∫ (d+ex)5∕2 ______________

Optimal. Leaf size=781 \[ \frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

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

d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)-(-a*e^2+3*c*d^2)*(a

*e^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)+1/2*e*arctanh((c^

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

^(3/2)*d^3+2*a*d*e^2*c^(1/2)-(-a*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1

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

^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)+(-a*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7

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

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

*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

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Rubi [A]
time = 2.01, antiderivative size = 781, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {718, 839, 841, 1183, 648, 632, 212, 642} \begin {gather*} \frac {e \left (\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (-\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (-\left (3 c d^2-a e^2\right ) \sqrt {a e^2+c d^2}+2 a \sqrt {c} d e^2+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {4 d e \sqrt {d+e x}}{c} \end {gather*}

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

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

Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])

+ (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sq

rt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(7/4)*S

qrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + (3*c*d^2 -

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

qrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]

]) - (e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + (3*c*d^2 - a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] +

Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(7/4)*S

qrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]

, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +

c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +

b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &

& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (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 + 2*c*d*e*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e}

, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 1]

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (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 + c*e*f)*x, x]/(a + c*x^2)), x], x] /

; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f

- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =

Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(

d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2

\begin {align*} \int \frac {(d+e x)^{5/2}}{a+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+2 c d e x\right )}{a+c x^2} \, dx}{c}\\ &=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\int \frac {c d \left (c d^2-3 a e^2\right )+c e \left (3 c d^2-a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{c^2}\\ &=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {2 \text {Subst}\left (\int \frac {c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )+c e \left (3 c d^2-a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c^2}\\ &=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )-\sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (c d e \left (c d^2-3 a e^2\right )-c d e \left (3 c d^2-a e^2\right )-\sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c^2 \sqrt {c d^2+a e^2}}-\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 c^2 \sqrt {c d^2+a e^2}}+\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{c^2 \sqrt {c d^2+a e^2}}+\frac {\left (e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{c^2 \sqrt {c d^2+a e^2}}\\ &=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.51, size = 250, normalized size = 0.32 \begin {gather*} \frac {2 \sqrt {c} e \sqrt {d+e x} (7 d+e x)+\frac {3 i \left (\sqrt {c} d+i \sqrt {a} e\right )^3 \tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {3 i \left (\sqrt {c} d-i \sqrt {a} e\right )^3 \tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{3 c^{3/2}} \end {gather*}

(2*Sqrt[c]*e*Sqrt[d + e*x]*(7*d + e*x) + ((3*I)*(Sqrt[c]*d + I*Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sq

rt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[a]*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - ((3*I)*(Sqr

t[c]*d - I*Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/

(Sqrt[a]*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/(3*c^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1598\) vs. \(2(632)=1264\).
time = 0.65, size = 1599, normalized size = 2.05


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1599\)
default	\(\text {Expression too large to display}\)	\(1599\)
risch	\(\text {Expression too large to display}\)	\(3924\)

int((e*x+d)^(5/2)/(c*x^2+a),x,method=_RETURNVERBOSE)

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

)*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2+3*(2*(a*c*e^2+c^2*d^2)^(1/

2)+2*c*d)^(1/2)*c^2*d^3-2*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*e^2+c*d^2

)^(1/2)*d+(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*e^2-3*(a*c*e^2+c^2*d^2)^(1/2)*(2*(

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

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

)^(1/2)*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c*d*e^2+3*(2*(a*c*e^2+c^2*d^

2)^(1/2)+2*c*d)^(1/2)*c^2*d^3-2*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*e^2

+c*d^2)^(1/2)*d+(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*e^2-3*(a*c*e^2+c^2*d^2)^(1/2

)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c*d^2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)/c^(1/2))/(4*(a*e^2+c*

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

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

(-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^

(1/2)*a*c*d*e^2-3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^2*d^3+2*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^

2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*e^2+c*d^2)^(1/2)*d-(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^

(1/2)*a*e^2+3*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c*d^2)/c^(1/2)*ln((e*x+d)*c^(1/2

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

^2+1/2*(-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^2+(2*(a*c*e^2+c^2*d^2)^(1/2)+

