3.6.51 \(\int \frac {(d+e x)^{7/2}}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=887 \[ -\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {e \left (c^2 d^4-4 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c e^2 d^2-\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c e^2 d^2-\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

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Rubi [A]  time = 6.45, antiderivative size = 887, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {739, 825, 827, 1169, 634, 618, 206, 628} \begin {gather*} -\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {e \left (c^2 d^4-4 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\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 )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\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 )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c e^2 d^2-\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c e^2 d^2-\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(a + c*x^2)^2,x]

[Out]

-(e*(c*d^2 - 5*a*e^2)*Sqrt[d + e*x])/(2*a*c^2) - (d*e*(d + e*x)^(3/2))/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^(5/2
))/(2*a*c*(a + c*x^2)) + (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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]]])/(4*Sqrt[2]*a*c^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^2*d^4
- 4*a*c*d^2*e^2 - 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqr
t[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]]])/(4*Sqrt[2]*a*c^(9/4
)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 - Sqrt[
c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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)])/(8*Sqrt[2]*a*c^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + S
qrt[c*d^2 + a*e^2]]) + (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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)])/(8*Sqrt[2]*a*c^(9/4)*Sqrt[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 739

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

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)^{7/2}}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (2 c d^2+5 a e^2\right )-\frac {3}{2} c d e x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (c d \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) x\right )}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\frac {1}{2} c \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )+\frac {1}{2} c^2 d e \left (c d^2+13 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a c^3}\\ &=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )+\frac {1}{2} c^2 d e \left (c d^2+13 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 )}{a c^3}\\ &=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}-\left (\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )-\frac {1}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{2 \sqrt {2} a c^{13/4} \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} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}+\left (\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )-\frac {1}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{2 \sqrt {2} a c^{13/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}-\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 a c^{5/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 a c^{5/2} \sqrt {c d^2+a e^2}}\\ &=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{4 a c^{5/2} \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{4 a c^{5/2} \sqrt {c d^2+a e^2}}\\ &=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 )}{8 \sqrt {2} a c^{9/4} \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 = 0.82, size = 311, normalized size = 0.35 \begin {gather*} \frac {\frac {2 \sqrt [4]{c} \sqrt {d+e x} \left (5 a^2 e^3+a c e \left (-3 d^2-3 d e x+4 e^2 x^2\right )+c^2 d^3 x\right )}{a \left (a+c x^2\right )}+\frac {a \sqrt {\sqrt {c} d-\sqrt {-a} e} \left (\sqrt {-a} c d^2 e+8 a \sqrt {c} d e^2+5 (-a)^{3/2} e^3+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{(-a)^{5/2}}+\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \left (-\sqrt {-a} c d^2 e+8 a \sqrt {c} d e^2+5 \sqrt {-a} a e^3+2 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{(-a)^{3/2}}}{4 c^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(a + c*x^2)^2,x]

[Out]

((2*c^(1/4)*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + 4*e^2*x^2)))/(a*(a + c*x^2)) + (a
*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*c^(3/2)*d^3 + Sqrt[-a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + 5*(-a)^(3/2)*e^3)*ArcTan
h[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(-a)^(5/2) + (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(2*c^(3/2)
*d^3 - Sqrt[-a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + 5*Sqrt[-a]*a*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
+ Sqrt[-a]*e]])/(-a)^(3/2))/(4*c^(9/4))

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IntegrateAlgebraic [C]  time = 1.32, size = 444, normalized size = 0.50 \begin {gather*} \frac {e \sqrt {d+e x} \left (5 a^2 e^4+4 a c d^2 e^2-11 a c d e^2 (d+e x)+4 a c e^2 (d+e x)^2-c^2 d^4+c^2 d^3 (d+e x)\right )}{2 a c^2 \left (a e^2+c d^2-2 c d (d+e x)+c (d+e x)^2\right )}+\frac {i \left (13 i a^{3/2} \sqrt {c} d e^3-5 a^2 e^4+i \sqrt {a} c^{3/2} d^3 e+9 a c d^2 e^2+2 c^2 d^4\right ) \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 )}{4 a^{3/2} c^2 \sqrt {-i \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\right )}}-\frac {i \left (-13 i a^{3/2} \sqrt {c} d e^3-5 a^2 e^4-i \sqrt {a} c^{3/2} d^3 e+9 a c d^2 e^2+2 c^2 d^4\right ) \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 )}{4 a^{3/2} c^2 \sqrt {i \sqrt {c} \left (\sqrt {a} e+i \sqrt {c} d\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(7/2)/(a + c*x^2)^2,x]

