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

Optimal. Leaf size=315 \[ \frac {3 \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{64 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}-\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )} \]

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Rubi [A]  time = 0.28, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {819, 823, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {3 \left (\sqrt {a} B-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^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B-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^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}-\frac {\sqrt {x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac {\sqrt {x} (A+3 B x)}{16 a c \left (a+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 819

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

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 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 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 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 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)]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 360, normalized size = 1.14 \begin {gather*} \frac {-\frac {3 \sqrt {2} \sqrt [4]{a} A \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}+\frac {3 \sqrt {2} \sqrt [4]{a} A \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}-\frac {6 \sqrt {2} \sqrt [4]{a} A \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{c^{5/4}}+\frac {6 \sqrt {2} \sqrt [4]{a} A \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{c^{5/4}}+\frac {24 A x^{5/2}}{a+c x^2}+\frac {32 a A x^{5/2}}{\left (a+c x^2\right )^2}-\frac {12 (-a)^{3/4} B \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac {12 (-a)^{3/4} B \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac {8 B x^{7/2}}{a+c x^2}+\frac {32 a B x^{7/2}}{\left (a+c x^2\right )^2}-\frac {24 A \sqrt {x}}{c}-\frac {8 B x^{3/2}}{c}}{128 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-24*A*Sqrt[x])/c - (8*B*x^(3/2))/c + (32*a*A*x^(5/2))/(a + c*x^2)^2 + (32*a*B*x^(7/2))/(a + c*x^2)^2 + (24*A
*x^(5/2))/(a + c*x^2) + (8*B*x^(7/2))/(a + c*x^2) - (6*Sqrt[2]*a^(1/4)*A*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/
a^(1/4)])/c^(5/4) + (6*Sqrt[2]*a^(1/4)*A*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(5/4) - (12*(-a)^(3/
4)*B*ArcTan[(c^(1/4)*Sqrt[x])/(-a)^(1/4)])/c^(7/4) + (12*(-a)^(3/4)*B*ArcTanh[(c^(1/4)*Sqrt[x])/(-a)^(1/4)])/c
^(7/4) - (3*Sqrt[2]*a^(1/4)*A*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(5/4) + (3*Sqrt[2]
*a^(1/4)*A*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(5/4))/(128*a^2)

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IntegrateAlgebraic [A]  time = 1.17, size = 198, normalized size = 0.63 \begin {gather*} -\frac {3 \left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}-\frac {3 \left (\sqrt {a} B-A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{32 \sqrt {2} a^{7/4} c^{7/4}}+\frac {-3 a A \sqrt {x}-a B x^{3/2}+A c x^{5/2}+3 B c x^{7/2}}{16 a c \left (a+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*A*Sqrt[x] - a*B*x^(3/2) + A*c*x^(5/2) + 3*B*c*x^(7/2))/(16*a*c*(a + c*x^2)^2) - (3*(Sqrt[a]*B + A*Sqrt[c
])*ArcTan[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])])/(32*Sqrt[2]*a^(7/4)*c^(7/4)) - (3*(Sqrt[a]
*B - A*Sqrt[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/(32*Sqrt[2]*a^(7/4)*c^(7/4))

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fricas [B]  time = 0.47, size = 982, normalized size = 3.12 \begin {gather*} \frac {3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - A B^{2} a^{3} c^{2} + A^{3} a^{2} c^{3}\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}}\right ) - 3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - A B^{2} a^{3} c^{2} + A^{3} a^{2} c^{3}\right )} \sqrt {-\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + 2 \, A B}{a^{3} c^{3}}}\right ) - 3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + A B^{2} a^{3} c^{2} - A^{3} a^{2} c^{3}\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}}\right ) + 3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}} \log \left (-27 \, {\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - 27 \, {\left (B a^{6} c^{5} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} + A B^{2} a^{3} c^{2} - A^{3} a^{2} c^{3}\right )} \sqrt {\frac {a^{3} c^{3} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{7} c^{7}}} - 2 \, A B}{a^{3} c^{3}}}\right ) + 4 \, {\left (3 \, B c x^{3} + A c x^{2} - B a x - 3 \, A a\right )} \sqrt {x}}{64 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(3*(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)*sqrt(-(a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)
) + 2*A*B)/(a^3*c^3))*log(-27*(B^4*a^2 - A^4*c^2)*sqrt(x) + 27*(B*a^6*c^5*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4
*c^2)/(a^7*c^7)) - A*B^2*a^3*c^2 + A^3*a^2*c^3)*sqrt(-(a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*
c^7)) + 2*A*B)/(a^3*c^3))) - 3*(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)*sqrt(-(a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a
*c + A^4*c^2)/(a^7*c^7)) + 2*A*B)/(a^3*c^3))*log(-27*(B^4*a^2 - A^4*c^2)*sqrt(x) - 27*(B*a^6*c^5*sqrt(-(B^4*a^
2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) - A*B^2*a^3*c^2 + A^3*a^2*c^3)*sqrt(-(a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B
^2*a*c + A^4*c^2)/(a^7*c^7)) + 2*A*B)/(a^3*c^3))) - 3*(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)*sqrt((a^3*c^3*sqrt(-
(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) - 2*A*B)/(a^3*c^3))*log(-27*(B^4*a^2 - A^4*c^2)*sqrt(x) + 27*(B
*a^6*c^5*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) + A*B^2*a^3*c^2 - A^3*a^2*c^3)*sqrt((a^3*c^3*sqr
t(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) - 2*A*B)/(a^3*c^3))) + 3*(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)
*sqrt((a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) - 2*A*B)/(a^3*c^3))*log(-27*(B^4*a^2 - A^4
*c^2)*sqrt(x) - 27*(B*a^6*c^5*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) + A*B^2*a^3*c^2 - A^3*a^2*c
^3)*sqrt((a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) - 2*A*B)/(a^3*c^3))) + 4*(3*B*c*x^3 + A
*c*x^2 - B*a*x - 3*A*a)*sqrt(x))/(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)

