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

Optimal. Leaf size=275 \[ \frac{\sqrt{2} \sqrt{c} \left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \sqrt{c} \left (-d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a} \]

[Out]

(-2*Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/a + (Sqrt[2]*Sqrt[c]*(b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*ArcTanh[(
Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(a*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b
- Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*(b*d - Sqrt[b^2 - 4*a*c]*d - 2*a*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[
d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(a*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*
e])

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Rubi [A]  time = 1.12901, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {897, 1287, 206, 1166, 208} \[ \frac{\sqrt{2} \sqrt{c} \left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \sqrt{c} \left (-d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/a + (Sqrt[2]*Sqrt[c]*(b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*ArcTanh[(
Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(a*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b
- Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*Sqrt[c]*(b*d - Sqrt[b^2 - 4*a*c]*d - 2*a*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[
d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(a*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*
e])

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1287

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

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 1166

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

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.923132, size = 267, normalized size = 0.97 \[ \frac{\frac{\sqrt{2} \sqrt{c} \left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\sqrt{2} \sqrt{c} \left (d \sqrt{b^2-4 a c}+2 a e-b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.283, size = 581, normalized size = 2.1 \begin{align*} -2\,{\frac{\sqrt{d}}{a}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{c{e}^{2}\sqrt{2}}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c\sqrt{2}}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }+{\frac{ce\sqrt{2}bd}{a}\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-{\frac{c\sqrt{2}d}{a}\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-2\,{\frac{c{e}^{2}\sqrt{2}}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}{\it Artanh} \left ({\frac{\sqrt{ex+d}c\sqrt{2}}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }+{\frac{ce\sqrt{2}bd}{a}{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}+{\frac{c\sqrt{2}d}{a}{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.13064, size = 5018, normalized size = 18.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4
*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e + (a^2*b^3 - 4*a^3
*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4
*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d - a*c*e)*sq
rt(e*x + d)) - sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*
e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e + (a^2*b
^3 - 4*a^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a
^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d -
a*c*e)*sqrt(e*x + d)) + sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*
d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e
- (a^2*b^3 - 4*a^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (b^2 - 2*a*
c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4
*(b*c*d - a*c*e)*sqrt(e*x + d)) - sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2
- 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 -
4*a^2*c)*e - (a^2*b^3 - 4*a^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (
b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a
^3*c)) - 4*(b*c*d - a*c*e)*sqrt(e*x + d)) + 2*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x))/a, 1/2*(sq
rt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4
*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e + (a^2*b^3 - 4*a^3*b*c)*sq
rt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*
sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d - a*c*e)*sqrt(e*x +
d)) - sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^
4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e + (a^2*b^3 - 4*a
^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 -
4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d - a*c*e)*
sqrt(e*x + d)) + sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^
2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e - (a^2*
b^3 - 4*a^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (
a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d
- a*c*e)*sqrt(e*x + d)) - sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b
*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)
*e - (a^2*b^3 - 4*a^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b*e - (b^2 - 2*
a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) -
4*(b*c*d - a*c*e)*sqrt(e*x + d)) + 4*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d))/a]

