3.194 \(\int \frac{A+B x^2}{x^{9/2} (b x^2+c x^4)} \, dx\)

Optimal. Leaf size=278 \[ -\frac{c^{7/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}-\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{15/4}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{11 b x^{11/2}} \]

[Out]

(-2*A)/(11*b*x^(11/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(3*b^3*x^(3/2)) - (c^(7/4)*(b*B -
A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*ArcTan[1 + (Sqr
t[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(15/4)) - (c^(7/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4
)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt
[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4))

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Rubi [A]  time = 0.241391, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1584, 453, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{c^{7/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}-\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{15/4}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{11 b x^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^(9/2)*(b*x^2 + c*x^4)),x]

[Out]

(-2*A)/(11*b*x^(11/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(3*b^3*x^(3/2)) - (c^(7/4)*(b*B -
A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*ArcTan[1 + (Sqr
t[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(15/4)) - (c^(7/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4
)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt
[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4))

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{A+B x^2}{x^{9/2} \left (b x^2+c x^4\right )} \, dx &=\int \frac{A+B x^2}{x^{13/2} \left (b+c x^2\right )} \, dx\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{\left (2 \left (-\frac{11 b B}{2}+\frac{11 A c}{2}\right )\right ) \int \frac{1}{x^{9/2} \left (b+c x^2\right )} \, dx}{11 b}\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{(c (b B-A c)) \int \frac{1}{x^{5/2} \left (b+c x^2\right )} \, dx}{b^2}\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}+\frac{\left (c^2 (b B-A c)\right ) \int \frac{1}{\sqrt{x} \left (b+c x^2\right )} \, dx}{b^3}\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}+\frac{\left (2 c^2 (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}+\frac{\left (c^2 (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^{7/2}}+\frac{\left (c^2 (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{c} x^2}{b+c x^4} \, dx,x,\sqrt{x}\right )}{b^{7/2}}\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}+\frac{\left (c^{3/2} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{2 b^{7/2}}+\frac{\left (c^{3/2} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{2 b^{7/2}}-\frac{\left (c^{7/4} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{b}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} b^{15/4}}-\frac{\left (c^{7/4} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{b}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} b^{15/4}}\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{c^{7/4} (b B-A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}+\frac{\left (c^{7/4} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}-\frac{\left (c^{7/4} (b B-A c)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}\\ &=-\frac{2 A}{11 b x^{11/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}-\frac{c^{7/4} (b B-A c) \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}\\ \end{align*}

Mathematica [C]  time = 0.0164283, size = 47, normalized size = 0.17 \[ \frac{-22 x^2 (b B-A c) \, _2F_1\left (-\frac{7}{4},1;-\frac{3}{4};-\frac{c x^2}{b}\right )-14 A b}{77 b^2 x^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^(9/2)*(b*x^2 + c*x^4)),x]

[Out]

(-14*A*b - 22*(b*B - A*c)*x^2*Hypergeometric2F1[-7/4, 1, -3/4, -((c*x^2)/b)])/(77*b^2*x^(11/2))

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Maple [A]  time = 0.011, size = 336, normalized size = 1.2 \begin{align*} -{\frac{2\,A}{11\,b}{x}^{-{\frac{11}{2}}}}+{\frac{2\,Ac}{7\,{b}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,b}{x}^{-{\frac{7}{2}}}}-{\frac{2\,A{c}^{2}}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{2\,Bc}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{{c}^{3}\sqrt{2}A}{2\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{{c}^{3}\sqrt{2}A}{2\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{{c}^{3}\sqrt{2}A}{4\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{{c}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{{c}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{{c}^{2}\sqrt{2}B}{4\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^(9/2)/(c*x^4+b*x^2),x)

[Out]

