3.2289 \(\int \frac{(d+e x)^{5/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=459 \[ -\frac{\sqrt{2} \left (-3 c^2 d e \left (-d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\sqrt{2} \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c} \]

[Out]

(2*e*(2*c*d - b*e)*Sqrt[d + e*x])/c^2 + (2*e*(d + e*x)^(3/2))/(3*c) - (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 - Sqrt[b^2 - 4*a*c])*e]
])/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (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^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

________________________________________________________________________________________

Rubi [A]  time = 4.43191, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {703, 824, 826, 1166, 208} \[ -\frac{\sqrt{2} \left (-3 c^2 d e \left (-d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\sqrt{2} \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2),x]

[Out]

(2*e*(2*c*d - b*e)*Sqrt[d + e*x])/c^2 + (2*e*(d + e*x)^(3/2))/(3*c) - (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 - Sqrt[b^2 - 4*a*c])*e]
])/(c^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (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^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 703

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

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]

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{(d+e x)^{5/2}}{a+b x+c x^2} \, dx &=\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{\int \frac{\sqrt{d+e x} \left (c d^2-a e^2+e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{c}\\ &=\frac{2 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{\int \frac{c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{c^2}\\ &=\frac{2 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{2 \operatorname{Subst}\left (\int \frac{e \left (c^2 d^3-3 a c d e^2+a b e^3\right )-d e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )+e \left (3 c^2 d^2+b^2 e^2-c e (3 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 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}+\frac{\left (2 c^3 d^3-b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d-\sqrt{b^2-4 a c} d+2 a e\right )+c e^2 \left (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\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^2 \sqrt{b^2-4 a c}}-\frac{\left (2 c^3 d^3-b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d+\sqrt{b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d+a \sqrt{b^2-4 a c} e+3 b \left (\sqrt{b^2-4 a c} d+a e\right )\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^2 \sqrt{b^2-4 a c}}\\ &=\frac{2 e (2 c d-b e) \sqrt{d+e x}}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c}-\frac{\sqrt{2} \left (2 c^3 d^3-b^2 \left (b-\sqrt{b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d-\sqrt{b^2-4 a c} d+2 a e\right )+c e^2 \left (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\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^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}+\frac{\sqrt{2} \left (2 c^3 d^3-b^2 \left (b+\sqrt{b^2-4 a c}\right ) e^3-3 c^2 d e \left (b d+\sqrt{b^2-4 a c} d+2 a e\right )+c e^2 \left (3 b^2 d+a \sqrt{b^2-4 a c} e+3 b \left (\sqrt{b^2-4 a c} d+a e\right )\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^{5/2} \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 = 1.08186, size = 455, normalized size = 0.99 \[ -\frac{\sqrt{2} \left (3 c^2 d e \left (d \sqrt{b^2-4 a c}-2 a e-b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}-b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\sqrt{2} \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c^{5/2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 e \sqrt{d+e x} (b e-2 c d)}{c^2}+\frac{2 e (d+e x)^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2),x]

[Out]

(-2*e*(-2*c*d + b*e)*Sqrt[d + e*x])/c^2 + (2*e*(d + e*x)^(3/2))/(3*c) - (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^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (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^(5/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Maple [B]  time = 0.335, size = 1929, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*x^2+b*x+a),x)

[Out]

2/3*e*(e*x+d)^(3/2)/c-2/c^2*(e*x+d)^(1/2)*b*e^2+4*d*e*(e*x+d)^(1/2)/c-3/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))*a*b*e^4+6/(-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*d*e^3+1/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))*b^3*e^4-3/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*d*e^3+3/(-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^2*e^2-2*e*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))*d^3-1/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))*a*e^3+1/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))*b^2*e^3-3/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*d*e^2+3*e*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^2-3/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))*a*b*e^4+6/(-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))*a*d*e^3+1/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)*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^3*e^4-3/
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*d*e^3+3/(-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^2*e^2-2*e*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)*arcta
nh((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^3+1/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))
*a*e^3-1/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))*b^2*e^3+3/c*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))*b*d*e^2-3*e*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^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{c x^{2} + b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(5/2)/(c*x^2 + b*x + a), x)

