3.1613 \(\int \frac {(b+2 c x) (d+e x)^{3/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=422 \[ \frac {\sqrt {2} \left (-2 c^2 d \left (d \sqrt {b^2-4 a c}-4 a e\right )-2 c e \left (-b d \sqrt {b^2-4 a c}-a e \sqrt {b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (2 c^2 d \left (d \sqrt {b^2-4 a c}+4 a e\right )-2 c e \left (b d \sqrt {b^2-4 a c}+a e \sqrt {b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d+e x} (2 c d-b e)}{c}+\frac {4}{3} (d+e x)^{3/2} \]

[Out]

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

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Rubi [A]  time = 2.23, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {824, 826, 1166, 208} \[ \frac {\sqrt {2} \left (-2 c^2 d \left (d \sqrt {b^2-4 a c}-4 a e\right )-2 c e \left (-b d \sqrt {b^2-4 a c}-a e \sqrt {b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (2 c^2 d \left (d \sqrt {b^2-4 a c}+4 a e\right )-2 c e \left (b d \sqrt {b^2-4 a c}+a e \sqrt {b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d+e x} (2 c d-b e)}{c}+\frac {4}{3} (d+e x)^{3/2} \]

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)*Sqrt[d + e*x])/c + (4*(d + e*x)^(3/2))/3 + (Sqrt[2]*(b^2*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c^2*
d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) - 2*c*e*(b^2*d - b*Sqrt[b^2 - 4*a*c]*d + 2*a*b*e - a*Sqrt[b^2 - 4*a*c]*e))*Arc
Tanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt
[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[2]*(b^2*(b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c^2*d*(Sqrt[b^2 - 4*a*c]*
d + 4*a*e) - 2*c*e*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]
*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^
2 - 4*a*c])*e])

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]

Rule 824

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

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]

Rubi steps

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

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Mathematica [A]  time = 2.42, size = 782, normalized size = 1.85 \[ \frac {\frac {(b e-2 c d) \left (\frac {\sqrt {2} \left (2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}-b\right )-2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {\sqrt {2} \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+2 \sqrt {c} e \sqrt {d+e x}\right )}{c^{3/2}}-\frac {2 \sqrt {2} \left (3 c^2 d e \left (d \sqrt {b^2-4 a c}-2 a e-b d\right )+c e^2 \left (-3 b d \sqrt {b^2-4 a c}-a e \sqrt {b^2-4 a c}+3 a b e+3 b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}-b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {2 \sqrt {2} \left (-3 c^2 d e \left (d \sqrt {b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt {b^2-4 a c}+a e\right )+a e \sqrt {b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {4 e \sqrt {d+e x} (b e-2 c d)}{c}+\frac {4}{3} e (d+e x)^{3/2}}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.16, size = 2909, normalized size = 6.89 \[ \text {result too large to display} \]

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]

1/6*(3*sqrt(2)*c*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*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)
)/c^3)*log(sqrt(2)*(6*c^3*d^3 - 9*b*c^2*d^2*e + (5*b^2*c - 2*a*c^2)*d*e^2 - (b^3 - a*b*c)*e^3 - c^3*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))*sqr
t((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*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))/c^3) - 4*(3*c^3*d^
4 - 6*b*c^2*d^3*e + 2*(2*b^2*c + a*c^2)*d^2*e^2 - (b^3 + 2*a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*sqrt(e*x + d))
- 3*sqrt(2)*c*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*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))/c
^3)*log(-sqrt(2)*(6*c^3*d^3 - 9*b*c^2*d^2*e + (5*b^2*c - 2*a*c^2)*d*e^2 - (b^3 - a*b*c)*e^3 - c^3*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))*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*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))/c^3) - 4*(3*c^3*d^4
- 6*b*c^2*d^3*e + 2*(2*b^2*c + a*c^2)*d^2*e^2 - (b^3 + 2*a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*sqrt(e*x + d)) +
3*sqrt(2)*c*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*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))/c^3
)*log(sqrt(2)*(6*c^3*d^3 - 9*b*c^2*d^2*e + (5*b^2*c - 2*a*c^2)*d*e^2 - (b^3 - a*b*c)*e^3 + c^3*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))*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*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))/c^3) - 4*(3*c^3*d^4 - 6
*b*c^2*d^3*e + 2*(2*b^2*c + a*c^2)*d^2*e^2 - (b^3 + 2*a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*sqrt(e*x + d)) - 3*s
qrt(2)*c*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*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))/c^3)*l
og(-sqrt(2)*(6*c^3*d^3 - 9*b*c^2*d^2*e + (5*b^2*c - 2*a*c^2)*d*e^2 - (b^3 - a*b*c)*e^3 + c^3*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))*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*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))/c^3) - 4*(3*c^3*d^4 - 6*b
*c^2*d^3*e + 2*(2*b^2*c + a*c^2)*d^2*e^2 - (b^3 + 2*a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*sqrt(e*x + d)) + 4*(2*
c*e*x + 8*c*d - 3*b*e)*sqrt(e*x + d))/c

