3.621 \(\int \frac {1}{\sqrt {d+e x} (a+c x^2)} \, dx\)

Optimal. Leaf size=538 \[ -\frac {e \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}} \]

[Out]

1/2*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^
(1/2))^(1/2))/c^(1/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/2*e*arctanh((c^(1/4)
*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))/c^(1/4)*2
^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/4*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-
c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/c^(1/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^
(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/4*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d
*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/c^(1/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

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Rubi [A]  time = 0.45, antiderivative size = 538, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {708, 1094, 634, 618, 206, 628} \[ -\frac {e \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(a + c*x^2)),x]

[Out]

(e*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]]])/(Sqrt[2]*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*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]]])/(Sqrt[
2]*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*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)])/(2*Sqrt[2]*c^(1/4)*Sqrt[c*d^2
+ a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*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)])/(2*Sqrt[2]*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 708

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1094

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )\\ &=\frac {e \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-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 )}{\sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+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 )}{\sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {e \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 )}{2 \sqrt {c} \sqrt {c d^2+a e^2}}+\frac {e \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 )}{2 \sqrt {c} \sqrt {c d^2+a e^2}}-\frac {e \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 )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \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 )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {e \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 )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \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 )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \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 )}{\sqrt {c} \sqrt {c d^2+a e^2}}-\frac {e \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 )}{\sqrt {c} \sqrt {c d^2+a e^2}}\\ &=\frac {e \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 )}{\sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \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 )}{\sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \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 )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \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 )}{2 \sqrt {2} \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 = 0.11, size = 135, normalized size = 0.25 \[ \frac {\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}}{\sqrt {-a} \sqrt [4]{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)),x]

[Out]

