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

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

[Out]

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

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Rubi [A]  time = 0.271973, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {825, 827, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{a} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} c^{7/4}}-\frac{\sqrt [4]{a} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} c^{7/4}}+\frac{2 A \sqrt{x}}{c}+\frac{2 B x^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 825

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

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

Mathematica [A]  time = 0.0686937, size = 287, normalized size = 1.03 \[ \frac{\sqrt [4]{a} A \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} A \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{5/4}}+\frac{\sqrt [4]{a} A \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} A \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} c^{5/4}}+\frac{(-a)^{3/4} B \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}-\frac{(-a)^{3/4} B \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac{2 A \sqrt{x}}{c}+\frac{2 B x^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 289, normalized size = 1. \begin{align*}{\frac{2\,B}{3\,c}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{c}}-{\frac{A\sqrt{2}}{4\,c}\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{A\sqrt{2}}{2\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{A\sqrt{2}}{2\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{aB\sqrt{2}}{4\,{c}^{2}}\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{aB\sqrt{2}}{2\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{aB\sqrt{2}}{2\,{c}^{2}}\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(x^(3/2)*(B*x+A)/(c*x^2+a),x)

[Out]

2/3*B*x^(3/2)/c+2*A*x^(1/2)/c-1/4/c*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)))-1/2/c*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)-1/2/
c*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)-1/4*a/c^2*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)))-1/2*a/c^2*B/(a/c)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)-1/2*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 [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(x^(3/2)*(B*x+A)/(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60714, size = 1555, normalized size = 5.59 \begin{align*} -\frac{3 \, c \sqrt{-\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (B c^{5} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - A B^{2} a c^{2} + A^{3} c^{3}\right )} \sqrt{-\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}}\right ) - 3 \, c \sqrt{-\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (B c^{5} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - A B^{2} a c^{2} + A^{3} c^{3}\right )} \sqrt{-\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + 2 \, A B a}{c^{3}}}\right ) - 3 \, c \sqrt{\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (B c^{5} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + A B^{2} a c^{2} - A^{3} c^{3}\right )} \sqrt{\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}}\right ) + 3 \, c \sqrt{\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (B c^{5} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} + A B^{2} a c^{2} - A^{3} c^{3}\right )} \sqrt{\frac{c^{3} \sqrt{-\frac{B^{4} a^{3} - 2 \, A^{2} B^{2} a^{2} c + A^{4} a c^{2}}{c^{7}}} - 2 \, A B a}{c^{3}}}\right ) - 4 \,{\left (B x + 3 \, A\right )} \sqrt{x}}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 27.0834, size = 432, normalized size = 1.55 \begin{align*} \begin{cases} \tilde{\infty } \left (2 A \sqrt{x} + \frac{2 B x^{\frac{3}{2}}}{3}\right ) & \text{for}\: a = 0 \wedge c = 0 \\\frac{\frac{2 A x^{\frac{5}{2}}}{5} + \frac{2 B x^{\frac{7}{2}}}{7}}{a} & \text{for}\: c = 0 \\\frac{2 A \sqrt{x} + \frac{2 B x^{\frac{3}{2}}}{3}}{c} & \text{for}\: a = 0 \\\frac{\sqrt [4]{-1} A \sqrt [4]{a} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 c^{4} \left (\frac{1}{c}\right )^{\frac{11}{4}}} - \frac{\sqrt [4]{-1} A \sqrt [4]{a} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 c^{4} \left (\frac{1}{c}\right )^{\frac{11}{4}}} + \frac{\sqrt [4]{-1} A \sqrt [4]{a} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{c}}} \right )}}{c^{4} \left (\frac{1}{c}\right )^{\frac{11}{4}}} + \frac{2 A \sqrt{x}}{c} + \left (-1\right )^{\frac{3}{4}} B a^{\frac{3}{4}} c \left (\frac{1}{c}\right )^{\frac{11}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )} - \frac{\left (-1\right )^{\frac{3}{4}} B a^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 c^{4} \left (\frac{1}{c}\right )^{\frac{9}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} B a^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 c^{4} \left (\frac{1}{c}\right )^{\frac{9}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} B a^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{c}}} \right )}}{c^{4} \left (\frac{1}{c}\right )^{\frac{9}{4}}} + \frac{2 B x^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(2*A*sqrt(x) + 2*B*x**(3/2)/3), Eq(a, 0) & Eq(c, 0)), ((2*A*x**(5/2)/5 + 2*B*x**(7/2)/7)/a, Eq(
c, 0)), ((2*A*sqrt(x) + 2*B*x**(3/2)/3)/c, Eq(a, 0)), ((-1)**(1/4)*A*a**(1/4)*log(-(-1)**(1/4)*a**(1/4)*(1/c)*
*(1/4) + sqrt(x))/(2*c**4*(1/c)**(11/4)) - (-1)**(1/4)*A*a**(1/4)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt
(x))/(2*c**4*(1/c)**(11/4)) + (-1)**(1/4)*A*a**(1/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(c**4*(
1/c)**(11/4)) + 2*A*sqrt(x)/c + (-1)**(3/4)*B*a**(3/4)*c*(1/c)**(11/4)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4)
+ sqrt(x)) - (-1)**(3/4)*B*a**(3/4)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*c**4*(1/c)**(9/4)) -
(-1)**(3/4)*B*a**(3/4)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*c**4*(1/c)**(9/4)) - (-1)**(3/4)*B*
a**(3/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(c**4*(1/c)**(9/4)) + 2*B*x**(3/2)/(3*c), True))

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Giac [A]  time = 1.22777, size = 344, normalized size = 1.24 \begin{align*} -\frac{\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 )}{2 \, c^{4}} - \frac{\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 )}{2 \, c^{4}} - \frac{\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 )}{4 \, c^{4}} + \frac{\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 )}{4 \, c^{4}} + \frac{2 \,{\left (B c^{2} x^{\frac{3}{2}} + 3 \, A c^{2} \sqrt{x}\right )}}{3 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*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))/c^4 - 1/2*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))/c^4 - 1/4*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))/c^4 + 1/4*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))/c^4 + 2/3*(B*c^2*x^(3/2) + 3*A*c^2*sqrt(x))/c^3