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giac [B]  time = 0.48, size = 813, normalized size = 1.93 \[ \frac {{\left ({\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d^{2} e - 2 \, \sqrt {b^{2} - 4 \, a c} b c d e^{2} + {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} - 2 \, {\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e - a b c^{2} e^{3} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} - {\left (4 \, \sqrt {b^{2} - 4 \, a c} c^{4} d^{2} e - 4 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d e^{2} + \sqrt {b^{2} - 4 \, a c} b^{2} c^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{4} d - b c^{3} e + \sqrt {-4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{4} + {\left (2 \, c^{4} d - b c^{3} e\right )}^{2}}}{c^{4}}}}\right )}{4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{2}} - \frac {{\left ({\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d^{2} e - 2 \, \sqrt {b^{2} - 4 \, a c} b c d e^{2} + {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} + 2 \, {\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e - a b c^{2} e^{3} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} - {\left (4 \, \sqrt {b^{2} - 4 \, a c} c^{4} d^{2} e - 4 \, \sqrt {b^{2} - 4 \, a c} b c^{3} d e^{2} + \sqrt {b^{2} - 4 \, a c} b^{2} c^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{4} d - b c^{3} e - \sqrt {-4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{4} + {\left (2 \, c^{4} d - b c^{3} e\right )}^{2}}}{c^{4}}}}\right )}{4 \, {\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} c^{2}} + \frac {2 \, {\left (2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} + 6 \, \sqrt {x e + d} c^{3} d - 3 \, \sqrt {x e + d} b c^{2} e\right )}}{3 \, c^{3}} \]

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]

1/4*((2*sqrt(b^2 - 4*a*c)*c^2*d^2*e - 2*sqrt(b^2 - 4*a*c)*b*c*d*e^2 + (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*e^3)*sqr
t(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2 - 2*(2*c^4*d^3 - 3*b*c^3*d^2*e - a*b*c^2*e^3 + (b^2*c^2 + 2*
a*c^3)*d*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c) - (4*sqrt(b^2 - 4*a*c)*c^4*d^2*e - 4*sqr
t(b^2 - 4*a*c)*b*c^3*d*e^2 + sqrt(b^2 - 4*a*c)*b^2*c^2*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*
arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^4*d - b*c^3*e + sqrt(-4*(c^4*d^2 - b*c^3*d*e + a*c^3*e^2)*c^4 + (2
*c^4*d - b*c^3*e)^2))/c^4))/((c^4*d^2 - b*c^3*d*e + a*c^3*e^2)*c^2) - 1/4*((2*sqrt(b^2 - 4*a*c)*c^2*d^2*e - 2*
sqrt(b^2 - 4*a*c)*b*c*d*e^2 + (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*
c)*e)*c^2 + 2*(2*c^4*d^3 - 3*b*c^3*d^2*e - a*b*c^2*e^3 + (b^2*c^2 + 2*a*c^3)*d*e^2)*sqrt(-4*c^2*d + 2*(b*c + s
qrt(b^2 - 4*a*c)*c)*e)*abs(c) - (4*sqrt(b^2 - 4*a*c)*c^4*d^2*e - 4*sqrt(b^2 - 4*a*c)*b*c^3*d*e^2 + sqrt(b^2 -
4*a*c)*b^2*c^2*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(
2*c^4*d - b*c^3*e - sqrt(-4*(c^4*d^2 - b*c^3*d*e + a*c^3*e^2)*c^4 + (2*c^4*d - b*c^3*e)^2))/c^4))/((c^4*d^2 -
b*c^3*d*e + a*c^3*e^2)*c^2) + 2/3*(2*(x*e + d)^(3/2)*c^3 + 6*sqrt(x*e + d)*c^3*d - 3*sqrt(x*e + d)*b*c^2*e)/c^
3

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maple [B]  time = 0.10, size = 1494, normalized size = 3.54 \[ \text {result too large to display} \]

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]

4/3*(e*x+d)^(3/2)-2/c*b*e*(e*x+d)^(1/2)+4*(e*x+d)^(1/2)*d-4/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4
*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c
)*a*b*e^3+8*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)
^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*d*e^2+1/c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)
/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)
^(1/2))*c)^(1/2)*c)*b^3*e^3-2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)
*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*d*e^2+2*2^(1/2)/((-b*e+2
*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c
)^(1/2)*c)*a*e^2-1/c*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-
b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*e^2+2*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1
/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d*e-2*c*2^(1/2)/((-b*e+
2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*
c)^(1/2)*c)*d^2-4/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+
d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*b*e^3+8*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2
)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(
1/2))*c)^(1/2)*c)*a*d*e^2+1/c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*
arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^3*e^3-2/(-(4*a*c-b^2)*e^2)^(1
/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b
^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*d*e^2-2*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^
(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a*e^2+1/c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^
2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2*e^2-2*2^
(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^
2)^(1/2))*c)^(1/2)*c)*b*d*e+2*c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^
(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} + b x + a}\,{d x} \]