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giac [A]  time = 0.22, size = 289, normalized size = 0.92 \begin {gather*} \frac {3 \, B c x^{\frac {7}{2}} + A c x^{\frac {5}{2}} - B a x^{\frac {3}{2}} - 3 \, A a \sqrt {x}}{16 \, {\left (c x^{2} + a\right )}^{2} a c} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\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^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac {3}{4}} B\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^{2} c^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{2} c^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{128 \, a^{2} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 334, normalized size = 1.06 \begin {gather*} \frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{64 a^{2} c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{64 a^{2} c}+\frac {3 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{128 a^{2} c}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{c}\right )^{\frac {1}{4}} a \,c^{2}}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{c}\right )^{\frac {1}{4}} a \,c^{2}}+\frac {3 \sqrt {2}\, B \ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{128 \left (\frac {a}{c}\right )^{\frac {1}{4}} a \,c^{2}}+\frac {\frac {3 B \,x^{\frac {7}{2}}}{16 a}+\frac {A \,x^{\frac {5}{2}}}{16 a}-\frac {B \,x^{\frac {3}{2}}}{16 c}-\frac {3 A \sqrt {x}}{16 c}}{\left (c \,x^{2}+a \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(3/32*B/a*x^(7/2)+1/32*A/a*x^(5/2)-1/32*B/c*x^(3/2)-3/32*A/c*x^(1/2))/(c*x^2+a)^2+3/128/a^2/c*A*(a/c)^(1/4)*
2^(1/2)*ln((x+(a/c)^(1/4)*2^(1/2)*x^(1/2)+(a/c)^(1/2))/(x-(a/c)^(1/4)*2^(1/2)*x^(1/2)+(a/c)^(1/2)))+3/64/a^2/c
*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+3/64/a^2/c*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
a/c)^(1/4)*x^(1/2)-1)+3/128/a/c^2*B/(a/c)^(1/4)*2^(1/2)*ln((x-(a/c)^(1/4)*2^(1/2)*x^(1/2)+(a/c)^(1/2))/(x+(a/c
)^(1/4)*2^(1/2)*x^(1/2)+(a/c)^(1/2)))+3/64/a/c^2*B/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+3
/64/a/c^2*B/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)

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maxima [A]  time = 1.35, size = 289, normalized size = 0.92 \begin {gather*} \frac {3 \, B c x^{\frac {7}{2}} + A c x^{\frac {5}{2}} - B a x^{\frac {3}{2}} - 3 \, A a \sqrt {x}}{16 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (B \sqrt {a} - A \sqrt {c}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{128 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(3*B*c*x^(7/2) + A*c*x^(5/2) - B*a*x^(3/2) - 3*A*a*sqrt(x))/(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c) + 3/128*(
2*sqrt(2)*(B*sqrt(a) + A*sqrt(c))*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a
)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(B*sqrt(a) + A*sqrt(c))*arctan(-1/2*sqrt(2)*(s
qrt(2)*a^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) - s
qrt(2)*(B*sqrt(a) - A*sqrt(c))*log(sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)) +
sqrt(2)*(B*sqrt(a) - A*sqrt(c))*log(-sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)))
/(a*c)