2*c*d)^(1/2)*a*c*d*e^2-3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^2*d^3+2*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2

+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*e^2+c*d^2)^(1/2)*d-(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+

2*c*d)^(1/2)*a*e^2+3*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c*d^2)*(2*((a*e^2+c*d^2)*

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

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1546 vs. \(2 (620) = 1240\).
time = 1.63, size = 1546, normalized size = 1.98 \begin {gather*} -\frac {3 \, c \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} - 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} + {\left (10 \, a c^{4} d^{5} e^{2} - 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} + {\left (a c^{6} d^{2} - a^{2} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) - 3 \, c \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} - 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} - {\left (10 \, a c^{4} d^{5} e^{2} - 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} + {\left (a c^{6} d^{2} - a^{2} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) + 3 \, c \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} - 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} + {\left (10 \, a c^{4} d^{5} e^{2} - 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} - {\left (a c^{6} d^{2} - a^{2} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) - 3 \, c \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} - 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} - {\left (10 \, a c^{4} d^{5} e^{2} - 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} - {\left (a c^{6} d^{2} - a^{2} c^{5} e^{2}\right )} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {-\frac {c^{2} d^{5} - 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {-\frac {25 \, c^{4} d^{8} e^{2} - 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} - 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) - 4 \, {\left (x e^{2} + 7 \, d e\right )} \sqrt {x e + d}}{6 \, c} \end {gather*}

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

-1/6*(3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*

e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)) + 5*a^2*d*e^4)/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*

a^3*c*d^2*e^7 + a^4*e^9)*sqrt(x*e + d) + (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2

- a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)

/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4

*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)) + 5*a^2*d*e^4)/(a*c^3))) - 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 +

a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7))

+ 5*a^2*d*e^4)/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(x*e + d) - (10

*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*

a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 +

a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7))

+ 5*a^2*d*e^4)/(a*c^3))) + 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*

e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)) + 5*a^2*d*e^4)/(a*c^3))*log((5*c^4*d^8*e - 1

4*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(x*e + d) + (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*

c^2*d*e^6 - (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3

*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6

*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)) + 5*a^2*d*e^4)/(a*c^3))) - 3*c*sqrt(-(c^2*d

^5 - 10*a*c*d^3*e^2 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8

+ a^4*e^10)/(a*c^7)) + 5*a^2*d*e^4)/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^

9)*sqrt(x*e + d) - (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-

(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*

d^5 - 10*a*c*d^3*e^2 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^

8 + a^4*e^10)/(a*c^7)) + 5*a^2*d*e^4)/(a*c^3))) - 4*(x*e^2 + 7*d*e)*sqrt(x*e + d))/c

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Sympy [A]
time = 40.89, size = 498, normalized size = 0.64 \begin {gather*} - \frac {4 a d e^{3} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c e^{6} + 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} + 1, \left ( t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} - 64 t^{3} a c^{2} d^{3} e^{2} + 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )}}{c} - \frac {2 a e^{3} \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left ( t \mapsto t \log {\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )}}{c} - 4 d^{3} e \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c e^{6} + 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} + 1, \left ( t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} - 64 t^{3} a c^{2} d^{3} e^{2} + 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )} + 2 d^{2} e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left ( t \mapsto t \log {\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} + 4 d^{2} e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left ( t \mapsto t \log {\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} + \frac {4 d e \sqrt {d + e x}}{c} + \frac {2 e \left (d + e x\right )^{\frac {3}{2}}}{3 c} \end {gather*}

-4*a*d*e**3*RootSum(_t**4*(256*a**3*c*e**6 + 256*a**2*c**2*d**2*e**4) + 32*_t**2*a*c*d*e**2 + 1, Lambda(_t, _t

*log(-64*_t**3*a**2*c*d*e**4 - 64*_t**3*a*c**2*d**3*e**2 + 4*_t*a*e**2 - 4*_t*c*d**2 + sqrt(d + e*x))))/c - 2*

a*e**3*RootSum(256*_t**4*a**2*c**3*e**4 + 32*_t**2*a*c**2*d*e**2 + a*e**2 + c*d**2, Lambda(_t, _t*log(64*_t**3

