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

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

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Rubi [A]  time = 0.27, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {825, 827, 1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\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} \end {gather*}

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 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 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 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)}{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*}

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Mathematica [A]  time = 0.06, size = 287, normalized size = 1.03 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.34, size = 160, normalized size = 0.58 \begin {gather*} \frac {\left (a^{3/4} B+\sqrt [4]{a} A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} c^{7/4}}+\frac {\left (a^{3/4} B-\sqrt [4]{a} A \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} c^{7/4}}+\frac {2 \left (3 A \sqrt {x}+B x^{3/2}\right )}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.46, size = 772, normalized size = 2.78 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.19, size = 255, normalized size = 0.92 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.05, size = 289, normalized size = 1.04 \begin {gather*} \frac {2 B \,x^{\frac {3}{2}}}{3 c}-\frac {\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 )}{2 c}-\frac {\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 )}{2 c}-\frac {\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 )}{4 c}-\frac {\sqrt {2}\, B a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {\sqrt {2}\, B a \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}-\frac {\sqrt {2}\, B 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 )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {2 A \sqrt {x}}{c} \end {gather*}

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/c*x^(3/2)+2*A/c*x^(1/2)-1/4/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)))-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
)*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)))-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 [A]  time = 1.29, size = 247, normalized size = 0.89 \begin {gather*} -\frac {a {\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 )}}{4 \, c} + \frac {2 \, {\left (B x^{\frac {3}{2}} + 3 \, A \sqrt {x}\right )}}{3 \, 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),x, algorithm="maxima")

[Out]

-1/4*a*(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))/sqr
t(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*sq
rt(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)) - 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)) + 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)))/c + 2/3*(B*x^(3/2) + 3*A*sqrt(x))/c

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mupad [B]  time = 0.25, size = 601, normalized size = 2.16 \begin {gather*} \frac {2\,A\,\sqrt {x}}{c}+\frac {2\,B\,x^{3/2}}{3\,c}-\mathrm {atan}\left (\frac {B^2\,a^3\,\sqrt {x}\,\sqrt {\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}-\frac {A\,B\,a}{2\,c^3}-\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}-\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}+\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}-\frac {A^2\,a^2\,c\,\sqrt {x}\,\sqrt {\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}-\frac {A\,B\,a}{2\,c^3}-\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}-\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}+\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a\,c^7}-B^2\,a\,\sqrt {-a\,c^7}+2\,A\,B\,a\,c^4}{4\,c^7}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {B^2\,a^3\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}-\frac {A\,B\,a}{2\,c^3}-\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}+\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}-\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}-\frac {A^2\,a^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a\,c^7}}{4\,c^6}-\frac {A\,B\,a}{2\,c^3}-\frac {B^2\,a\,\sqrt {-a\,c^7}}{4\,c^7}}\,32{}\mathrm {i}}{\frac {16\,B^3\,a^4}{c^2}+\frac {16\,A^3\,a^2\,\sqrt {-a\,c^7}}{c^4}-\frac {16\,A^2\,B\,a^3}{c}-\frac {16\,A\,B^2\,a^3\,\sqrt {-a\,c^7}}{c^5}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a\,c^7}-A^2\,c\,\sqrt {-a\,c^7}+2\,A\,B\,a\,c^4}{4\,c^7}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*A*x^(1/2))/c - atan((B^2*a^3*x^(1/2)*((A^2*(-a*c^7)^(1/2))/(4*c^6) - (A*B*a)/(2*c^3) - (B^2*a*(-a*c^7)^(1/2
))/(4*c^7))^(1/2)*32i)/((16*B^3*a^4)/c^2 + (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c - (16*A*B^2*a^3*
(-a*c^7)^(1/2))/c^5) - (A^2*a^2*c*x^(1/2)*((A^2*(-a*c^7)^(1/2))/(4*c^6) - (A*B*a)/(2*c^3) - (B^2*a*(-a*c^7)^(1
/2))/(4*c^7))^(1/2)*32i)/((16*B^3*a^4)/c^2 + (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c - (16*A*B^2*a^
3*(-a*c^7)^(1/2))/c^5))*(-(B^2*a*(-a*c^7)^(1/2) - A^2*c*(-a*c^7)^(1/2) + 2*A*B*a*c^4)/(4*c^7))^(1/2)*2i - atan
((B^2*a^3*x^(1/2)*((B^2*a*(-a*c^7)^(1/2))/(4*c^7) - (A*B*a)/(2*c^3) - (A^2*(-a*c^7)^(1/2))/(4*c^6))^(1/2)*32i)
/((16*B^3*a^4)/c^2 - (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c + (16*A*B^2*a^3*(-a*c^7)^(1/2))/c^5) -
(A^2*a^2*c*x^(1/2)*((B^2*a*(-a*c^7)^(1/2))/(4*c^7) - (A*B*a)/(2*c^3) - (A^2*(-a*c^7)^(1/2))/(4*c^6))^(1/2)*32
i)/((16*B^3*a^4)/c^2 - (16*A^3*a^2*(-a*c^7)^(1/2))/c^4 - (16*A^2*B*a^3)/c + (16*A*B^2*a^3*(-a*c^7)^(1/2))/c^5)
)*(-(A^2*c*(-a*c^7)^(1/2) - B^2*a*(-a*c^7)^(1/2) + 2*A*B*a*c^4)/(4*c^7))^(1/2)*2i + (2*B*x^(3/2))/(3*c)

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sympy [A]  time = 10.06, size = 379, normalized size = 1.36 \begin {gather*} \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} \sqrt [4]{\frac {1}{c}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c} - \frac {\sqrt [4]{-1} A \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} \log {\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} + \sqrt {x} \right )}}{2 c} + \frac {\sqrt [4]{-1} A \sqrt [4]{a} \sqrt [4]{\frac {1}{c}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {x}}{\sqrt [4]{a} \sqrt [4]{\frac {1}{c}}} \right )}}{c} + \frac {2 A \sqrt {x}}{c} + \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^{2} \sqrt [4]{\frac {1}{c}}} - \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^{2} \sqrt [4]{\frac {1}{c}}} - \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^{2} \sqrt [4]{\frac {1}{c}}} + \frac {2 B x^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases} \end {gather*}

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)*(1/c)**(1/4)*log(-(-1)**(1/4)*a*
*(1/4)*(1/c)**(1/4) + sqrt(x))/(2*c) - (-1)**(1/4)*A*a**(1/4)*(1/c)**(1/4)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/
4) + sqrt(x))/(2*c) + (-1)**(1/4)*A*a**(1/4)*(1/c)**(1/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/c
+ 2*A*sqrt(x)/c + (-1)**(3/4)*B*a**(3/4)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*c**2*(1/c)**(1/4
)) - (-1)**(3/4)*B*a**(3/4)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*c**2*(1/c)**(1/4)) - (-1)**(3/
4)*B*a**(3/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(c**2*(1/c)**(1/4)) + 2*B*x**(3/2)/(3*c), True
))

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