3.6.54 \(\int \frac {\sqrt {d+e x}}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=675 \[ \frac {e \left (\sqrt {a e^2+c d^2}+\sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \left (\sqrt {a e^2+c d^2}+\sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \left (d-\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \left (d-\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )} \]

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Rubi [A]  time = 1.02, antiderivative size = 675, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {737, 827, 1169, 634, 618, 206, 628} \begin {gather*} \frac {e \left (\sqrt {a e^2+c d^2}+\sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \left (\sqrt {a e^2+c d^2}+\sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \left (d-\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \left (d-\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a + c*x^2)^2,x]

[Out]

(x*Sqrt[d + e*x])/(2*a*(a + c*x^2)) + (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^
2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(3/4)*Sqrt
[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sq
rt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqr
t[2]*a*c^(3/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c
])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d
+ e*x)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(d - Sqrt[c*d^2
+ a*e^2]/Sqrt[c])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x
] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 737

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

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 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^2} \, dx &=\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )}-\frac {\int \frac {-d-\frac {e x}{2}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {d e}{2}-\frac {e x^2}{2}}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {d e \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt {2} \sqrt [4]{c}}-\left (-\frac {d e}{2}+\frac {e \sqrt {c d^2+a e^2}}{2 \sqrt {c}}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {d e \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt {2} \sqrt [4]{c}}+\left (-\frac {d e}{2}+\frac {e \sqrt {c d^2+a e^2}}{2 \sqrt {c}}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )}+\frac {\left (e \left (1+\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a c}+\frac {\left (e \left (1+\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a c}-\frac {\left (e \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )}-\frac {e \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (1+\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a c}-\frac {\left (e \left (1+\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a c}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a+c x^2\right )}+\frac {e \left (1+\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{3/4} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (1+\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{3/4} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.52, size = 265, normalized size = 0.39 \begin {gather*} \frac {-\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (2 \sqrt {-a} c d^2-a \sqrt {c} d e+\sqrt {-a} a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{2 a c^{3/4}}+\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \left (2 \sqrt {-a} c d^2+a \sqrt {c} d e+\sqrt {-a} a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{2 a c^{3/4}}+\frac {x \sqrt {d+e x} \left (a e^2+c d^2\right )}{a+c x^2}}{2 a \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/(a + c*x^2)^2,x]

[Out]

(((c*d^2 + a*e^2)*x*Sqrt[d + e*x])/(a + c*x^2) - (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*Sqrt[-a]*c*d^2 - a*Sqrt[c]*d
*e + Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(2*a*c^(3/4)) + (Sqrt[Sqrt
[c]*d + Sqrt[-a]*e]*(2*Sqrt[-a]*c*d^2 + a*Sqrt[c]*d*e + Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d + Sqrt[-a]*e]])/(2*a*c^(3/4)))/(2*a*(c*d^2 + a*e^2))

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IntegrateAlgebraic [C]  time = 0.56, size = 294, normalized size = 0.44 \begin {gather*} \frac {i \left (2 \sqrt {c} d+i \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d+i \sqrt {a} e}\right )}{4 a^{3/2} \sqrt {c} \sqrt {-i \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\right )}}-\frac {i \left (2 \sqrt {c} d-i \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d-i \sqrt {a} e}\right )}{4 a^{3/2} \sqrt {c} \sqrt {i \sqrt {c} \left (\sqrt {a} e+i \sqrt {c} d\right )}}+\frac {e^2 x \sqrt {d+e x}}{2 a \left (a e^2+c d^2-2 c d (d+e x)+c (d+e x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[d + e*x]/(a + c*x^2)^2,x]

[Out]

(e^2*x*Sqrt[d + e*x])/(2*a*(c*d^2 + a*e^2 - 2*c*d*(d + e*x) + c*(d + e*x)^2)) + ((I/4)*(2*Sqrt[c]*d + I*Sqrt[a
]*e)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(a^(3/2)*Sqrt[c]*Sq
rt[(-I)*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)]) - ((I/4)*(2*Sqrt[c]*d - I*Sqrt[a]*e)*ArcTan[(Sqrt[-(c*d) + I*Sq
rt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(a^(3/2)*Sqrt[c]*Sqrt[I*Sqrt[c]*(I*Sqrt[c]*d + Sqr
t[a]*e)])

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fricas [B]  time = 0.43, size = 1371, normalized size = 2.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a*c*x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4
*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 -
(2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*
sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^
4)))/(a^3*c^2*d^2 + a^4*c*e^2))) - (a*c*x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt
(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3 + a*e^5)*s
qrt(e*x + d) - (a^2*c*d*e^4 - (2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4
*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 +
2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) + (a*c*x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 -
(a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2
))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 + (2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*
sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c*d^3 + 3*a*d*e^2 - (a^3*c^2*d^2 + a^4*c*
e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) - (a*c*x^2 + a^2)
*sqrt(-(4*c*d^3 + 3*a*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e
^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) - (a^2*c*d*e^4 + (2*a^3*c^4*d^4 + 3*a
^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c*d^3 + 3*a
*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 +
a^4*c*e^2))) + 4*sqrt(e*x + d)*x)/(a*c*x^2 + a^2)