*a*c**2*e**2 + 4*_t*c*d + sqrt(d + e*x))))/c - 4*d**3*e*RootSum(_t**4*(256*a**3*c*e**6 + 256*a**2*c**2*d**2*e*

*4) + 32*_t**2*a*c*d*e**2 + 1, Lambda(_t, _t*log(-64*_t**3*a**2*c*d*e**4 - 64*_t**3*a*c**2*d**3*e**2 + 4*_t*a*

e**2 - 4*_t*c*d**2 + sqrt(d + e*x)))) + 2*d**2*e*RootSum(256*_t**4*a**2*c**3*e**4 + 32*_t**2*a*c**2*d*e**2 + a

*e**2 + c*d**2, Lambda(_t, _t*log(64*_t**3*a*c**2*e**2 + 4*_t*c*d + sqrt(d + e*x)))) + 4*d**2*e*RootSum(256*_t

**4*a**2*c**3*e**4 + 32*_t**2*a*c**2*d*e**2 + a*e**2 + c*d**2, Lambda(_t, _t*log(64*_t**3*a*c**2*e**2 + 4*_t*c

*d + sqrt(d + e*x)))) + 4*d*e*sqrt(d + e*x)/c + 2*e*(d + e*x)**(3/2)/(3*c)

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Giac [A]
time = 53.68, size = 373, normalized size = 0.48 \begin {gather*} \frac {{\left (c^{4} d^{4} - 3 \, a c^{3} d^{2} e^{2} + {\left (3 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} c^{2} + 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{3} e + \sqrt {-a c} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e}} + \frac {{\left (c^{4} d^{4} - 3 \, a c^{3} d^{2} e^{2} + {\left (3 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} c^{2} - 2 \, {\left (\sqrt {-a c} c^{2} d^{3} e + \sqrt {-a c} a c d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{3} e - \sqrt {-a c} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {x e + d} c^{2} d e\right )}}{3 \, c^{3}} \end {gather*}

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

(c^4*d^4 - 3*a*c^3*d^2*e^2 + (3*a*c*d^2*e^2 - a^2*e^4)*c^2 + 2*(sqrt(-a*c)*c^2*d^3*e + sqrt(-a*c)*a*c*d*e^3)*a

bs(c))*arctan(sqrt(x*e + d)/sqrt(-(c^4*d + sqrt(c^8*d^2 - (c^4*d^2 + a*c^3*e^2)*c^4))/c^4))/((a*c^3*e + sqrt(-

a*c)*c^3*d)*sqrt(-c^2*d - sqrt(-a*c)*c*e)) + (c^4*d^4 - 3*a*c^3*d^2*e^2 + (3*a*c*d^2*e^2 - a^2*e^4)*c^2 - 2*(s

qrt(-a*c)*c^2*d^3*e + sqrt(-a*c)*a*c*d*e^3)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(c^4*d - sqrt(c^8*d^2 - (c^4*d^

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

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Mupad [B]
time = 0.48, size = 2500, normalized size = 3.20 \begin {gather*} \text {Too large to display} \end {gather*}

(2*e*(d + e*x)^(3/2))/(3*c) - atan((a^3*e^8*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5

*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2)

)/(2*a*c^6))^(1/2)*32i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9)/c + 160*a*c*d

^6*e^5 - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e^10*(-a^3*c^7)

^(1/2))/c^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) - (d^5*e^3*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(-a^3*

c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*

a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^5*e^11)/c - 160*a^4*d^2*e^9 - 80*a*c^3*d

^8*e^3 + 64*a^3*c*d^4*e^7 + 160*a^2*c^2*d^6*e^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/c - (160*a*d^5*e^6*(-a^3*c^7)

^(1/2))/c^2 + (32*a^3*d*e^10*(-a^3*c^7)^(1/2))/c^4 - (288*a^2*d^3*e^8*(-a^3*c^7)^(1/2))/c^3) + (d^3*e^5*(-a^3*

c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(

4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/(16*a^4*

e^11 - 80*c^4*d^8*e^3 + 160*a*c^3*d^6*e^5 - 160*a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e^7 + (160*d^7*e^4*(-a^3*c^7)^(

