3.422 \(\int \frac {\sqrt {x} (A+B x)}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=292 \[ -\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )} \]

[Out]

-1/16*ln(a^(1/2)+x*c^(1/2)-a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(B*a^(1/2)-A*c^(1/2))/a^(5/4)/c^(5/4)*2^(1/2)+1/16
*ln(a^(1/2)+x*c^(1/2)+a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(B*a^(1/2)-A*c^(1/2))/a^(5/4)/c^(5/4)*2^(1/2)-1/8*arcta
n(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)+A*c^(1/2))/a^(5/4)/c^(5/4)*2^(1/2)+1/8*arctan(1+c^(1/4)*2^(1/2
)*x^(1/2)/a^(1/4))*(B*a^(1/2)+A*c^(1/2))/a^(5/4)/c^(5/4)*2^(1/2)-1/2*(-A*c*x+B*a)*x^(1/2)/a/c/(c*x^2+a)

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Rubi [A]  time = 0.23, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {821, 827, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[x]*(a*B - A*c*x))/(2*a*c*(a + c*x^2)) - ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a
^(1/4)])/(4*Sqrt[2]*a^(5/4)*c^(5/4)) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])
/(4*Sqrt[2]*a^(5/4)*c^(5/4)) - ((Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/(8*Sqrt[2]*a^(5/4)*c^(5/4)) + ((Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + S
qrt[c]*x])/(8*Sqrt[2]*a^(5/4)*c^(5/4))

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 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*
x^2)^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*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, 0] && (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 {\sqrt {x} (A+B x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\frac {a B}{2}+\frac {A c x}{2}}{\sqrt {x} \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {a B}{2}+\frac {1}{2} A c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{a c}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a c}+\frac {\left (A+\frac {\sqrt {a} B}{\sqrt {c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx,x,\sqrt {x}\right )}{4 a c}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\left (A+\frac {\sqrt {a} B}{\sqrt {c}}\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 )}{8 a c}+\frac {\left (A+\frac {\sqrt {a} B}{\sqrt {c}}\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 )}{8 a c}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\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 )}{8 \sqrt {2} a^{5/4} c^{3/4}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\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 )}{8 \sqrt {2} a^{5/4} c^{3/4}}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}+\frac {\left (\sqrt {a} B+A \sqrt {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 )}{4 \sqrt {2} a^{5/4} c^{5/4}}-\frac {\left (\sqrt {a} B+A \sqrt {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 )}{4 \sqrt {2} a^{5/4} c^{5/4}}\\ &=-\frac {\sqrt {x} (a B-A c x)}{2 a c \left (a+c x^2\right )}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} c^{5/4}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{5/4} c^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 315, normalized size = 1.08 \[ \frac {-\frac {\sqrt {2} a^{5/4} B \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}+\frac {\sqrt {2} a^{5/4} B \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{c^{5/4}}-\frac {2 \sqrt {2} a^{5/4} B \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{c^{5/4}}+\frac {2 \sqrt {2} a^{5/4} B \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{c^{5/4}}-\frac {4 (-a)^{3/4} A \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {4 (-a)^{3/4} A \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-a}}\right )}{c^{3/4}}+\frac {8 a A x^{3/2}}{a+c x^2}+\frac {8 a B x^{5/2}}{a+c x^2}-\frac {8 a B \sqrt {x}}{c}}{16 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*a*B*Sqrt[x])/c + (8*a*A*x^(3/2))/(a + c*x^2) + (8*a*B*x^(5/2))/(a + c*x^2) - (2*Sqrt[2]*a^(5/4)*B*ArcTan[
1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(5/4) + (2*Sqrt[2]*a^(5/4)*B*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^
(1/4)])/c^(5/4) - (4*(-a)^(3/4)*A*ArcTan[(c^(1/4)*Sqrt[x])/(-a)^(1/4)])/c^(3/4) + (4*(-a)^(3/4)*A*ArcTanh[(c^(
1/4)*Sqrt[x])/(-a)^(1/4)])/c^(3/4) - (Sqrt[2]*a^(5/4)*B*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/c^(5/4) + (Sqrt[2]*a^(5/4)*B*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(5/4))/(16*a^
2)