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giac [A]  time = 0.49, size = 332, normalized size = 0.49 \begin {gather*} -\frac {{\left (2 \, a c d^{2} {\left | c \right |} - \sqrt {-a c} d {\left | a \right |} {\left | c \right |} e + a^{2} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c d + \sqrt {a^{2} c^{2} d^{2} - {\left (a c d^{2} + a^{2} e^{2}\right )} a c}}{a c}}}\right )}{4 \, {\left (a^{2} c e - \sqrt {-a c} a c d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a \right |}} - \frac {{\left (2 \, a c d^{2} {\left | c \right |} + \sqrt {-a c} d {\left | a \right |} {\left | c \right |} e + a^{2} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c d - \sqrt {a^{2} c^{2} d^{2} - {\left (a c d^{2} + a^{2} e^{2}\right )} a c}}{a c}}}\right )}{4 \, {\left (a^{2} c e + \sqrt {-a c} a c d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a \right |}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} e - \sqrt {x e + d} d e}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*c*d^2*abs(c) - sqrt(-a*c)*d*abs(a)*abs(c)*e + a^2*abs(c)*e^2)*arctan(sqrt(x*e + d)/sqrt(-(a*c*d + sq
rt(a^2*c^2*d^2 - (a*c*d^2 + a^2*e^2)*a*c))/(a*c)))/((a^2*c*e - sqrt(-a*c)*a*c*d)*sqrt(-c^2*d + sqrt(-a*c)*c*e)
*abs(a)) - 1/4*(2*a*c*d^2*abs(c) + sqrt(-a*c)*d*abs(a)*abs(c)*e + a^2*abs(c)*e^2)*arctan(sqrt(x*e + d)/sqrt(-(
a*c*d - sqrt(a^2*c^2*d^2 - (a*c*d^2 + a^2*e^2)*a*c))/(a*c)))/((a^2*c*e + sqrt(-a*c)*a*c*d)*sqrt(-c^2*d - sqrt(
-a*c)*c*e)*abs(a)) + 1/2*((x*e + d)^(3/2)*e - sqrt(x*e + d)*d*e)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + a
*e^2)*a)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x +d}}{\left (c \,x^{2}+a \right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(c*x^2+a)^2,x)

[Out]

int((e*x+d)^(1/2)/(c*x^2+a)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(e*x + d)/(c*x^2 + a)^2, x)

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mupad [B]  time = 2.17, size = 2380, normalized size = 3.53

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(a + c*x^2)^2,x)

[Out]

atan((((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2
)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2))*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6
*c^4*d^2 + a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 - 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(e^3*(-a^9*c^3)^(1/2) +
4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2)*1i - ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*
(d + e*x)^(1/2)*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(
1/2))*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2) - ((a
*c^2*e^4 - 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64
*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2)*1i)/((4*c^2*d^2*e^3 + a*c*e^5)/(4*a^3) + ((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(
d + e*x)^(1/2)*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1
/2))*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2) + ((a*
c^2*e^4 - 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*
(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2) + ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(-a^9*c^3)^(1/2) +
4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2))*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3
*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 - 4*c^3*d^2*e^2)*(d + e*x)^(1/2)
)/a^2)*(-(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2)))*(-
(e^3*(-a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2)*2i + atan((((
8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a
^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2))*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2
+ a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 - 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)
^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2)*1i - ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x
)^(1/2)*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2))*(-
(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4
- 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^
4*d^2 + a^7*c^3*e^2)))^(1/2)*1i)/((4*c^2*d^2*e^3 + a*c*e^5)/(4*a^3) + ((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)
^(1/2)*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2))*(-(
4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4
- 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4
*d^2 + a^7*c^3*e^2)))^(1/2) + ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)
^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2))*(-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3
*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 - 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(
-(4*a^3*c^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2)))*(-(4*a^3*c
^3*d^3 - e^3*(-a^9*c^3)^(1/2) + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 + a^7*c^3*e^2)))^(1/2)*2i + ((e*(d + e*x)^(3
/2))/(2*a) - (d*e*(d + e*x)^(1/2))/(2*a))/(c*(d + e*x)^2 + a*e^2 + c*d^2 - 2*c*d*(d + e*x))

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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((e*x+d)**(1/2)/(c*x**2+a)**2,x)

[Out]

Timed out

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