[Out]

(e*Sqrt[d + e*x]*(-(c^2*d^4) + 4*a*c*d^2*e^2 + 5*a^2*e^4 + c^2*d^3*(d + e*x) - 11*a*c*d*e^2*(d + e*x) + 4*a*c*
e^2*(d + e*x)^2))/(2*a*c^2*(c*d^2 + a*e^2 - 2*c*d*(d + e*x) + c*(d + e*x)^2)) + ((I/4)*(2*c^2*d^4 + I*Sqrt[a]*
c^(3/2)*d^3*e + 9*a*c*d^2*e^2 + (13*I)*a^(3/2)*Sqrt[c]*d*e^3 - 5*a^2*e^4)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt
[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(a^(3/2)*c^2*Sqrt[(-I)*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)]
) - ((I/4)*(2*c^2*d^4 - I*Sqrt[a]*c^(3/2)*d^3*e + 9*a*c*d^2*e^2 - (13*I)*a^(3/2)*Sqrt[c]*d*e^3 - 5*a^2*e^4)*Ar
cTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(a^(3/2)*c^2*Sqrt[I*Sqrt[c
]*(I*Sqrt[c]*d + Sqrt[a]*e)])

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fricas [B]  time = 0.61, size = 2091, normalized size = 2.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/8*((a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*s
qrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a
^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9
+ 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*
d^2*e^8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 +
21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2
+ 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4
*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 +
35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 +
21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 +
1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*
x + d) - (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 +
4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12
+ 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*s
qrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a
^3*c^9)))/(a^3*c^4))) + (a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3
*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12
+ 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 83
66*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*
e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + 2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 107
80*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 +
35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 +
21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 + a^2*c^2)*
sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 107
80*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(1
40*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625
*a^5*e^13)*sqrt(e*x + d) - (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 +
2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 77
00*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3
*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12
+ 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - 4*(4*a*c*e^3*x^2 - 3*a*c*d^2*e + 5*a^2*e^3 + (c^2*d^3 - 3*a*c*d*e^2)
*x)*sqrt(e*x + d))/(a*c^3*x^2 + a^2*c^2)

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giac [A]  time = 0.67, size = 581, normalized size = 0.66 \begin {gather*} -\frac {{\left ({\left (c^{2} d^{3} e^{2} + 13 \, a c d e^{4}\right )} a^{2} {\left | c \right |} - {\left (\sqrt {-a c} c^{2} d^{4} e - 4 \, \sqrt {-a c} a c d^{2} e^{3} - 5 \, \sqrt {-a c} a^{2} e^{5}\right )} {\left | a \right |} {\left | c \right |} + {\left (2 \, a c^{3} d^{5} + 9 \, a^{2} c^{2} d^{3} e^{2} - 5 \, a^{3} c d e^{4}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d + \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} e - \sqrt {-a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a \right |}} + \frac {{\left ({\left (\sqrt {-a c} c d^{3} e^{2} + 13 \, \sqrt {-a c} a d e^{4}\right )} a^{2} {\left | c \right |} - {\left (a c^{2} d^{4} e - 4 \, a^{2} c d^{2} e^{3} - 5 \, a^{3} e^{5}\right )} {\left | a \right |} {\left | c \right |} + {\left (2 \, \sqrt {-a c} a c^{2} d^{5} + 9 \, \sqrt {-a c} a^{2} c d^{3} e^{2} - 5 \, \sqrt {-a c} a^{3} d e^{4}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d - \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} d - \sqrt {-a c} a^{2} c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a \right |}} + \frac {2 \, \sqrt {x e + d} e^{3}}{c^{2}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - \sqrt {x e + d} c^{2} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d e^{3} + \sqrt {x e + d} a^{2} e^{5}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )} a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+a)^2,x, algorithm="giac")

[Out]