1/2))/a - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^2 + (32*a^2*d*e^10*(-a^3*c^7)^

(1/2))/c^3) - (a*d*e^7*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3

*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2

*a*c^6))^(1/2)*32i)/(16*a^4*c*e^11 - 160*d^5*e^6*(-a^3*c^7)^(1/2) - 80*c^5*d^8*e^3 + 160*a*c^4*d^6*e^5 + 64*a^

2*c^3*d^4*e^7 - 160*a^3*c^2*d^2*e^9 - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c + (160*c*d^7*e^4*(-a^3*c^7)^(1/2))/a

+ (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^2) + (a*c^2*d^4*e^4*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5

/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a

^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9)

/c + 160*a*c*d^6*e^5 - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e

^10*(-a^3*c^7)^(1/2))/c^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) - (a^2*c*d^2*e^6*(d + e*x)^(1/2)*((e^5*(-a

^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/

(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*320i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2

*d^8*e^3 - (160*a^3*d^2*e^9)/c + 160*a*c*d^6*e^5 - (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 - (288*a*d^3*e^8*(-a^3*c

^7)^(1/2))/c^4 + (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)))*((a^2*e^5*(-a

^3*c^7)^(1/2) - a*c^6*d^5 - 5*a^3*c^4*d*e^4 + 10*a^2*c^5*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^7)^(1/2) - 10*a*c*d^2*e

^3*(-a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*2i - atan((a^3*e^8*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c)

- (e^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(

-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9

)/c + 160*a*c*d^6*e^5 + (160*d^5*e^6*(-a^3*c^7)^(1/2))/c^3 + (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^4 - (32*a^2*d*

e^10*(-a^3*c^7)^(1/2))/c^5 - (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) + (d^5*e^3*(-a^3*c^7)^(1/2)*(d + e*x)^(1/

2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c) - (e^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*

c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^5*e^11)/c - 160*a^4*d^2*e

^9 - 80*a*c^3*d^8*e^3 + 64*a^3*c*d^4*e^7 + 160*a^2*c^2*d^6*e^5 - (160*d^7*e^4*(-a^3*c^7)^(1/2))/c + (160*a*d^5

*e^6*(-a^3*c^7)^(1/2))/c^2 - (32*a^3*d*e^10*(-a^3*c^7)^(1/2))/c^4 + (288*a^2*d^3*e^8*(-a^3*c^7)^(1/2))/c^3) -

(d^3*e^5*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c) - (e^5*(-a^3*c^7)^(1/2))/(4*c^7)

- (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)

*320i)/(16*a^4*e^11 - 80*c^4*d^8*e^3 + 160*a*c^3*d^6*e^5 - 160*a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e^7 - (160*d^7*e

^4*(-a^3*c^7)^(1/2))/a + (160*d^5*e^6*(-a^3*c^7)^(1/2))/c + (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^2 - (32*a^2*d*e

^10*(-a^3*c^7)^(1/2))/c^3) + (a*d*e^7*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) - d^5/(4*a*c) - (e

^5*(-a^3*c^7)^(1/2))/(4*c^7) - (5*a*d*e^4)/(4*c^3) - (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) + (5*d^2*e^3*(-a^3

*c^7)^(1/2))/(2*a*c^6))^(1/2)*32i)/(160*d^5*e^6*(-a^3*c^7)^(1/2) + 16*a^4*c*e^11 - 80*c^5*d^8*e^3 + 160*a*c^4*

d^6*e^5 + 64*a^2*c^3*d^4*e^7 - 160*a^3*c^2*d^2*e^9 + (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c - (160*c*d^7*e^4*(-a^3

*c^7)^(1/2))/a - (32*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^2) + (a*c^2*d^4*e^4*(d + e*x)^(1/2)*((5*d^3*e^2)/(2*c^2) -

d^5/(4*a*c) - (e^5*(-a^3*c^7)^(1/2))/(4*c^7) -...

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3.7.18 \(\int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx\) [618]