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fricas [B]  time = 0.93, size = 901, normalized size = 3.09 \[ -\frac {{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + B^{3} a^{3} c - A^{2} B a^{2} c^{2}\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + B^{3} a^{3} c - A^{2} B a^{2} c^{2}\right )} \sqrt {-\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} + 2 \, A B}{a^{2} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - B^{3} a^{3} c + A^{2} B a^{2} c^{2}\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}}\right ) + {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c^{4} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - B^{3} a^{3} c + A^{2} B a^{2} c^{2}\right )} \sqrt {\frac {a^{2} c^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c^{5}}} - 2 \, A B}{a^{2} c^{2}}}\right ) - 4 \, {\left (A c x - B a\right )} \sqrt {x}}{8 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(B*x+A)/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^
2))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + B^3*a
^3*c - A^2*B*a^2*c^2)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2)))
- (a*c^2*x^2 + a^2*c)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))
*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + B^3*a^3*
c - A^2*B*a^2*c^2)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))) -
(a*c^2*x^2 + a^2*c)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))*log
(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - B^3*a^3*c +
A^2*B*a^2*c^2)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))) + (a*c^
2*x^2 + a^2*c)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))*log(-(B^
4*a^2 - A^4*c^2)*sqrt(x) - (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - B^3*a^3*c + A^2*B
*a^2*c^2)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))) - 4*(A*c*x -
B*a)*sqrt(x))/(a*c^2*x^2 + a^2*c)

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giac [A]  time = 0.19, size = 271, normalized size = 0.93 \[ \frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} a c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(B*x+A)/(c*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*(A*c*x^(3/2) - B*a*sqrt(x))/((c*x^2 + a)*a*c) + 1/8*sqrt(2)*((a*c^3)^(1/4)*B*a*c + (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^2*c^3) + 1/8*sqrt(2)*((a*c^3)^(1/4)*B*a*c + (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^2*c^3) + 1/16*sqrt(2)*((a*
c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^3) - 1/16*sqrt(2)*
((a*c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^3)

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maple [A]  time = 0.05, size = 316, normalized size = 1.08 \[ \frac {\sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {\sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {\sqrt {2}\, A \ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}} a c}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{8 a c}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{8 a c}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{c}}}\right )}{16 a c}+\frac {\frac {A \,x^{\frac {3}{2}}}{2 a}-\frac {B \sqrt {x}}{2 c}}{c \,x^{2}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(B*x+A)/(c*x^2+a)^2,x)

[Out]

2*(1/4*A/a*x^(3/2)-1/4*B/c*x^(1/2))/(c*x^2+a)+1/8/a/c*B*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)
+1)+1/8/a/c*B*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+1/16/a/c*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)))+1/16/a/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)))+1/8/a/c*A/(a
/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+1/8/a/c*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 [A]  time = 1.24, size = 272, normalized size = 0.93 \[ \frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{16 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(B*x+A)/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*(A*c*x^(3/2) - B*a*sqrt(x))/(a*c^2*x^2 + a^2*c) + 1/16*(2*sqrt(2)*(B*a*sqrt(c) + A*sqrt(a)*c)*arctan(1/2*s
qrt(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))*sqr
t(c)) + 2*sqrt(2)*(B*a*sqrt(c) + A*sqrt(a)*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)) + sqrt(2)*(B*a*sqrt(c) - A*sqrt(a)*c)*log(sqr
t(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(B*a*sqrt(c) - A*sqrt(a)*c)*lo
g(-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 = 1.28, size = 652, normalized size = 2.23 \[ 2\,\mathrm {atanh}\left (\frac {2\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}-\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}}}{\frac {A\,B^2}{4}-\frac {A^3\,c}{4\,a}-\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^2\,c^3}+\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^2}}-\frac {2\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}-\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}}}{\frac {A\,B^2}{4\,a}-\frac {A^3\,c}{4\,a^2}-\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^3}+\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^4\,c^2}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a^5\,c^5}-B^2\,a\,\sqrt {-a^5\,c^5}+2\,A\,B\,a^3\,c^3}{64\,a^5\,c^5}}+2\,\mathrm {atanh}\left (\frac {2\,A^2\,c^2\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}-\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}}}{\frac {A\,B^2}{4}-\frac {A^3\,c}{4\,a}+\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^2\,c^3}-\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^2}}-\frac {2\,B^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c^5}}{64\,a^5\,c^4}-\frac {A\,B}{32\,a^2\,c^2}-\frac {B^2\,\sqrt {-a^5\,c^5}}{64\,a^4\,c^5}}}{\frac {A\,B^2}{4\,a}-\frac {A^3\,c}{4\,a^2}+\frac {B^3\,\sqrt {-a^5\,c^5}}{4\,a^3\,c^3}-\frac {A^2\,B\,\sqrt {-a^5\,c^5}}{4\,a^4\,c^2}}\right )\,\sqrt {-\frac {B^2\,a\,\sqrt {-a^5\,c^5}-A^2\,c\,\sqrt {-a^5\,c^5}+2\,A\,B\,a^3\,c^3}{64\,a^5\,c^5}}+\frac {\frac {A\,x^{3/2}}{2\,a}-\frac {B\,\sqrt {x}}{2\,c}}{c\,x^2+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2)*(A + B*x))/(a + c*x^2)^2,x)

[Out]