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Fricas [B]  time = 17.1642, size = 13779, normalized size = 30.02 \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)^(5/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/6*(3*sqrt(2)*c^2*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)
*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 + (b^2*c^5 - 4*a*c^
6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5
*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*
e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*
c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*
c^11)))/(b^2*c^5 - 4*a*c^6))*log(sqrt(2)*(10*(b^2*c^5 - 4*a*c^6)*d^5*e^2 - 25*(b^3*c^4 - 4*a*b*c^5)*d^4*e^3 +
10*(3*b^4*c^3 - 14*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^4 - 10*(2*b^5*c^2 - 11*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*e^5 + (7*
b^6*c - 44*a*b^4*c^2 + 66*a^2*b^2*c^3 - 8*a^3*c^4)*d*e^6 - (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b*c^3)*e^
7 - (2*(b^2*c^7 - 4*a*c^8)*d^2 - 2*(b^3*c^6 - 4*a*b*c^7)*d*e + (b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c^7)*e^2)*sqrt((
25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10
*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(
9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a
^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))*s
qrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c
- 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 + (b^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*
e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^
4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 -
36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d
*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4
*a*c^6)) + 4*(5*c^6*d^8*e - 20*b*c^5*d^7*e^2 + 35*b^2*c^4*d^6*e^3 - 35*b^3*c^3*d^5*e^4 + 7*(3*b^4*c^2 + a*b^2*
c^3 - 2*a^2*c^4)*d^4*e^5 - 7*(b^5*c + 2*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^6 + (b^6 + 9*a*b^4*c - 15*a^2*b^2*c^2 -
8*a^3*c^3)*d^2*e^7 - (2*a*b^5 - a^2*b^3*c - 8*a^3*b*c^2)*d*e^8 + (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^9)*sqrt(
e*x + d)) - 3*sqrt(2)*c^2*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a
*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 + (b^2*c^5 -
4*a*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c
^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^
5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a
^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10
- 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(-sqrt(2)*(10*(b^2*c^5 - 4*a*c^6)*d^5*e^2 - 25*(b^3*c^4 - 4*a*b*c^5)*d^
4*e^3 + 10*(3*b^4*c^3 - 14*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^4 - 10*(2*b^5*c^2 - 11*a*b^3*c^3 + 12*a^2*b*c^4)*d^2*e
^5 + (7*b^6*c - 44*a*b^4*c^2 + 66*a^2*b^2*c^3 - 8*a^3*c^4)*d*e^6 - (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b
*c^3)*e^7 - (2*(b^2*c^7 - 4*a*c^8)*d^2 - 2*(b^3*c^6 - 4*a*b*c^7)*d*e + (b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c^7)*e^2
)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*
e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e
^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c
^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c
^11)))*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5
*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 + (b^2*c^5 - 4*a*c^6)*sqrt((25*
c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(2
1*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b
^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*
b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)))/(b^2
*c^5 - 4*a*c^6)) + 4*(5*c^6*d^8*e - 20*b*c^5*d^7*e^2 + 35*b^2*c^4*d^6*e^3 - 35*b^3*c^3*d^5*e^4 + 7*(3*b^4*c^2
+ a*b^2*c^3 - 2*a^2*c^4)*d^4*e^5 - 7*(b^5*c + 2*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^6 + (b^6 + 9*a*b^4*c - 15*a^2*b
^2*c^2 - 8*a^3*c^3)*d^2*e^7 - (2*a*b^5 - a^2*b^3*c - 8*a^3*b*c^2)*d*e^8 + (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^
9)*sqrt(e*x + d)) + 3*sqrt(2)*c^2*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c
^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 - (b
