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

Optimal. Leaf size=689 \[ \frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {2 e \sqrt {d+e x}}{c} \]

[Out]

2*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))*(c*d^2+a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/c^(5/4)*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))*(c*d^2+a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/c^(5/4)
*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))*(c*d^2+a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^
(1/2))/c^(5/4)*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))*(c*d^2+a*e^2+2*d*c^(1/2)*
(a*e^2+c*d^2)^(1/2))/c^(5/4)*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 = 1.48, antiderivative size = 689, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {704, 827, 1169, 634, 618, 206, 628} \[ \frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}-\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {2 e \sqrt {d+e x}}{c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a + c*x^2),x]

[Out]

(2*e*Sqrt[d + e*x])/c - (e*(c*d^2 + a*e^2 - 2*Sqrt[c]*d*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^(5/4)*Sqrt[c
*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Ar
cTanh[(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^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + a*e^2 + 2*Sq
rt[c]*d*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^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]
]) - (e*(c*d^2 + a*e^2 + 2*Sqrt[c]*d*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^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sq
rt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

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]

Rule 628

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]

Rule 634

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]

Rule 704

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]

Rule 827

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]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1169

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
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{a+c x^2} \, dx &=\frac {2 e \sqrt {d+e x}}{c}+\frac {\int \frac {c d^2-a e^2+2 c d e x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{c}\\ &=\frac {2 e \sqrt {d+e x}}{c}+\frac {2 \operatorname {Subst}\left (\int \frac {-2 c d^2 e+e \left (c d^2-a e^2\right )+2 c d e x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c}\\ &=\frac {2 e \sqrt {d+e x}}{c}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \left (-2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (-2 c d^2 e+e \left (c d^2-a e^2\right )-2 \sqrt {c} d e \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \left (-2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (-2 c d^2 e+e \left (c d^2-a e^2\right )-2 \sqrt {c} d e \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {2 e \sqrt {d+e x}}{c}-\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \operatorname {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^{3/2} \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \operatorname {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^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \operatorname {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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \operatorname {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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {2 e \sqrt {d+e x}}{c}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \operatorname {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^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \operatorname {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^{3/2} \sqrt {c d^2+a e^2}}\\ &=\frac {2 e \sqrt {d+e x}}{c}-\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2-2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+2 \sqrt {c} d \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^{5/4} \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 [A]  time = 0.14, size = 159, normalized size = 0.23 \[ \frac {2 \sqrt {-a} \sqrt [4]{c} e \sqrt {d+e x}+\left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )-\left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} c^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-a]*c^(1/4)*e*Sqrt[d + e*x] + (Sqrt[c]*d - Sqrt[-a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt
[c]*d - Sqrt[-a]*e]] - (Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a
]*e]])/(Sqrt[-a]*c^(5/4))

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fricas [A]  time = 1.20, size = 998, normalized size = 1.45 \[ \frac {c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} + a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} + a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} + a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} - a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4} - a c^{4} d \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {-\frac {c d^{3} - 3 \, a d e^{2} - a c^{2} \sqrt {-\frac {9 \, c^{2} d^{4} e^{2} - 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + 4 \, \sqrt {e x + d} e}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log
(-(3*c^2*d^4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) + (3*a*c^2*d^2*e^2 - a^2*c*e^4 + a*c^4*d*sqrt(-(9*c^2*
d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2
*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4
+ a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 -
a^2*c*e^4 + a*c^4*d*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt(-(c*d^3 - 3*a*d*e^2 + a*c^2
*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) + c*sqrt(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt
(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sq
rt(e*x + d) + (3*a*c^2*d^2*e^2 - a^2*c*e^4 - a*c^4*d*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))
*sqrt(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt
(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^
4*e + 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 - a^2*c*e^4 - a*c^4*d*sqrt(-(9*c^2*d^4*e^2 - 6
*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt(-(c*d^3 - 3*a*d*e^2 - a*c^2*sqrt(-(9*c^2*d^4*e^2 - 6*a*c*d^2*e^4 + a^2*
e^6)/(a*c^5)))/(a*c^2))) + 4*sqrt(e*x + d)*e)/c

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

[Out]

Timed out

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maple [B]  time = 0.18, size = 2763, normalized size = 4.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.68, size = 1625, normalized size = 2.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(a + c*x^2),x)