-2/11*A/b/x^(11/2)+2/7/b^2/x^(7/2)*A*c-2/7/b/x^(7/2)*B-2/3/b^3*c^2/x^(3/2)*A+2/3/b^2*c/x^(3/2)*B-1/2*c^3/b^4*(
b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-1/2*c^3/b^4*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b
/c)^(1/4)*x^(1/2)-1)-1/4*c^3/b^4*(b/c)^(1/4)*2^(1/2)*A*ln((x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x-(b/c)
^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))+1/2*c^2/b^3*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+1
/2*c^2/b^3*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+1/4*c^2/b^3*(b/c)^(1/4)*2^(1/2)*B*ln((x
+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^(9/2)/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.52287, size = 1540, normalized size = 5.54 \begin{align*} -\frac{924 \, b^{3} x^{6} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{b^{8} \sqrt{-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}} +{\left (B^{2} b^{2} c^{4} - 2 \, A B b c^{5} + A^{2} c^{6}\right )} x} b^{11} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{3}{4}} +{\left (B b^{12} c^{2} - A b^{11} c^{3}\right )} \sqrt{x} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{3}{4}}}{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}\right ) + 231 \, b^{3} x^{6} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} \log \left (b^{4} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} -{\left (B b c^{2} - A c^{3}\right )} \sqrt{x}\right ) - 231 \, b^{3} x^{6} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} \log \left (-b^{4} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} -{\left (B b c^{2} - A c^{3}\right )} \sqrt{x}\right ) - 4 \,{\left (77 \,{\left (B b c - A c^{2}\right )} x^{4} - 21 \, A b^{2} - 33 \,{\left (B b^{2} - A b c\right )} x^{2}\right )} \sqrt{x}}{462 \, b^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^(9/2)/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

-1/462*(924*b^3*x^6*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*A^3*B*b*c^10 + A^4*c^11)/b^15)^(1
/4)*arctan((sqrt(b^8*sqrt(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*A^3*B*b*c^10 + A^4*c^11)/b^1
5) + (B^2*b^2*c^4 - 2*A*B*b*c^5 + A^2*c^6)*x)*b^11*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*A^
3*B*b*c^10 + A^4*c^11)/b^15)^(3/4) + (B*b^12*c^2 - A*b^11*c^3)*sqrt(x)*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^
2*B^2*b^2*c^9 - 4*A^3*B*b*c^10 + A^4*c^11)/b^15)^(3/4))/(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4
*A^3*B*b*c^10 + A^4*c^11)) + 231*b^3*x^6*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*A^3*B*b*c^10
+ A^4*c^11)/b^15)^(1/4)*log(b^4*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*A^3*B*b*c^10 + A^4*c
^11)/b^15)^(1/4) - (B*b*c^2 - A*c^3)*sqrt(x)) - 231*b^3*x^6*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c
^9 - 4*A^3*B*b*c^10 + A^4*c^11)/b^15)^(1/4)*log(-b^4*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*
A^3*B*b*c^10 + A^4*c^11)/b^15)^(1/4) - (B*b*c^2 - A*c^3)*sqrt(x)) - 4*(77*(B*b*c - A*c^2)*x^4 - 21*A*b^2 - 33*
(B*b^2 - A*b*c)*x^2)*sqrt(x))/(b^3*x^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**(9/2)/(c*x**4+b*x**2),x)

[Out]

Timed out

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Giac [A]  time = 1.21939, size = 393, normalized size = 1.41 \begin{align*} \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4}} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4}} + \frac{2 \,{\left (77 \, B b c x^{4} - 77 \, A c^{2} x^{4} - 33 \, B b^{2} x^{2} + 33 \, A b c x^{2} - 21 \, A b^{2}\right )}}{231 \, b^{3} x^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^(9/2)/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((b*c^3)^(1/4)*B*b*c - (b*c^3)^(1/4)*A*c^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(
b/c)^(1/4))/b^4 + 1/2*sqrt(2)*((b*c^3)^(1/4)*B*b*c - (b*c^3)^(1/4)*A*c^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(
1/4) - 2*sqrt(x))/(b/c)^(1/4))/b^4 + 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b*c - (b*c^3)^(1/4)*A*c^2)*log(sqrt(2)*sqrt(
x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 - 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b*c - (b*c^3)^(1/4)*A*c^2)*log(-sqrt(2)*sqr
t(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 + 2/231*(77*B*b*c*x^4 - 77*A*c^2*x^4 - 33*B*b^2*x^2 + 33*A*b*c*x^2 - 21*
A*b^2)/(b^3*x^(11/2))