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

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fricas [B]  time = 0.83, size = 941, normalized size = 1.75 \[ \frac {1}{2} \, \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}} \log \left (\sqrt {e x + d} e + {\left (a e^{2} + {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}}\right ) - \frac {1}{2} \, \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}} \log \left (\sqrt {e x + d} e - {\left (a e^{2} + {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {-\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} + d}{a c d^{2} + a^{2} e^{2}}}\right ) + \frac {1}{2} \, \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}} \log \left (\sqrt {e x + d} e + {\left (a e^{2} - {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}}\right ) - \frac {1}{2} \, \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}} \log \left (\sqrt {e x + d} e - {\left (a e^{2} - {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}\right )} \sqrt {\frac {{\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {-\frac {e^{2}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}} - d}{a c d^{2} + a^{2} e^{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) + d)/(a*c*d^2 + a^2*e^2)
)*log(sqrt(e*x + d)*e + (a*e^2 + (a*c^2*d^3 + a^2*c*d*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^
4)))*sqrt(-((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) + d)/(a*c*d^2 + a^2*e^2
))) - 1/2*sqrt(-((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) + d)/(a*c*d^2 + a^
2*e^2))*log(sqrt(e*x + d)*e - (a*e^2 + (a*c^2*d^3 + a^2*c*d*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^
3*c*e^4)))*sqrt(-((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) + d)/(a*c*d^2 + a
^2*e^2))) + 1/2*sqrt(((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) - d)/(a*c*d^2
+ a^2*e^2))*log(sqrt(e*x + d)*e + (a*e^2 - (a*c^2*d^3 + a^2*c*d*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2
+ a^3*c*e^4)))*sqrt(((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) - d)/(a*c*d^2
+ a^2*e^2))) - 1/2*sqrt(((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) - d)/(a*c
*d^2 + a^2*e^2))*log(sqrt(e*x + d)*e - (a*e^2 - (a*c^2*d^3 + a^2*c*d*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2
*e^2 + a^3*c*e^4)))*sqrt(((a*c*d^2 + a^2*e^2)*sqrt(-e^2/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)) - d)/(a*c
*d^2 + a^2*e^2)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [B]  time = 0.25, size = 1548, normalized size = 2.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/(a*e^2+c*d^2)^(1/2)/a/e*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*
d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*d-1/4/(a*e^2+c*d^2)^(1/2)/c/a/e*ln((e*x+d)*c^(1/2)-(e*x+d)
^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a
*c*e^2+c^2*d^2)^(1/2)+2*e/(a*e^2+c*d^2)^(1/2)/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))
^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*
c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))+1/2/(a*e^2+c*d^2)^(1/2)/a/e/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*
((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*
d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*(2*c
*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*d-1/2/(a*e^2+c*d^2)^(1/2)/c/a/e/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((
a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+
4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*(2*c*d
+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)-1/4/(a*e^2+c*d^2)^(1/2)/a/e*ln((e*x+d)*c^(1/2)+(e*x+
d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*
d+1/4/(a*e^2+c*d^2)^(1/2)/c/a/e*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^
2+c*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)+2*e/(a*e^2+c*d^2)^(1/2)/(-2*c*
d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)+(2*c*d+2*((a*
e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))+1/2/(a*e^2
+c*d^2)^(1/2)/a/e/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/
2)*c^(1/2)+(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^
(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*d-1/2/(a*e^2+c*d
^2)^(1/2)/c/a/e/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)
*c^(1/2)+(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1
/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^
(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + a\right )} \sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.65, size = 1366, normalized size = 2.54 \[ 2\,\mathrm {atanh}\left (\frac {32\,a^2\,c^5\,d^2\,e^2\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}-\frac {32\,c^3\,e^2\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^2\,c^4\,d\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a\,c^3\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}+\frac {32\,a\,c^4\,d\,e^3\,\sqrt {-a^3\,c}\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}\right )\,\sqrt {-\frac {e\,\sqrt {-a^3\,c}+a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}-2\,\mathrm {atanh}\left (\frac {32\,c^3\,e^2\,\sqrt {\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^2\,c^4\,d\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a\,c^3\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}-\frac {32\,a^2\,c^5\,d^2\,e^2\,\sqrt {\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}+\frac {32\,a\,c^4\,d\,e^3\,\sqrt {-a^3\,c}\,\sqrt {\frac {e\,\sqrt {-a^3\,c}}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}-\frac {a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}}\,\sqrt {d+e\,x}}{\frac {16\,a^4\,c^6\,d^3\,e^3}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^4\,c^4\,e^6\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}+\frac {16\,a^5\,c^5\,d\,e^5}{a^3\,c\,e^2+a^2\,c^2\,d^2}-\frac {16\,a^3\,c^5\,d^2\,e^4\,\sqrt {-a^3\,c}}{a^3\,c\,e^2+a^2\,c^2\,d^2}}\right )\,\sqrt {\frac {e\,\sqrt {-a^3\,c}-a\,c\,d}{4\,\left (a^3\,c\,e^2+a^2\,c^2\,d^2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atanh((32*a^2*c^5*d^2*e^2*(- (e*(-a^3*c)^(1/2))/(4*(a^3*c*e^2 + a^2*c^2*d^2)) - (a*c*d)/(4*(a^3*c*e^2 + a^2*
c^2*d^2)))^(1/2)*(d + e*x)^(1/2))/((16*a^4*c^6*d^3*e^3)/(a^3*c*e^2 + a^2*c^2*d^2) + (16*a^4*c^4*e^6*(-a^3*c)^(
1/2))/(a^3*c*e^2 + a^2*c^2*d^2) + (16*a^5*c^5*d*e^5)/(a^3*c*e^2 + a^2*c^2*d^2) + (16*a^3*c^5*d^2*e^4*(-a^3*c)^
(1/2))/(a^3*c*e^2 + a^2*c^2*d^2)) - (32*c^3*e^2*(- (e*(-a^3*c)^(1/2))/(4*(a^3*c*e^2 + a^2*c^2*d^2)) - (a*c*d)/
(4*(a^3*c*e^2 + a^2*c^2*d^2)))^(1/2)*(d + e*x)^(1/2))/((16*a^2*c^4*d*e^3)/(a^3*c*e^2 + a^2*c^2*d^2) + (16*a*c^
3*e^4*(-a^3*c)^(1/2))/(a^3*c*e^2 + a^2*c^2*d^2)) + (32*a*c^4*d*e^3*(-a^3*c)^(1/2)*(- (e*(-a^3*c)^(1/2))/(4*(a^
3*c*e^2 + a^2*c^2*d^2)) - (a*c*d)/(4*(a^3*c*e^2 + a^2*c^2*d^2)))^(1/2)*(d + e*x)^(1/2))/((16*a^4*c^6*d^3*e^3)/
(a^3*c*e^2 + a^2*c^2*d^2) + (16*a^4*c^4*e^6*(-a^3*c)^(1/2))/(a^3*c*e^2 + a^2*c^2*d^2) + (16*a^5*c^5*d*e^5)/(a^
3*c*e^2 + a^2*c^2*d^2) + (16*a^3*c^5*d^2*e^4*(-a^3*c)^(1/2))/(a^3*c*e^2 + a^2*c^2*d^2)))*(-(e*(-a^3*c)^(1/2) +
a*c*d)/(4*(a^3*c*e^2 + a^2*c^2*d^2)))^(1/2) - 2*atanh((32*c^3*e^2*((e*(-a^3*c)^(1/2))/(4*(a^3*c*e^2 + a^2*c^2
*d^2)) - (a*c*d)/(4*(a^3*c*e^2 + a^2*c^2*d^2)))^(1/2)*(d + e*x)^(1/2))/((16*a^2*c^4*d*e^3)/(a^3*c*e^2 + a^2*c^
2*d^2) - (16*a*c^3*e^4*(-a^3*c)^(1/2))/(a^3*c*e^2 + a^2*c^2*d^2)) - (32*a^2*c^5*d^2*e^2*((e*(-a^3*c)^(1/2))/(4
*(a^3*c*e^2 + a^2*c^2*d^2)) - (a*c*d)/(4*(a^3*c*e^2 + a^2*c^2*d^2)))^(1/2)*(d + e*x)^(1/2))/((16*a^4*c^6*d^3*e
^3)/(a^3*c*e^2 + a^2*c^2*d^2) - (16*a^4*c^4*e^6*(-a^3*c)^(1/2))/(a^3*c*e^2 + a^2*c^2*d^2) + (16*a^5*c^5*d*e^5)
/(a^3*c*e^2 + a^2*c^2*d^2) - (16*a^3*c^5*d^2*e^4*(-a^3*c)^(1/2))/(a^3*c*e^2 + a^2*c^2*d^2)) + (32*a*c^4*d*e^3*
(-a^3*c)^(1/2)*((e*(-a^3*c)^(1/2))/(4*(a^3*c*e^2 + a^2*c^2*d^2)) - (a*c*d)/(4*(a^3*c*e^2 + a^2*c^2*d^2)))^(1/2
)*(d + e*x)^(1/2))/((16*a^4*c^6*d^3*e^3)/(a^3*c*e^2 + a^2*c^2*d^2) - (16*a^4*c^4*e^6*(-a^3*c)^(1/2))/(a^3*c*e^
2 + a^2*c^2*d^2) + (16*a^5*c^5*d*e^5)/(a^3*c*e^2 + a^2*c^2*d^2) - (16*a^3*c^5*d^2*e^4*(-a^3*c)^(1/2))/(a^3*c*e
^2 + a^2*c^2*d^2)))*((e*(-a^3*c)^(1/2) - a*c*d)/(4*(a^3*c*e^2 + a^2*c^2*d^2)))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + c x^{2}\right ) \sqrt {d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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