3.431 \(\int \frac{A+B x}{x^{3/2} (a+c x^2)^3} \, dx\)

Optimal. Leaf size=333 \[ \frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2} \]

[Out]

(-45*A)/(16*a^3*Sqrt[x]) + (A + B*x)/(4*a*Sqrt[x]*(a + c*x^2)^2) + (9*A + 7*B*x)/(16*a^2*Sqrt[x]*(a + c*x^2))
- (3*(7*Sqrt[a]*B - 15*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*c^(1/4))
+ (3*(7*Sqrt[a]*B - 15*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*c^(1/4)
) - (3*(7*Sqrt[a]*B + 15*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^
(13/4)*c^(1/4)) + (3*(7*Sqrt[a]*B + 15*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/
(64*Sqrt[2]*a^(13/4)*c^(1/4))

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Rubi [A]  time = 0.354311, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {823, 829, 827, 1168, 1162, 617, 204, 1165, 628} \[ \frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B-15 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*A)/(16*a^3*Sqrt[x]) + (A + B*x)/(4*a*Sqrt[x]*(a + c*x^2)^2) + (9*A + 7*B*x)/(16*a^2*Sqrt[x]*(a + c*x^2))
- (3*(7*Sqrt[a]*B - 15*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*c^(1/4))
+ (3*(7*Sqrt[a]*B - 15*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*c^(1/4)
) - (3*(7*Sqrt[a]*B + 15*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^
(13/4)*c^(1/4)) + (3*(7*Sqrt[a]*B + 15*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/
(64*Sqrt[2]*a^(13/4)*c^(1/4))

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 829

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

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1168

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

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])

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]

Rubi steps

\begin{align*} \int \frac{A+B x}{x^{3/2} \left (a+c x^2\right )^3} \, dx &=\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}-\frac{\int \frac{-\frac{9}{2} a A c-\frac{7}{2} a B c x}{x^{3/2} \left (a+c x^2\right )^2} \, dx}{4 a^2 c}\\ &=\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{\int \frac{\frac{45}{4} a^2 A c^2+\frac{21}{4} a^2 B c^2 x}{x^{3/2} \left (a+c x^2\right )} \, dx}{8 a^4 c^2}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{\int \frac{\frac{21}{4} a^3 B c^2-\frac{45}{4} a^2 A c^3 x}{\sqrt{x} \left (a+c x^2\right )} \, dx}{8 a^5 c^2}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{21}{4} a^3 B c^2-\frac{45}{4} a^2 A c^3 x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{4 a^5 c^2}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{32 a^3}+\frac{\left (3 \left (15 A+\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{32 a^3}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 a^3}-\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 a^3}-\frac{\left (3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{\left (3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}-\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}+\frac{\left (3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}\\ &=-\frac{45 A}{16 a^3 \sqrt{x}}+\frac{A+B x}{4 a \sqrt{x} \left (a+c x^2\right )^2}+\frac{9 A+7 B x}{16 a^2 \sqrt{x} \left (a+c x^2\right )}+\frac{3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}-\frac{3 \left (15 A-\frac{7 \sqrt{a} B}{\sqrt{c}}\right ) \sqrt [4]{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}-\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}+\frac{3 \left (7 \sqrt{a} B+15 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{13/4} \sqrt [4]{c}}\\ \end{align*}

Mathematica [C]  time = 0.302213, size = 300, normalized size = 0.9 \[ \frac{\sqrt [4]{a} \left (\frac{32 a^{7/4} A}{\sqrt{x} \left (a+c x^2\right )^2}+\frac{72 a^{3/4} A}{\sqrt{x} \left (a+c x^2\right )}+\frac{32 a^{7/4} B \sqrt{x}}{\left (a+c x^2\right )^2}+\frac{56 a^{3/4} B \sqrt{x}}{a+c x^2}-\frac{21 \sqrt{2} B \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{21 \sqrt{2} B \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{42 \sqrt{2} B \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{42 \sqrt{2} B \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}\right )-\frac{360 A \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{c x^2}{a}\right )}{\sqrt{x}}}{128 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-360*A*Hypergeometric2F1[-1/4, 1, 3/4, -((c*x^2)/a)])/Sqrt[x] + a^(1/4)*((32*a^(7/4)*A)/(Sqrt[x]*(a + c*x^2)
^2) + (32*a^(7/4)*B*Sqrt[x])/(a + c*x^2)^2 + (72*a^(3/4)*A)/(Sqrt[x]*(a + c*x^2)) + (56*a^(3/4)*B*Sqrt[x])/(a
+ c*x^2) - (42*Sqrt[2]*B*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(1/4) + (42*Sqrt[2]*B*ArcTan[1 + (Sq
rt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(1/4) - (21*Sqrt[2]*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[
c]*x])/c^(1/4) + (21*Sqrt[2]*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4)))/(128*a^3)

