3.1016 \(\int \frac{A+B x}{x^{9/2} (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=381 \[ -\frac{\sqrt{2} \sqrt{c} \left (\frac{a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt{b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^4 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \left (-\frac{a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt{b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^4 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 \left (-a A c-a b B+A b^2\right )}{3 a^3 x^{3/2}}-\frac{2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt{x}}+\frac{2 (A b-a B)}{5 a^2 x^{5/2}}-\frac{2 A}{7 a x^{7/2}} \]

[Out]

(-2*A)/(7*a*x^(7/2)) + (2*(A*b - a*B))/(5*a^2*x^(5/2)) - (2*(A*b^2 - a*b*B - a*A*c))/(3*a^3*x^(3/2)) - (2*(a*B
*(b^2 - a*c) - A*(b^3 - 2*a*b*c)))/(a^4*Sqrt[x]) - (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - a*c) - A*(b^3 - 2*a*b*c) + (a*
b*B*(b^2 - 3*a*c) - A*(b^4 - 4*a*b^2*c + 2*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[
b - Sqrt[b^2 - 4*a*c]]])/(a^4*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - a*c) - A*(b^3 - 2*a*
b*c) - (a*b*B*(b^2 - 3*a*c) - A*(b^4 - 4*a*b^2*c + 2*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt
[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^4*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 1.67087, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {828, 826, 1166, 205} \[ -\frac{\sqrt{2} \sqrt{c} \left (\frac{a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt{b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^4 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} \left (-\frac{a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt{b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^4 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{2 \left (-a A c-a b B+A b^2\right )}{3 a^3 x^{3/2}}-\frac{2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt{x}}+\frac{2 (A b-a B)}{5 a^2 x^{5/2}}-\frac{2 A}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(7*a*x^(7/2)) + (2*(A*b - a*B))/(5*a^2*x^(5/2)) - (2*(A*b^2 - a*b*B - a*A*c))/(3*a^3*x^(3/2)) - (2*(a*B
*(b^2 - a*c) - A*(b^3 - 2*a*b*c)))/(a^4*Sqrt[x]) - (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - a*c) - A*(b^3 - 2*a*b*c) + (a*
b*B*(b^2 - 3*a*c) - A*(b^4 - 4*a*b^2*c + 2*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[
b - Sqrt[b^2 - 4*a*c]]])/(a^4*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - a*c) - A*(b^3 - 2*a*
b*c) - (a*b*B*(b^2 - 3*a*c) - A*(b^4 - 4*a*b^2*c + 2*a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt
[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^4*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 828

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 1166

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

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 1.03806, size = 430, normalized size = 1.13 \[ \frac{\frac{105 \sqrt{2} \sqrt{c} \left (\frac{\left (A \left (2 a^2 c^2+b^3 \sqrt{b^2-4 a c}-4 a b^2 c-2 a b c \sqrt{b^2-4 a c}+b^4\right )+a B \left (-b^2 \sqrt{b^2-4 a c}+a c \sqrt{b^2-4 a c}+3 a b c-b^3\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (A \left (-2 a^2 c^2+b^3 \sqrt{b^2-4 a c}+4 a b^2 c-2 a b c \sqrt{b^2-4 a c}-b^4\right )+a B \left (-b^2 \sqrt{b^2-4 a c}+a c \sqrt{b^2-4 a c}-3 a b c+b^3\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c}}+\frac{42 a^2 (A b-a B)}{x^{5/2}}-\frac{30 a^3 A}{x^{7/2}}+\frac{70 a \left (a A c+a b B-A b^2\right )}{x^{3/2}}+\frac{210 \left (A \left (b^3-2 a b c\right )+a B \left (a c-b^2\right )\right )}{\sqrt{x}}}{105 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-30*a^3*A)/x^(7/2) + (42*a^2*(A*b - a*B))/x^(5/2) + (70*a*(-(A*b^2) + a*b*B + a*A*c))/x^(3/2) + (210*(a*B*(-
b^2 + a*c) + A*(b^3 - 2*a*b*c)))/Sqrt[x] + (105*Sqrt[2]*Sqrt[c]*(((a*B*(-b^3 + 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c]
+ a*c*Sqrt[b^2 - 4*a*c]) + A*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 2*a*b*c*Sqrt[b^2 - 4*a*c]
))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]] + ((a*B*(b^3 - 3
*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4*a*c]) + A*(-b^4 + 4*a*b^2*c - 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a
*c] - 2*a*b*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt
[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c])/(105*a^4)

