b^2*c^2 - 1
5040*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^3*a^7*c^3 - 30720*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*a^6*b
^3*c^(3/2) - 12288*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2*a^7*b*c^(5/2) + 105*(sqrt(c)*x^2 - sqrt(c*x^4 + b
*x^2 + a))*a^5*b^6 - 900*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a^6*b^4*c - 13200*(sqrt(c)*x^2 - sqrt(c*x^4 +
b*x^2 + a))*a^7*b^2*c^2 - 960*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*a^8*c^3 - 3072*a^8*b*c^(5/2))/(((sqrt(c
)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2 - a)^6*a^4)

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maple [B]  time = 0.03, size = 457, normalized size = 2.12 \begin {gather*} \frac {c^{3} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {9 b^{2} c^{2} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{128 a^{\frac {5}{2}}}+\frac {15 b^{4} c \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {7 b^{6} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{2048 a^{\frac {9}{2}}}+\frac {27 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,c^{2}}{320 a^{2} x^{2}}-\frac {19 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3} c}{384 a^{3} x^{2}}+\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{5}}{1024 a^{4} x^{2}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}}{32 a \,x^{4}}+\frac {9 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c}{320 a^{2} x^{4}}-\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{4}}{1536 a^{3} x^{4}}-\frac {3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b c}{160 a \,x^{6}}+\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3}}{1920 a^{2} x^{6}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2}}{320 a \,x^{8}}-\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c}{48 x^{8}}-\frac {13 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b}{120 x^{10}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, a}{12 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^(3/2)/x^13,x)

[Out]

-1/320/a*b^2/x^8*(c*x^4+b*x^2+a)^(1/2)+7/1920/a^2*b^3/x^6*(c*x^4+b*x^2+a)^(1/2)-7/1536/a^3*b^4/x^4*(c*x^4+b*x^
2+a)^(1/2)+7/1024/a^4*b^5/x^2*(c*x^4+b*x^2+a)^(1/2)-7/2048/a^(9/2)*b^6*ln((b*x^2+2*a+2*(c*x^4+b*x^2+a)^(1/2)*a
^(1/2))/x^2)-9/128*c^2*b^2/a^(5/2)*ln((b*x^2+2*a+2*(c*x^4+b*x^2+a)^(1/2)*a^(1/2))/x^2)-1/32*c^2/a/x^4*(c*x^4+b
*x^2+a)^(1/2)-1/12*a/x^12*(c*x^4+b*x^2+a)^(1/2)-7/48*c/x^8*(c*x^4+b*x^2+a)^(1/2)-13/120*b/x^10*(c*x^4+b*x^2+a)
^(1/2)+1/32*c^3/a^(3/2)*ln((b*x^2+2*a+2*(c*x^4+b*x^2+a)^(1/2)*a^(1/2))/x^2)+27/320*c^2*b/a^2/x^2*(c*x^4+b*x^2+
a)^(1/2)+15/512/a^(7/2)*b^4*c*ln((b*x^2+2*a+2*(c*x^4+b*x^2+a)^(1/2)*a^(1/2))/x^2)-19/384/a^3*b^3*c/x^2*(c*x^4+
b*x^2+a)^(1/2)+9/320/a^2*b^2*c/x^4*(c*x^4+b*x^2+a)^(1/2)-3/160/a*b*c/x^6*(c*x^4+b*x^2+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((c*x^4+b*x^2+a)^(3/2)/x^13,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 details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^{13}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^(3/2)/x^13,x)

[Out]

int((a + b*x^2 + c*x^4)^(3/2)/x^13, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{13}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**(3/2)/x**13,x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/x**13, x)

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