2*atanh((2*A^2*c^2*x^(1/2)*((B^2*(-a^5*c^5)^(1/2))/(64*a^4*c^5) - (A^2*(-a^5*c^5)^(1/2))/(64*a^5*c^4) - (A*B)/
(32*a^2*c^2))^(1/2))/((A*B^2)/4 - (A^3*c)/(4*a) - (B^3*(-a^5*c^5)^(1/2))/(4*a^2*c^3) + (A^2*B*(-a^5*c^5)^(1/2)
)/(4*a^3*c^2)) - (2*B^2*c*x^(1/2)*((B^2*(-a^5*c^5)^(1/2))/(64*a^4*c^5) - (A^2*(-a^5*c^5)^(1/2))/(64*a^5*c^4) -
(A*B)/(32*a^2*c^2))^(1/2))/((A*B^2)/(4*a) - (A^3*c)/(4*a^2) - (B^3*(-a^5*c^5)^(1/2))/(4*a^3*c^3) + (A^2*B*(-a
^5*c^5)^(1/2))/(4*a^4*c^2)))*(-(A^2*c*(-a^5*c^5)^(1/2) - B^2*a*(-a^5*c^5)^(1/2) + 2*A*B*a^3*c^3)/(64*a^5*c^5))
^(1/2) + 2*atanh((2*A^2*c^2*x^(1/2)*((A^2*(-a^5*c^5)^(1/2))/(64*a^5*c^4) - (A*B)/(32*a^2*c^2) - (B^2*(-a^5*c^5
)^(1/2))/(64*a^4*c^5))^(1/2))/((A*B^2)/4 - (A^3*c)/(4*a) + (B^3*(-a^5*c^5)^(1/2))/(4*a^2*c^3) - (A^2*B*(-a^5*c
^5)^(1/2))/(4*a^3*c^2)) - (2*B^2*c*x^(1/2)*((A^2*(-a^5*c^5)^(1/2))/(64*a^5*c^4) - (A*B)/(32*a^2*c^2) - (B^2*(-
a^5*c^5)^(1/2))/(64*a^4*c^5))^(1/2))/((A*B^2)/(4*a) - (A^3*c)/(4*a^2) + (B^3*(-a^5*c^5)^(1/2))/(4*a^3*c^3) - (
A^2*B*(-a^5*c^5)^(1/2))/(4*a^4*c^2)))*(-(B^2*a*(-a^5*c^5)^(1/2) - A^2*c*(-a^5*c^5)^(1/2) + 2*A*B*a^3*c^3)/(64*
a^5*c^5))^(1/2) + ((A*x^(3/2))/(2*a) - (B*x^(1/2))/(2*c))/(a + c*x^2)

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sympy [A]  time = 52.79, size = 1266, normalized size = 4.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)*(B*x+A)/(c*x**2+a)**2,x)

[Out]

Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(c, 0)), ((-2*A/(5*x**(5/2)) - 2*B/(3*x**(
3/2)))/c**2, Eq(a, 0)), ((2*A*x**(3/2)/3 + 2*B*x**(5/2)/5)/a**2, Eq(c, 0)), (4*(-1)**(1/4)*A*a**(1/4)*c*x**(3/
2)*(1/c)**(1/4)/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) + A*a*
log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5
/4)*c**2*x**2*(1/c)**(1/4)) - A*a*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(9/4)*c*(
1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) - 2*A*a*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)*
*(1/4)))/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) + A*c*x**2*lo
g(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4
)*c**2*x**2*(1/c)**(1/4)) - A*c*x**2*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(9/4)*
c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) - 2*A*c*x**2*atan((-1)**(3/4)*sqrt(x)/(a**(1/4
)*(1/c)**(1/4)))/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) - 4*(
-1)**(1/4)*B*a**(5/4)*sqrt(x)*(1/c)**(1/4)/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**
2*x**2*(1/c)**(1/4)) - I*B*a**(3/2)*sqrt(1/c)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)
*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) + I*B*a**(3/2)*sqrt(1/c)*log((-1)**(
1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**
2*(1/c)**(1/4)) - 2*I*B*a**(3/2)*sqrt(1/c)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(8*(-1)**(1/4)*a*
*(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) - I*B*sqrt(a)*c*x**2*sqrt(1/c)*log(-(-1
)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2
*x**2*(1/c)**(1/4)) + I*B*sqrt(a)*c*x**2*sqrt(1/c)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(8*(-1)**(
1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(5/4)*c**2*x**2*(1/c)**(1/4)) - 2*I*B*sqrt(a)*c*x**2*sqrt(1/c)
*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(8*(-1)**(1/4)*a**(9/4)*c*(1/c)**(1/4) + 8*(-1)**(1/4)*a**(
5/4)*c**2*x**2*(1/c)**(1/4)), True))

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