3.1616 \(\int \frac{b+2 c x}{(d+e x)^{3/2} (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=354 \[ \frac{\sqrt{2} \sqrt{c} \left (b e \left (\sqrt{b^2-4 a c}+b\right )-2 c \left (d \sqrt{b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{c} \left (2 c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac{2 (2 c d-b e)}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.703616, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {828, 826, 1166, 208} \[ \frac{\sqrt{2} \sqrt{c} \left (b e \left (\sqrt{b^2-4 a c}+b\right )-2 c \left (d \sqrt{b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{c} \left (2 c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac{2 (2 c d-b e)}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x]

[Out]

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

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 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{b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx &=\frac{2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}+\frac{\int \frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{c d^2-b d e+a e^2}\\ &=\frac{2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}+\frac{2 \operatorname{Subst}\left (\int \frac{-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )+c (2 c d-b e) 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 )}{c d^2-b d e+a e^2}\\ &=\frac{2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}+\frac{\left (2 \left (\frac{1}{2} c (2 c d-b e)-\frac{-c (2 c d-b e) (-2 c d+b e)+2 c \left (-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )\right )}{2 \sqrt{b^2-4 a c} 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 )}{c d^2-b d e+a e^2}+\frac{\left (2 \left (\frac{1}{2} c (2 c d-b e)+\frac{-c (2 c d-b e) (-2 c d+b e)+2 c \left (-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )\right )}{2 \sqrt{b^2-4 a c} 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 )}{c d^2-b d e+a e^2}\\ &=\frac{2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}+\frac{\sqrt{2} \sqrt{c} \left (b \left (b+\sqrt{b^2-4 a c}\right ) e-2 c \left (\sqrt{b^2-4 a c} d+2 a e\right )\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 )}{\sqrt{b^2-4 a c} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac{\sqrt{2} \sqrt{c} \left (b \left (b-\sqrt{b^2-4 a c}\right ) e+c \left (2 \sqrt{b^2-4 a c} d-4 a e\right )\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 )}{\sqrt{b^2-4 a c} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.549658, size = 317, normalized size = 0.9 \[ \frac{2 \left (\frac{\sqrt{c} \left (2 c \left (d \sqrt{b^2-4 a c}+2 a e\right )-b e \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\sqrt{c} \left (2 c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{b e-2 c d}{\sqrt{d+e x}}\right )}{e (b d-a e)-c d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/((d + e*x)^(3/2)*(a + b*x + c*x^2)),x]

[Out]

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

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Maple [B]  time = 0.032, size = 927, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, c x + b}{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((2*c*x + b)/((c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)

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Fricas [B]  time = 3.04912, size = 17538, normalized size = 49.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(
b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c +
a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 1
8*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 +
4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d
*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b
^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c
^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c
^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*
e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)
)*log(sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 - (5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^
4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 - (2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*
(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*
b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 -
22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2
*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e
^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c
^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*
c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9
+ 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a
*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d
^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3
*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4
*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2
*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*
c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5
+ 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4
*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^
2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)) - 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b
^2*c^2 - a*c^3)*e^2)*sqrt(e*x + d)) - sqrt(2)*(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt
((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a
^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((
9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^
2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*
d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c
^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 +
(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 +
15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^
2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3
*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))*log(-sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 -
(5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 - (2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*
(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*
b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*
a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*
e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^1
2 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^
4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3
)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 1
0*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b
^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c +
a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18
*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4
*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*
e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^
2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^
2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^
2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e
- 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))
- 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b^2*c^2 - a*c^3)*e^2)*sqrt(e*x + d)) + sqrt(2)*(c*d^3 - b*d^2*e + a*d*e^2 + (
c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^
3 - (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 +
3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*
c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a
^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*
d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a
*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^
2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^
3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*
c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))*log(sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e +
(11*b^2*c^2 - 8*a*c^3)*d^2*e^2 - (5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 + (2*c^4*d^7
- 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*
a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*
c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*
d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 1
5*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^
4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4
*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt(
(2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^
2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9
*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2
*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d
^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^
4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 +
(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 1
5*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2
*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*
e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)) - 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b^2*c^2 - a*c^3)*e^2)*sqrt(e*x + d)) - sqrt(
2)*(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 + a*e^3)*x)*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*
a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^
4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3
- 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^
3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^
6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a
^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^
3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^
8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*
b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4))*log(-sqr
t(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 - (5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^4 - 3*a*b
^2*c + 2*a^2*c^2)*e^4 + (2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c +
3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a
^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*
c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*
a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(
2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a
^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^
3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^
4*b^2 + 2*a^5*c)*d^2*e^10)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3
- (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 +
3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c
^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^
2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e - 6*a^5*b*d*e^11 + a^6*e^12 + 3*(5*b^2*c^4 + 2*a*c^5)*d
^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*
b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 - 6*(a*b^5 + 10*a^2
*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3
*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c
^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)) - 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b^2*c^2 -
a*c^3)*e^2)*sqrt(e*x + d)) + 4*(2*c*d - b*e)*sqrt(e*x + d))/(c*d^3 - b*d^2*e + a*d*e^2 + (c*d^2*e - b*d*e^2 +
a*e^3)*x)

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Sympy [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((2*c*x+b)/(e*x+d)**(3/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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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((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

Timed out