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

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

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Rubi [A]  time = 3.04, 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 = {704, 825, 827, 1169, 634, 618, 206, 628} \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*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a + c*x^2),x]

[Out]

(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]])

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 825

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]

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)^{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 \operatorname {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 {\operatorname {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 {\operatorname {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 ) \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^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 ) \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^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 ) \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^{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 ) \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^{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 ) \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^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 ) \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^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 [A]  time = 0.35, size = 226, normalized size = 0.29 \begin {gather*} \frac {2 \sqrt {-a} c^{3/4} e \sqrt {d+e x} (7 d+e x)+3 \sqrt {\sqrt {c} d-\sqrt {-a} e} \left (-2 \sqrt {-a} \sqrt {c} d e-a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )-3 \sqrt {\sqrt {-a} e+\sqrt {c} d} \left (2 \sqrt {-a} \sqrt {c} d e-a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{3 \sqrt {-a} c^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-a]*c^(3/4)*e*Sqrt[d + e*x]*(7*d + e*x) + 3*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d^2 - 2*Sqrt[-a]*Sqrt[c]*d
*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - 3*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*
d^2 + 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(3*Sqrt[-
a]*c^(7/4))

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IntegrateAlgebraic [C]  time = 0.59, size = 271, normalized size = 0.35 \begin {gather*} -\frac {i \left (\sqrt {c} d-i \sqrt {a} e\right )^3 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} c^{3/2} \sqrt {i \sqrt {c} \left (\sqrt {a} e+i \sqrt {c} d\right )}}+\frac {i \left (\sqrt {c} d+i \sqrt {a} e\right )^3 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} c^{3/2} \sqrt {-i \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\right )}}+\frac {2 e \left ((d+e x)^{3/2}+6 d \sqrt {d+e x}\right )}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(6*d*Sqrt[d + e*x] + (d + e*x)^(3/2)))/(3*c) + (I*(Sqrt[c]*d + I*Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) - I*Sqr
t[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[a]*c^(3/2)*Sqrt[(-I)*Sqrt[c]*((-I)*Sqrt[c]*d
+ Sqrt[a]*e)]) - (I*(Sqrt[c]*d - I*Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqr
t[c]*d - I*Sqrt[a]*e)])/(Sqrt[a]*c^(3/2)*Sqrt[I*Sqrt[c]*(I*Sqrt[c]*d + Sqrt[a]*e)])

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fricas [B]  time = 0.51, size = 1641, normalized size = 2.10

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 11
0*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(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(e*x + 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 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 1
10*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 +
5*a^2*d*e^4 + 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)))/(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(e*x + 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 +
5*a^2*d*e^4 + 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)))/(a*c^3))) + 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - 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)))/(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(e*x + 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 + 5*a^2*d*e^4 - 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)))/(a*c^3))) - 3*c*sqrt(-(c^2*d
^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - 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)))/(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(e*x + 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 + 5*a^2*d*e^4 - 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 - 2
0*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 4*(e^2*x + 7*d*e)*sqrt(e*x + d))/c

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [B]  time = 0.39, size = 3931, normalized size = 5.03 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{c x^{2} + a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.48, size = 3481, normalized size = 4.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(a + c*x^2),x)

[Out]

(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) - (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 - (1
60*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)*((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)/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 + (4*d*e*(d + e*x)^(1/2))/c

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sympy [A]  time = 118.39, size = 498, normalized size = 0.64 \begin {gather*} - \frac {4 a d 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} - \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} \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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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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