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

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mupad [B]  time = 0.96, size = 6933, normalized size = 16.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(d + e*x)^(3/2))/3 - atan(((((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 + 8*a*c^5*d^3*e^2 + 8*a^2*c^4*d*e^4 - b^4*
c^2*d*e^4 - 2*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^3 + 2*a*b^2*c^3*d*e^4))/c - (8*(d + e*x)^
(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c
^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/
(2*c^3))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*(-(b^3*e^3 - 2*c^3*d^3 + b^
2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*
c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2) - (8*(d + e*x)^(1/
2)*(b^6*e^6 - 8*a^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 + 48*a^2*c^4*d^2*e^4 + 2*b^2*c^4*d^4*e^2 -
4*b^3*c^3*d^3*e^3 + 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 16*a*b*c^4*d^3*e^3 + 28*a*b^3*c^2*d*e^
5 - 48*a^2*b*c^3*d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*
b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a
*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*1i - (((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 + 8*a*c
^5*d^3*e^2 + 8*a^2*c^4*d*e^4 - b^4*c^2*d*e^4 - 2*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^3 + 2*
a*b^2*c^3*d*e^4))/c + (8*(d + e*x)^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 -
a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2)
- 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e
^2))/c)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a
*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2)
)/(2*c^3))^(1/2) + (8*(d + e*x)^(1/2)*(b^6*e^6 - 8*a^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 + 48*a^2
*c^4*d^2*e^4 + 2*b^2*c^4*d^4*e^2 - 4*b^3*c^3*d^3*e^3 + 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 16*
a*b*c^4*d^3*e^3 + 28*a*b^3*c^2*d*e^5 - 48*a^2*b*c^3*d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3*d^3 +
b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b
^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*1i)/((16*(a^2*b
^4*e^8 + 8*a^4*c^2*e^8 + b^6*d^2*e^6 - 6*a^3*b^2*c*e^8 - 8*a*c^5*d^6*e^2 - 4*b^5*c*d^3*e^5 - 8*a^2*c^4*d^4*e^4
+ 8*a^3*c^3*d^2*e^6 + 2*b^2*c^4*d^6*e^2 - 6*b^3*c^3*d^5*e^3 + 7*b^4*c^2*d^4*e^4 - 2*a*b^5*d*e^7 - 18*a^2*b^2*
c^2*d^2*e^6 + 24*a*b*c^4*d^5*e^3 + 10*a^2*b^3*c*d*e^7 - 8*a^3*b*c^2*d*e^7 - 26*a*b^2*c^3*d^4*e^4 + 12*a*b^3*c^
2*d^3*e^5 + 16*a^2*b*c^3*d^3*e^5))/c + (((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 + 8*a*c^5*d^3*e^2 + 8*a^2*c^4*d*e
^4 - b^4*c^2*d*e^4 - 2*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^3 + 2*a*b^2*c^3*d*e^4))/c - (8*(
d + e*x)^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2
) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c
)^(1/2))/(2*c^3))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*(-(b^3*e^3 - 2*c^3
*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e
- 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2) - (8*(d +
e*x)^(1/2)*(b^6*e^6 - 8*a^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 + 48*a^2*c^4*d^2*e^4 + 2*b^2*c^4*d
^4*e^2 - 4*b^3*c^3*d^3*e^3 + 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 16*a*b*c^4*d^3*e^3 + 28*a*b^3
*c^2*d*e^5 - 48*a^2*b*c^3*d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/
2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(
b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2) + (((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 +
8*a*c^5*d^3*e^2 + 8*a^2*c^4*d*e^4 - b^4*c^2*d*e^4 - 2*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^
3 + 2*a*b^2*c^3*d*e^4))/c + (8*(d + e*x)^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*
e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^
(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c
^5*d*e^2))/c)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2)
+ 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)
^(1/2))/(2*c^3))^(1/2) + (8*(d + e*x)^(1/2)*(b^6*e^6 - 8*a^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 +
48*a^2*c^4*d^2*e^4 + 2*b^2*c^4*d^4*e^2 - 4*b^3*c^3*d^3*e^3 + 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5
+ 16*a*b*c^4*d^3*e^3 + 28*a*b^3*c^2*d*e^5 - 48*a^2*b*c^3*d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3
*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e
- 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)))*(-(b^3*
e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*
b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2