[Out]

2*atanh((32*a^2*c*e^6*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3
*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(48*c^2*d^5*e^3 - 16*a^2*d*e^7 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^3
+ 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c^2 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)) + (32*d*e^5*(-a^3*
c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(-a^
3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(48*c^3*d^5*e^3 + 32*a*c^2*d^3*e^5 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^2 - 16*a^
2*c*d*e^7 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/a + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c) - (96*d^3*e^3*(-a^3*c^5)^(1/2)*
(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(-a^3*c^5)^(1/2
))/(4*a^2*c^4))^(1/2))/(48*a*c^2*d^5*e^3 - 16*a^3*d*e^7 + 32*a^2*c*d^3*e^5 - (16*a^2*e^8*(-a^3*c^5)^(1/2))/c^3
+ (48*d^4*e^4*(-a^3*c^5)^(1/2))/c + (32*a*d^2*e^6*(-a^3*c^5)^(1/2))/c^2) - (96*a*c^2*d^2*e^4*(d + e*x)^(1/2)*
((3*d*e^2)/(4*c^2) - d^3/(4*a*c) + (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) - (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^
(1/2))/(48*c^2*d^5*e^3 - 16*a^2*d*e^7 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^3 + 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c
^5)^(1/2))/c^2 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)))*(-(a*c^4*d^3 - a*e^3*(-a^3*c^5)^(1/2) - 3*a^2*c^3*d*e^2
+ 3*c*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^5))^(1/2) - 2*atanh((32*a^2*c*e^6*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) -
d^3/(4*a*c) - (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^7
- 48*c^2*d^5*e^3 - (16*a*e^8*(-a^3*c^5)^(1/2))/c^3 - 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c^2 + (48
*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)) + (32*d*e^5*(-a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c
) - (e^3*(-a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(48*c^3*d^5*e^3 + 32*a*c
^2*d^3*e^5 + (16*a*e^8*(-a^3*c^5)^(1/2))/c^2 - 16*a^2*c*d*e^7 - (48*d^4*e^4*(-a^3*c^5)^(1/2))/a - (32*d^2*e^6*
(-a^3*c^5)^(1/2))/c) + (96*d^3*e^3*(-a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) - (e^3*(-
a^3*c^5)^(1/2))/(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^3*d*e^7 - 48*a*c^2*d^5*e^3 -
32*a^2*c*d^3*e^5 - (16*a^2*e^8*(-a^3*c^5)^(1/2))/c^3 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/c + (32*a*d^2*e^6*(-a^3*c
^5)^(1/2))/c^2) - (96*a*c^2*d^2*e^4*(d + e*x)^(1/2)*((3*d*e^2)/(4*c^2) - d^3/(4*a*c) - (e^3*(-a^3*c^5)^(1/2))/
(4*a*c^5) + (3*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^4))^(1/2))/(16*a^2*d*e^7 - 48*c^2*d^5*e^3 - (16*a*e^8*(-a^3*c^
5)^(1/2))/c^3 - 32*a*c*d^3*e^5 + (32*d^2*e^6*(-a^3*c^5)^(1/2))/c^2 + (48*d^4*e^4*(-a^3*c^5)^(1/2))/(a*c)))*(-(
a*c^4*d^3 + a*e^3*(-a^3*c^5)^(1/2) - 3*a^2*c^3*d*e^2 - 3*c*d^2*e*(-a^3*c^5)^(1/2))/(4*a^2*c^5))^(1/2) + (2*e*(
d + e*x)^(1/2))/c

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sympy [A]  time = 61.73, size = 394, normalized size = 0.57 \[ - \frac {2 a e^{3} \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} c e^{6} + 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} + 1, \left (t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} - 64 t^{3} a c^{2} d^{3} e^{2} + 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )}}{c} - 2 d^{2} e \operatorname {RootSum} {\left (t^{4} \left (256 a^{3} c e^{6} + 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} + 1, \left (t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} - 64 t^{3} a c^{2} d^{3} e^{2} + 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )} + 2 d e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left (t \mapsto t \log {\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} + 2 d e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left (t \mapsto t \log {\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} + \frac {2 e \sqrt {d + e x}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*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*l
og(-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*d*
*2*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(-6
4*_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*e*RootS
um(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)))) + 2*d*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)))) + 2*e*sqrt(d + e*x)/c

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