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Maple [B]  time = 0.043, size = 1210, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*A/a/x^(7/2)-2/5*B/a/x^(5/2)+2*B*c/a^2/x^(1/2)+2/a^3*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh
(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b-1/a^4*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*ar
ctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3-2/a^2*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4
*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A+1/a^3*c*2^(1/2)/((-b+
(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B*b^2-2/a^3*c^2*2^(1
/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b+1/a^4*c*2^
(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3-2/a^2*
c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2
))*c)^(1/2))*A-1/a^3*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2
))*c)^(1/2))*B*b^2-1/a^2*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+
b^2)^(1/2))*c)^(1/2))*B+1/a^2*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*
c+b^2)^(1/2))*c)^(1/2))*B+2/3*A*c/a^2/x^(3/2)+4/a^3*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)
^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^2-1/a^4*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/(
(-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^4-3/a^2*c^2/
(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2))*b*B+1/a^3*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x^(1/2)*c*2^(1/2)
/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B*b^3+4/a^3*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2
)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^2-1/a^4*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*
a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^4-3/a^2*c^2/(-4*a*c+b^
2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b
*B+1/a^3*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x^(1/2)*c*2^(1/2)/((b+(-4*a*c+b^
2)^(1/2))*c)^(1/2))*B*b^3+2/a^4/x^(1/2)*A*b^3-2/a^3/x^(1/2)*B*b^2+2/5/a^2/x^(5/2)*A*b-2/3/a^3/x^(3/2)*A*b^2+2/
3/a^2/x^(3/2)*b*B-4/a^3/x^(1/2)*A*b*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left (\frac{15 \, A a^{4}}{x^{\frac{7}{2}}} - 105 \,{\left ({\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} A -{\left (a b^{3} - 2 \, a^{2} b c\right )} B\right )} \sqrt{x} - \frac{105 \,{\left ({\left (a b^{3} - 2 \, a^{2} b c\right )} A -{\left (a^{2} b^{2} - a^{3} c\right )} B\right )}}{\sqrt{x}} - \frac{35 \,{\left (B a^{3} b -{\left (a^{2} b^{2} - a^{3} c\right )} A\right )}}{x^{\frac{3}{2}}} + \frac{21 \,{\left (B a^{4} - A a^{3} b\right )}}{x^{\frac{5}{2}}}\right )}}{105 \, a^{5}} - \int \frac{{\left ({\left (b^{4} c - 3 \, a b^{2} c^{2} + a^{2} c^{3}\right )} A -{\left (a b^{3} c - 2 \, a^{2} b c^{2}\right )} B\right )} x^{\frac{3}{2}} +{\left ({\left (b^{5} - 4 \, a b^{3} c + 3 \, a^{2} b c^{2}\right )} A -{\left (a b^{4} - 3 \, a^{2} b^{2} c + a^{3} c^{2}\right )} B\right )} \sqrt{x}}{a^{5} c x^{2} + a^{5} b x + a^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(15*A*a^4/x^(7/2) - 105*((b^4 - 3*a*b^2*c + a^2*c^2)*A - (a*b^3 - 2*a^2*b*c)*B)*sqrt(x) - 105*((a*b^3 -
2*a^2*b*c)*A - (a^2*b^2 - a^3*c)*B)/sqrt(x) - 35*(B*a^3*b - (a^2*b^2 - a^3*c)*A)/x^(3/2) + 21*(B*a^4 - A*a^3*
b)/x^(5/2))/a^5 - integrate((((b^4*c - 3*a*b^2*c^2 + a^2*c^3)*A - (a*b^3*c - 2*a^2*b*c^2)*B)*x^(3/2) + ((b^5 -
4*a*b^3*c + 3*a^2*b*c^2)*A - (a*b^4 - 3*a^2*b^2*c + a^3*c^2)*B)*sqrt(x))/(a^5*c*x^2 + a^5*b*x + a^6), x)