)*2i - atan(((((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 + 8*a*c^5*d^3*e^2 + 8*a^2*c^4*d*e^4 - b^4*c^2*d*e^4 - 2*b^2
*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^3 + 2*a*b^2*c^3*d*e^4))/c - (8*(d + e*x)^(1/2)*(-(b^3*e^3
- 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^
2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*(b
^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*
c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d
^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2) - (8*(d + e*x)^(1/2)*(b^6*e^6 - 8*a
^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 + 48*a^2*c^4*d^2*e^4 + 2*b^2*c^4*d^4*e^2 - 4*b^3*c^3*d^3*e^3
+ 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 16*a*b*c^4*d^3*e^3 + 28*a*b^3*c^2*d*e^5 - 48*a^2*b*c^3*
d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3
*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c
*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*1i - (((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 + 8*a*c^5*d^3*e^2 + 8*a^
2*c^4*d*e^4 - b^4*c^2*d*e^4 - 2*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^3 + 2*a*b^2*c^3*d*e^4))
/c + (8*(d + e*x)^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*
a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^
2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*(-(b^3*e^
3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*
c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)
+ (8*(d + e*x)^(1/2)*(b^6*e^6 - 8*a^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 + 48*a^2*c^4*d^2*e^4 + 2*
b^2*c^4*d^4*e^2 - 4*b^3*c^3*d^3*e^3 + 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 16*a*b*c^4*d^3*e^3 +
28*a*b^3*c^2*d*e^5 - 48*a^2*b*c^3*d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4
*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^
2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*1i)/((16*(a^2*b^4*e^8 + 8*a^4*c^
2*e^8 + b^6*d^2*e^6 - 6*a^3*b^2*c*e^8 - 8*a*c^5*d^6*e^2 - 4*b^5*c*d^3*e^5 - 8*a^2*c^4*d^4*e^4 + 8*a^3*c^3*d^2*
e^6 + 2*b^2*c^4*d^6*e^2 - 6*b^3*c^3*d^5*e^3 + 7*b^4*c^2*d^4*e^4 - 2*a*b^5*d*e^7 - 18*a^2*b^2*c^2*d^2*e^6 + 24*
a*b*c^4*d^5*e^3 + 10*a^2*b^3*c*d*e^7 - 8*a^3*b*c^2*d*e^7 - 26*a*b^2*c^3*d^4*e^4 + 12*a*b^3*c^2*d^3*e^5 + 16*a^
2*b*c^3*d^3*e^5))/c + (((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 + 8*a*c^5*d^3*e^2 + 8*a^2*c^4*d*e^4 - b^4*c^2*d*e^
4 - 2*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^3 + 2*a*b^2*c^3*d*e^4))/c - (8*(d + e*x)^(1/2)*(-
(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2
+ 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))
^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b
^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2
- 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2) - (8*(d + e*x)^(1/2)*(b^6*
e^6 - 8*a^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 + 48*a^2*c^4*d^2*e^4 + 2*b^2*c^4*d^4*e^2 - 4*b^3*c^
3*d^3*e^3 + 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 16*a*b*c^4*d^3*e^3 + 28*a*b^3*c^2*d*e^5 - 48*a
^2*b*c^3*d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3
+ a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2
) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2) + (((8*(a*b^3*c^2*e^5 - 4*a^2*b*c^3*e^5 + 8*a*c^5*d^3*e^2
+ 8*a^2*c^4*d*e^4 - b^4*c^2*d*e^4 - 2*b^2*c^4*d^3*e^2 + 3*b^3*c^3*d^2*e^3 - 12*a*b*c^4*d^2*e^3 + 2*a*b^2*c^3*d
*e^4))/c + (8*(d + e*x)^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^
2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e
^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*(-(
b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2
+ 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^
(1/2) + (8*(d + e*x)^(1/2)*(b^6*e^6 - 8*a^3*c^3*e^6 - 8*a*c^5*d^4*e^2 + 18*a^2*b^2*c^2*e^6 + 48*a^2*c^4*d^2*e^
4 + 2*b^2*c^4*d^4*e^2 - 4*b^3*c^3*d^3*e^3 + 6*b^4*c^2*d^2*e^4 - 8*a*b^4*c*e^6 - 4*b^5*c*d*e^5 + 16*a*b*c^4*d^3
*e^3 + 28*a*b^3*c^2*d*e^5 - 48*a^2*b*c^3*d*e^5 - 36*a*b^2*c^3*d^2*e^4))/c)*(-(b^3*e^3 - 2*c^3*d^3 - b^2*e^3*(b
^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2
- 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)))*(-(b^3*e^3 - 2*c^3*d^3 -
b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 + a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b
^2*c*d*e^2 - 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) + 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(2*c^3))^(1/2)*2i - ((4*(b*e -
2*c*d))/c - (2*b*e - 4*c*d)/c)*(d + e*x)^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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