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Maple [A]  time = 0.017, size = 354, normalized size = 1.1 \begin{align*} -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{13\,A{c}^{2}}{16\,{a}^{3} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{7\,Bc}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{17\,Ac}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{11\,B}{16\,a \left ( c{x}^{2}+a \right ) ^{2}}\sqrt{x}}+{\frac{21\,B\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{21\,B\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{21\,B\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{45\,A\sqrt{2}}{128\,{a}^{3}}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{45\,A\sqrt{2}}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{45\,A\sqrt{2}}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A/a^3/x^(1/2)-13/16/a^3/(c*x^2+a)^2*A*x^(7/2)*c^2+7/16/a^2/(c*x^2+a)^2*B*x^(5/2)*c-17/16/a^2/(c*x^2+a)^2*A*
x^(3/2)*c+11/16/a/(c*x^2+a)^2*B*x^(1/2)+21/64/a^3*B*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+
21/128/a^3*B*(a/c)^(1/4)*2^(1/2)*ln((x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x-(a/c)^(1/4)*x^(1/2)*2^(1/2)
+(a/c)^(1/2)))+21/64/a^3*B*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)-45/128/a^3*A/(a/c)^(1/4)*
2^(1/2)*ln((x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))-45/64/a^3*
A/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)-45/64/a^3*A/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/
c)^(1/4)*x^(1/2)-1)

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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+A)/x^(3/2)/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.91955, size = 2283, normalized size = 6.86 \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^(3/2)/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/64*(3*(a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt((a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2
)/(a^13*c)) + 210*A*B)/a^6)*log(-9*(2401*B^4*a^2 - 50625*A^4*c^2)*sqrt(x) + 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2
- 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 343*B^3*a^5 - 1575*A^2*B*a^4*c)*sqrt((a^6*sqrt(-(2401*B^4*a^
2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)) - 3*(a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt
((a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) + 210*A*B)/a^6)*log(-9*(2401*B^4*a^2
- 50625*A^4*c^2)*sqrt(x) - 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) +
343*B^3*a^5 - 1575*A^2*B*a^4*c)*sqrt((a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c))
+ 210*A*B)/a^6)) - 3*(a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt(-(a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c +
50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)*log(-9*(2401*B^4*a^2 - 50625*A^4*c^2)*sqrt(x) + 9*(15*A*a^10*c*sqrt(-
(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 343*B^3*a^5 + 1575*A^2*B*a^4*c)*sqrt(-(a^6*sqrt
(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)) + 3*(a^3*c^2*x^5 + 2*a^4*c*x^3
+ a^5*x)*sqrt(-(a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c^2)/(a^13*c)) - 210*A*B)/a^6)*log(-9
*(2401*B^4*a^2 - 50625*A^4*c^2)*sqrt(x) - 9*(15*A*a^10*c*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4*c
^2)/(a^13*c)) - 343*B^3*a^5 + 1575*A^2*B*a^4*c)*sqrt(-(a^6*sqrt(-(2401*B^4*a^2 - 22050*A^2*B^2*a*c + 50625*A^4
*c^2)/(a^13*c)) - 210*A*B)/a^6)) + 4*(45*A*c^2*x^4 - 7*B*a*c*x^3 + 81*A*a*c*x^2 - 11*B*a^2*x + 32*A*a^2)*sqrt(
x))/(a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)

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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**(3/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.4102, size = 410, normalized size = 1.23 \begin{align*} -\frac{2 \, A}{a^{3} \sqrt{x}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} c^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac{3 \, \sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 15 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{4} c^{2}} - \frac{13 \, A c^{2} x^{\frac{7}{2}} - 7 \, B a c x^{\frac{5}{2}} + 17 \, A a c x^{\frac{3}{2}} - 11 \, B a^{2} \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(c*x^2+a)^3,x, algorithm="giac")

[Out]

-2*A/(a^3*sqrt(x)) + 3/64*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c - 15*(a*c^3)^(3/4)*A)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/
c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^4*c^2) + 3/64*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c - 15*(a*c^3)^(3/4)*A)*arcta
n(-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^4*c^2) + 3/128*sqrt(2)*(7*(a*c^3)^(1/4)*B*a*c
+ 15*(a*c^3)^(3/4)*A)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^4*c^2) - 3/128*sqrt(2)*(7*(a*c^3)^(
1/4)*B*a*c + 15*(a*c^3)^(3/4)*A)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^4*c^2) - 1/16*(13*A*c^2*
x^(7/2) - 7*B*a*c*x^(5/2) + 17*A*a*c*x^(3/2) - 11*B*a^2*sqrt(x))/((c*x^2 + a)^2*a^3)