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mupad [B]  time = 1.29, size = 695, normalized size = 2.21 \begin {gather*} \frac {\frac {A\,x^{5/2}}{16\,a}+\frac {3\,B\,x^{7/2}}{16\,a}-\frac {3\,A\,\sqrt {x}}{16\,c}-\frac {B\,x^{3/2}}{16\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}-\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}}}{32\,\left (\frac {27\,B^3}{2048\,a\,c^2}-\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^6\,c^4}-\frac {27\,A^2\,B}{2048\,a^2\,c}+\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^5}\right )}-\frac {9\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}-\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}}}{32\,\left (\frac {27\,B^3}{2048\,c^2}-\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^4}-\frac {27\,A^2\,B}{2048\,a\,c}+\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^4\,c^5}\right )}\right )\,\sqrt {-\frac {9\,\left (A^2\,c\,\sqrt {-a^7\,c^7}-B^2\,a\,\sqrt {-a^7\,c^7}+2\,A\,B\,a^4\,c^4\right )}{4096\,a^7\,c^7}}-2\,\mathrm {atanh}\left (\frac {9\,B^2\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}-\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}}}{32\,\left (\frac {27\,B^3}{2048\,a\,c^2}+\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^6\,c^4}-\frac {27\,A^2\,B}{2048\,a^2\,c}-\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^5}\right )}-\frac {9\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,A^2\,\sqrt {-a^7\,c^7}}{4096\,a^7\,c^6}-\frac {9\,A\,B}{2048\,a^3\,c^3}-\frac {9\,B^2\,\sqrt {-a^7\,c^7}}{4096\,a^6\,c^7}}}{32\,\left (\frac {27\,B^3}{2048\,c^2}+\frac {27\,A^3\,\sqrt {-a^7\,c^7}}{2048\,a^5\,c^4}-\frac {27\,A^2\,B}{2048\,a\,c}-\frac {27\,A\,B^2\,\sqrt {-a^7\,c^7}}{2048\,a^4\,c^5}\right )}\right )\,\sqrt {-\frac {9\,\left (B^2\,a\,\sqrt {-a^7\,c^7}-A^2\,c\,\sqrt {-a^7\,c^7}+2\,A\,B\,a^4\,c^4\right )}{4096\,a^7\,c^7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((A*x^(5/2))/(16*a) + (3*B*x^(7/2))/(16*a) - (3*A*x^(1/2))/(16*c) - (B*x^(3/2))/(16*c))/(a^2 + c^2*x^4 + 2*a*c
*x^2) - 2*atanh((9*B^2*x^(1/2)*((9*B^2*(-a^7*c^7)^(1/2))/(4096*a^6*c^7) - (9*A^2*(-a^7*c^7)^(1/2))/(4096*a^7*c
^6) - (9*A*B)/(2048*a^3*c^3))^(1/2))/(32*((27*B^3)/(2048*a*c^2) - (27*A^3*(-a^7*c^7)^(1/2))/(2048*a^6*c^4) - (
27*A^2*B)/(2048*a^2*c) + (27*A*B^2*(-a^7*c^7)^(1/2))/(2048*a^5*c^5))) - (9*A^2*c*x^(1/2)*((9*B^2*(-a^7*c^7)^(1
/2))/(4096*a^6*c^7) - (9*A^2*(-a^7*c^7)^(1/2))/(4096*a^7*c^6) - (9*A*B)/(2048*a^3*c^3))^(1/2))/(32*((27*B^3)/(
2048*c^2) - (27*A^3*(-a^7*c^7)^(1/2))/(2048*a^5*c^4) - (27*A^2*B)/(2048*a*c) + (27*A*B^2*(-a^7*c^7)^(1/2))/(20
48*a^4*c^5))))*(-(9*(A^2*c*(-a^7*c^7)^(1/2) - B^2*a*(-a^7*c^7)^(1/2) + 2*A*B*a^4*c^4))/(4096*a^7*c^7))^(1/2) -
2*atanh((9*B^2*x^(1/2)*((9*A^2*(-a^7*c^7)^(1/2))/(4096*a^7*c^6) - (9*A*B)/(2048*a^3*c^3) - (9*B^2*(-a^7*c^7)^
(1/2))/(4096*a^6*c^7))^(1/2))/(32*((27*B^3)/(2048*a*c^2) + (27*A^3*(-a^7*c^7)^(1/2))/(2048*a^6*c^4) - (27*A^2*
B)/(2048*a^2*c) - (27*A*B^2*(-a^7*c^7)^(1/2))/(2048*a^5*c^5))) - (9*A^2*c*x^(1/2)*((9*A^2*(-a^7*c^7)^(1/2))/(4
096*a^7*c^6) - (9*A*B)/(2048*a^3*c^3) - (9*B^2*(-a^7*c^7)^(1/2))/(4096*a^6*c^7))^(1/2))/(32*((27*B^3)/(2048*c^
2) + (27*A^3*(-a^7*c^7)^(1/2))/(2048*a^5*c^4) - (27*A^2*B)/(2048*a*c) - (27*A*B^2*(-a^7*c^7)^(1/2))/(2048*a^4*
c^5))))*(-(9*(B^2*a*(-a^7*c^7)^(1/2) - A^2*c*(-a^7*c^7)^(1/2) + 2*A*B*a^4*c^4))/(4096*a^7*c^7))^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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