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Fricas [B]  time = 22.2148, size = 22005, normalized size = 57.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(105*sqrt(2)*a^4*x^4*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2
*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^
2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A
^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c
^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9
*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 3
44*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*
B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^
5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^
11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c))*
log(sqrt(2)*(B^3*a^3*b^11 - 3*A*B^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*a^8 - 40*A
^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6 - (8*B^3*a^8*b - 101*A*B^2*a^7*b^2 + 270*A^2*B*a^6*b^3 - 197*A^3*a^5*b^4)*c^5
+ (54*B^3*a^7*b^3 - 313*A*B^2*a^6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5 - 336*A*B^
2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 210*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*b^7 - 162*A*B^2*a^4*b^8 + 195*A^2*B*a^3*b^9
- 77*A^3*a^2*b^10)*c^2 - (11*B^3*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c - (B*a^10
*b^6 - A*a^9*b^7 - 4*(2*B*a^13 - 5*A*a^12*b)*c^3 + (18*B*a^12*b^2 - 25*A*a^11*b^3)*c^2 - (8*B*a^11*b^4 - 9*A*a
^10*b^5)*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c
^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 -
200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3
*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*
b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 +
115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*
A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*
b^14)*c)/(a^18*b^2 - 4*a^19*c)))*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 -
(7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 -
(7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13
+ 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*
b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B
^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b
^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 -
182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*
B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*
a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^1
0*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^2*a^4*b^2
+ 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 - 3*A^3*B*a^
2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 - A^4*b^8)*c
^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*c^4)*sqrt(x)) - 105*sqrt(2)*a^4*x^4*sqrt(-(
B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a
^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2
*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 +
A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b +
96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*
B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6
*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8
- 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268
*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*
B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c))*log(-sqrt(2)*(B^3*a^3*b^11 - 3*A*B
^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*a^8 - 40*A^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6
- (8*B^3*a^8*b - 101*A*B^2*a^7*b^2 + 270*A^2*B*a^6*b^3 - 197*A^3*a^5*b^4)*c^5 + (54*B^3*a^7*b^3 - 313*A*B^2*a^
6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5 - 336*A*B^2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 2
10*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*b^7 - 162*A*B^2*a^4*b^8 + 195*A^2*B*a^3*b^9 - 77*A^3*a^2*b^10)*c^2 - (11*B^3
*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c - (B*a^10*b^6 - A*a^9*b^7 - 4*(2*B*a^13 -
5*A*a^12*b)*c^3 + (18*B*a^12*b^2 - 25*A*a^11*b^3)*c^2 - (8*B*a^11*b^4 - 9*A*a^10*b^5)*c)*sqrt((B^4*a^4*b^12 -
4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8
*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b
^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5
+ (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(3
1*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*
a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*
b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))*
sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 3
0*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6
+ 9*A^2*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a
*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a
^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 2
40*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*
B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a
^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^1
0 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 -
26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B
*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^2*a^4*b^2 + 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b
^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4
*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 - A^4*b^8)*c^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b
^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*c^4)*sqrt(x)) + 105*sqrt(2)*a^4*x^4*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b
^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 4
0*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^10*c)*
sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*
B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^
7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 +
157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^
4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3
*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12
)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^
18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c))*log(sqrt(2)*(B^3*a^3*b^11 - 3*A*B^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3
*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*a^8 - 40*A^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6 - (8*B^3*a^8*b - 101*A*B^2*a^7*b^2
+ 270*A^2*B*a^6*b^3 - 197*A^3*a^5*b^4)*c^5 + (54*B^3*a^7*b^3 - 313*A*B^2*a^6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^
3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5 - 336*A*B^2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 210*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*
b^7 - 162*A*B^2*a^4*b^8 + 195*A^2*B*a^3*b^9 - 77*A^3*a^2*b^10)*c^2 - (11*B^3*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*
A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c + (B*a^10*b^6 - A*a^9*b^7 - 4*(2*B*a^13 - 5*A*a^12*b)*c^3 + (18*B*a^12*b^2 -
25*A*a^11*b^3)*c^2 - (8*B*a^11*b^4 - 9*A*a^10*b^5)*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b
^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^
10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*
B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7
*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7
+ 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*
A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*
B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 +
A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b
^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^
10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2
*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^
3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6
*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 +
367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A
^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^
2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*
c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^
4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^2*a^4*b^2 + 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^
3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A
^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 - A^4*b^8)*c^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*
c^4)*sqrt(x)) - 105*sqrt(2)*a^4*x^4*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4
- (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^
2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b
^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a
^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(
6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^
8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6
- 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184
*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B
^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*
a^10*c))*log(-sqrt(2)*(B^3*a^3*b^11 - 3*A*B^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*
a^8 - 40*A^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6 - (8*B^3*a^8*b - 101*A*B^2*a^7*b^2 + 270*A^2*B*a^6*b^3 - 197*A^3*a^
5*b^4)*c^5 + (54*B^3*a^7*b^3 - 313*A*B^2*a^6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5
- 336*A*B^2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 210*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*b^7 - 162*A*B^2*a^4*b^8 + 195*A^2
*B*a^3*b^9 - 77*A^3*a^2*b^10)*c^2 - (11*B^3*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c
+ (B*a^10*b^6 - A*a^9*b^7 - 4*(2*B*a^13 - 5*A*a^12*b)*c^3 + (18*B*a^12*b^2 - 25*A*a^11*b^3)*c^2 - (8*B*a^11*b
^4 - 9*A*a^10*b^5)*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 +
A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*
a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4
- 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*
A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B
*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*
b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13
+ 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4
*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*
b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^
3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 1
0*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^
6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46
*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*
a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^
8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 -
22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b
^2 - 4*a^10*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^
2*a^4*b^2 + 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 -
3*A^3*B*a^2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 -
A^4*b^8)*c^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*c^4)*sqrt(x)) - 4*(15*A*a^3 + 105
*(B*a*b^2 - A*b^3 - (B*a^2 - 2*A*a*b)*c)*x^3 - 35*(B*a^2*b - A*a*b^2 + A*a^2*c)*x^2 + 21*(B*a^3 - A*a^2*b)*x)*
sqrt(x))/(a^4*x^4)

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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+A)/x**(9/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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Giac [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+A)/x^(9/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

Timed out