-1/4*((c^2*d^3*e^2 + 13*a*c*d*e^4)*a^2*abs(c) - (sqrt(-a*c)*c^2*d^4*e - 4*sqrt(-a*c)*a*c*d^2*e^3 - 5*sqrt(-a*c
)*a^2*e^5)*abs(a)*abs(c) + (2*a*c^3*d^5 + 9*a^2*c^2*d^3*e^2 - 5*a^3*c*d*e^4)*abs(c))*arctan(sqrt(x*e + d)/sqrt
(-(a*c^3*d + sqrt(a^2*c^6*d^2 - (a*c^3*d^2 + a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((a^2*c^3*e - sqrt(-a*c)*a*c^3*d)*
sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(a)) + 1/4*((sqrt(-a*c)*c*d^3*e^2 + 13*sqrt(-a*c)*a*d*e^4)*a^2*abs(c) - (a*c^
2*d^4*e - 4*a^2*c*d^2*e^3 - 5*a^3*e^5)*abs(a)*abs(c) + (2*sqrt(-a*c)*a*c^2*d^5 + 9*sqrt(-a*c)*a^2*c*d^3*e^2 -
5*sqrt(-a*c)*a^3*d*e^4)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d - sqrt(a^2*c^6*d^2 - (a*c^3*d^2 + a^2*c^2*
e^2)*a*c^3))/(a*c^3)))/((a^2*c^3*d - sqrt(-a*c)*a^2*c^2*e)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(a)) + 2*sqrt(x*e
+ d)*e^3/c^2 + 1/2*((x*e + d)^(3/2)*c^2*d^3*e - sqrt(x*e + d)*c^2*d^4*e - 3*(x*e + d)^(3/2)*a*c*d*e^3 + sqrt(x
*e + d)*a^2*e^5)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + a*e^2)*a*c^2)

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maple [B]  time = 0.21, size = 5657, normalized size = 6.38 \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)^(7/2)/(c*x^2+a)^2,x)

[Out]

result too large to display

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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 {7}{2}}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + a)^2, x)

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mupad [B]  time = 0.87, size = 4192, normalized size = 4.73

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(a + c*x^2)^2,x)

[Out]