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Sympy [B]  time = 60.5149, size = 1295, normalized size = 4.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e**2*RootSum(_t**4*(256*a**3*c**2*e**6 - 128*a**2*b**2*c*e**6 - 256*a**2*b*c**2*d*e**5 + 256*a**2*c**3*d**2*
e**4 + 16*a*b**4*e**6 + 128*a*b**3*c*d*e**5 - 128*a*b**2*c**2*d**2*e**4 - 16*b**5*d*e**5 + 16*b**4*c*d**2*e**4
) + _t**2*(-16*a*b*c*e**3 + 32*a*c**2*d*e**2 + 4*b**3*e**3 - 8*b**2*c*d*e**2) + c, Lambda(_t, _t*log(32*_t**3*
a**2*b*e**5 - 64*_t**3*a**2*c*d*e**4 - 8*_t**3*a*b**3*e**5/c - 16*_t**3*a*b**2*d*e**4 + 96*_t**3*a*b*c*d**2*e*
*3 - 64*_t**3*a*c**2*d**3*e**2 + 8*_t**3*b**4*d*e**4/c - 24*_t**3*b**3*d**2*e**3 + 16*_t**3*b**2*c*d**3*e**2 +
4*_t*a*e**2 - 2*_t*b**2*e**2/c + 4*_t*b*d*e - 4*_t*c*d**2 + sqrt(d + e*x)))) - 2*b*d*e*RootSum(_t**4*(256*a**
3*c**2*e**6 - 128*a**2*b**2*c*e**6 - 256*a**2*b*c**2*d*e**5 + 256*a**2*c**3*d**2*e**4 + 16*a*b**4*e**6 + 128*a
*b**3*c*d*e**5 - 128*a*b**2*c**2*d**2*e**4 - 16*b**5*d*e**5 + 16*b**4*c*d**2*e**4) + _t**2*(-16*a*b*c*e**3 + 3
2*a*c**2*d*e**2 + 4*b**3*e**3 - 8*b**2*c*d*e**2) + c, Lambda(_t, _t*log(32*_t**3*a**2*b*e**5 - 64*_t**3*a**2*c
*d*e**4 - 8*_t**3*a*b**3*e**5/c - 16*_t**3*a*b**2*d*e**4 + 96*_t**3*a*b*c*d**2*e**3 - 64*_t**3*a*c**2*d**3*e**
2 + 8*_t**3*b**4*d*e**4/c - 24*_t**3*b**3*d**2*e**3 + 16*_t**3*b**2*c*d**3*e**2 + 4*_t*a*e**2 - 2*_t*b**2*e**2
/c + 4*_t*b*d*e - 4*_t*c*d**2 + sqrt(d + e*x))))/a + 2*c*d**2*RootSum(_t**4*(256*a**3*c**2*e**6 - 128*a**2*b**
2*c*e**6 - 256*a**2*b*c**2*d*e**5 + 256*a**2*c**3*d**2*e**4 + 16*a*b**4*e**6 + 128*a*b**3*c*d*e**5 - 128*a*b**
2*c**2*d**2*e**4 - 16*b**5*d*e**5 + 16*b**4*c*d**2*e**4) + _t**2*(-16*a*b*c*e**3 + 32*a*c**2*d*e**2 + 4*b**3*e
**3 - 8*b**2*c*d*e**2) + c, Lambda(_t, _t*log(32*_t**3*a**2*b*e**5 - 64*_t**3*a**2*c*d*e**4 - 8*_t**3*a*b**3*e
**5/c - 16*_t**3*a*b**2*d*e**4 + 96*_t**3*a*b*c*d**2*e**3 - 64*_t**3*a*c**2*d**3*e**2 + 8*_t**3*b**4*d*e**4/c
- 24*_t**3*b**3*d**2*e**3 + 16*_t**3*b**2*c*d**3*e**2 + 4*_t*a*e**2 - 2*_t*b**2*e**2/c + 4*_t*b*d*e - 4*_t*c*d
**2 + sqrt(d + e*x))))/a - 2*c*d*RootSum(_t**4*(256*a**2*c**3*e**4 - 128*a*b**2*c**2*e**4 + 16*b**4*c*e**4) +
_t**2*(-16*a*b*c*e**3 + 32*a*c**2*d*e**2 + 4*b**3*e**3 - 8*b**2*c*d*e**2) + a*e**2 - b*d*e + c*d**2, Lambda(_t
, _t*log(64*_t**3*a*c**2*e**2 - 16*_t**3*b**2*c*e**2 - 2*_t*b*e + 4*_t*c*d + sqrt(d + e*x))))/a + 2*d*atan(sqr
t(d + e*x)/sqrt(-d))/(a*sqrt(-d))

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out