^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 -
6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*
a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c
^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(
b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(sqrt(2)*(10*(b^2*c^5 - 4*a*c^6)*d^5*e^2 - 25*(b^3*c^4 - 4*a*b*
c^5)*d^4*e^3 + 10*(3*b^4*c^3 - 14*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^4 - 10*(2*b^5*c^2 - 11*a*b^3*c^3 + 12*a^2*b*c^4
)*d^2*e^5 + (7*b^6*c - 44*a*b^4*c^2 + 66*a^2*b^2*c^3 - 8*a^3*c^4)*d*e^6 - (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 -
4*a^3*b*c^3)*e^7 + (2*(b^2*c^7 - 4*a*c^8)*d^2 - 2*(b^3*c^6 - 4*a*b*c^7)*d*e + (b^4*c^5 - 6*a*b^2*c^6 + 8*a^2*c
^7)*e^2)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^
6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5
)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^
2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10
- 4*a*c^11)))*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^2*
e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 - (b^2*c^5 - 4*a*c^6)*sq
rt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*d^5*e^5
+ 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d^3*e^7 +
5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 -
2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4*a*c^11)
))/(b^2*c^5 - 4*a*c^6)) + 4*(5*c^6*d^8*e - 20*b*c^5*d^7*e^2 + 35*b^2*c^4*d^6*e^3 - 35*b^3*c^3*d^5*e^4 + 7*(3*b
^4*c^2 + a*b^2*c^3 - 2*a^2*c^4)*d^4*e^5 - 7*(b^5*c + 2*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^6 + (b^6 + 9*a*b^4*c - 1
5*a^2*b^2*c^2 - 8*a^3*c^3)*d^2*e^7 - (2*a*b^5 - a^2*b^3*c - 8*a^3*b*c^2)*d*e^8 + (a^2*b^4 - 3*a^3*b^2*c + a^4*
c^2)*e^9)*sqrt(e*x + d)) - 3*sqrt(2)*c^2*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10
*(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e
^5 - (b^2*c^5 - 4*a*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^
3*c^5 - 6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^
4 + 11*a^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*
a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*
e^10)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(-sqrt(2)*(10*(b^2*c^5 - 4*a*c^6)*d^5*e^2 - 25*(b^3*c^4
- 4*a*b*c^5)*d^4*e^3 + 10*(3*b^4*c^3 - 14*a*b^2*c^4 + 8*a^2*c^5)*d^3*e^4 - 10*(2*b^5*c^2 - 11*a*b^3*c^3 + 12*a
^2*b*c^4)*d^2*e^5 + (7*b^6*c - 44*a*b^4*c^2 + 66*a^2*b^2*c^3 - 8*a^3*c^4)*d*e^6 - (b^7 - 7*a*b^5*c + 13*a^2*b^
3*c^2 - 4*a^3*b*c^3)*e^7 + (2*(b^2*c^7 - 4*a*c^8)*d^2 - 2*(b^3*c^6 - 4*a*b*c^7)*d*e + (b^4*c^5 - 6*a*b^2*c^6 +
8*a^2*c^7)*e^2)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 -
6*a*b*c^6)*d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a
^2*b*c^5)*d^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^
2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b
^2*c^10 - 4*a*c^11)))*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*(b^3*c^2 - 3*a*b*c
^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^5 - (b^2*c^5 - 4*a
*c^6)*sqrt((25*c^8*d^8*e^2 - 100*b*c^7*d^7*e^3 + 100*(2*b^2*c^6 - a*c^7)*d^6*e^4 - 50*(5*b^3*c^5 - 6*a*b*c^6)*
d^5*e^5 + 10*(21*b^4*c^4 - 43*a*b^2*c^5 + 11*a^2*c^6)*d^4*e^6 - 20*(6*b^5*c^3 - 18*a*b^3*c^4 + 11*a^2*b*c^5)*d
^3*e^7 + 5*(9*b^6*c^2 - 36*a*b^4*c^3 + 36*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e^8 - 10*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b
^3*c^3 - 2*a^3*b*c^4)*d*e^9 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^10)/(b^2*c^10 - 4
*a*c^11)))/(b^2*c^5 - 4*a*c^6)) + 4*(5*c^6*d^8*e - 20*b*c^5*d^7*e^2 + 35*b^2*c^4*d^6*e^3 - 35*b^3*c^3*d^5*e^4
+ 7*(3*b^4*c^2 + a*b^2*c^3 - 2*a^2*c^4)*d^4*e^5 - 7*(b^5*c + 2*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^6 + (b^6 + 9*a*b
^4*c - 15*a^2*b^2*c^2 - 8*a^3*c^3)*d^2*e^7 - (2*a*b^5 - a^2*b^3*c - 8*a^3*b*c^2)*d*e^8 + (a^2*b^4 - 3*a^3*b^2*
c + a^4*c^2)*e^9)*sqrt(e*x + d)) + 4*(c*e^2*x + 7*c*d*e - 3*b*e^2)*sqrt(e*x + d))/c^2

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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((e*x+d)**(5/2)/(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((e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

Timed out