(((a^2*e^5 - c^2*d^4*e)*(d + e*x)^(1/2))/(2*a) + ((c^2*d^3*e - 3*a*c*d*e^3)*(d + e*x)^(3/2))/(2*a))/(c^3*(d +
e*x)^2 + c^3*d^2 + a*c^2*e^2 - 2*c^3*d*(d + e*x)) - atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7
/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (
77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((885*d^5*e
^9)/2 + (491*a*d^3*e^11)/(2*c) + (329*c*d^7*e^7)/(2*a) - (50*a^2*d*e^13)/c^2 + (35*c^2*d^9*e^5)/(2*a^2) + (125
*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^7) - (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(-a^9*c^9)^(
1/2))/(a^4*c^5) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^4) + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^6*c^3)) + (d^3
*e^7*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d
^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (
35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((35*a^2*c^5*d^9*e^5)/2 + (329*a^3*c^4*d^7*e^7)/2 + (88
5*a^4*c^3*d^5*e^9)/2 + (491*a^5*c^2*d^3*e^11)/2 - 50*a^6*c*d*e^13 + (125*a^2*e^14*(-a^9*c^9)^(1/2))/(4*c^4) +
(35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^2) - (204*d^4*e^10*(-a^9*c^9)^(1/2))/c^2 - (335*a*d^2*e^12*(-a^9*c^9)^(1/2)
)/(2*c^3) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a*c)) + (d^5*e^5*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64
*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a
^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/
((491*a^6*c*d^3*e^11)/2 - 50*a^7*d*e^13 + (35*a^3*c^4*d^9*e^5)/2 + (329*a^4*c^3*d^7*e^7)/2 + (885*a^5*c^2*d^5*
e^9)/2 + (125*a^3*e^14*(-a^9*c^9)^(1/2))/(4*c^5) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*c^2) - (204*a*d^4*e^10*(-a^
9*c^9)^(1/2))/c^3 + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a*c) - (335*a^2*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^4)) - (a*
d^2*e^8*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^
2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a
^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*d^7*e^7)/(2*a) + (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2)
+ (35*c*d^9*e^5)/(2*a^2) - (50*a^2*d*e^13)/c^3 + (125*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(-a^
9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^6) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^
5) + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^6*c^4)) - (c*d^4*e^6*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a
^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2
*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((329*d^7*e^7)/(2
*a) + (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) + (35*c*d^9*e^5)/(2*a^2) - (50*a^2*d*e^13)/c^3 + (125*e^1
4*(-a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^10*(-a^9*c^9)^(1/2)
)/(a^4*c^6) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^5) + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^6*c^4)) - (d*e^9*(
-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2
)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4
*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((35*a*c^6*d^9*e^5)/2 - 50*a^5*c^2*d*e^13 + (329*a^2*c^5*d^7*e
^7)/2 + (885*a^3*c^4*d^5*e^9)/2 + (491*a^4*c^3*d^3*e^11)/2 + (125*a*e^14*(-a^9*c^9)^(1/2))/(4*c^3) + (7*d^6*e^
8*(-a^9*c^9)^(1/2))/(2*a^2) - (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^2) + (35*c*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^3
) - (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a*c)))*(-(25*a^2*e^7*(-a^9*c^9)^(1/2) + 4*a^3*c^8*d^7 - 105*a^6*c^5*d*e^6
+ 35*a^4*c^7*d^5*e^2 + 70*a^5*c^6*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c^9)^(1/2) - 154*a*c*d^2*e^5*(-a^9*c^9)^(1/2
))/(64*a^6*c^9))^(1/2)*2i + atan((d^3*e^7*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3
*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e
^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((35*a^2*c^5*d^9*e
^5)/2 + (329*a^3*c^4*d^7*e^7)/2 + (885*a^4*c^3*d^5*e^9)/2 + (491*a^5*c^2*d^3*e^11)/2 - 50*a^6*c*d*e^13 - (125*
a^2*e^14*(-a^9*c^9)^(1/2))/(4*c^4) - (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^2) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/c
^2 + (335*a*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^3) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a*c)) - (a^2*e^10*(d + e*x)^(
1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-
a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*
a^6*c^7))^(1/2)*50i)/((885*d^5*e^9)/2 + (491*a*d^3*e^11)/(2*c) + (329*c*d^7*e^7)/(2*a) - (50*a^2*d*e^13)/c^2 +
(35*c^2*d^9*e^5)/(2*a^2) - (125*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^7) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c
^6) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^5) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^4) - (35*d^8*e^6*(-a^9
*c^9)^(1/2))/(4*a^6*c^3)) + (d^5*e^5*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) -
(35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-
a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((491*a^6*c*d^3*e^11)/2
- 50*a^7*d*e^13 + (35*a^3*c^4*d^9*e^5)/2 + (329*a^4*c^3*d^7*e^7)/2 + (885*a^5*c^2*d^5*e^9)/2 - (125*a^3*e^14*(
-a^9*c^9)^(1/2))/(4*c^5) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*c^2) + (204*a*d^4*e^10*(-a^9*c^9)^(1/2))/c^3 - (35*
d^8*e^6*(-a^9*c^9)^(1/2))/(4*a*c) + (335*a^2*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^4)) + (a*d^2*e^8*(d + e*x)^(1/2)*
((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c
^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c
^7))^(1/2)*308i)/((329*d^7*e^7)/(2*a) + (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) + (35*c*d^9*e^5)/(2*a^2
) - (50*a^2*d*e^13)/c^3 - (125*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^8) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^7
) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d^8*e^6*(-a^9*c
^9)^(1/2))/(4*a^6*c^4)) + (c*d^4*e^6*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32
*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(
32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((329*d^7*e^7)/(2*a) + (885*d^5*e^9)/(2*c
) + (491*a*d^3*e^11)/(2*c^2) + (35*c*d^9*e^5)/(2*a^2) - (50*a^2*d*e^13)/c^3 - (125*e^14*(-a^9*c^9)^(1/2))/(4*a
^2*c^8) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^7) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8
*(-a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^6*c^4)) - (d*e^9*(-a^9*c^9)^(1/2)*(d + e*x
)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7
*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(
64*a^6*c^7))^(1/2)*50i)/((35*a*c^6*d^9*e^5)/2 - 50*a^5*c^2*d*e^13 + (329*a^2*c^5*d^7*e^7)/2 + (885*a^3*c^4*d^5
*e^9)/2 + (491*a^4*c^3*d^3*e^11)/2 - (125*a*e^14*(-a^9*c^9)^(1/2))/(4*c^3) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a
^2) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^2) - (35*c*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^3) + (204*d^4*e^10*(-a^9*
c^9)^(1/2))/(a*c)))*(-(4*a^3*c^8*d^7 - 25*a^2*e^7*(-a^9*c^9)^(1/2) - 105*a^6*c^5*d*e^6 + 35*a^4*c^7*d^5*e^2 +
70*a^5*c^6*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c^9)^(1/2) + 154*a*c*d^2*e^5*(-a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*2
i + (2*e^3*(d + e*x)^(1/2))/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(c*x**2+a)**2,x)

[Out]

Timed out

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