3.6.63 \(\int \frac {(d+e x)^{7/2}}{(a+c x^2)^3} \, dx\)

Optimal. Leaf size=905 \[ -\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}-\frac {\left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt {d+e x}}{16 a^2 c^2 \left (c x^2+a\right )}+\frac {e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

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Rubi [A]  time = 5.64, antiderivative size = 905, 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 = {739, 819, 827, 1169, 634, 618, 206, 628} \begin {gather*} -\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}-\frac {\left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt {d+e x}}{16 a^2 c^2 \left (c x^2+a\right )}+\frac {e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (3 c d^2+4 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 )}{64 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (3 c d^2+4 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 )}{64 \sqrt {2} a^2 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)^3,x]

[Out]

-((a*e - c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a + c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(7*c*d^2 + 5*a*e^2) - 2*c*d*(3*c*
d^2 + 2*a*e^2)*x))/(16*a^2*c^2*(a + c*x^2)) + (e*(6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^
2 + a*e^2]*(6*c*d^2 + 8*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]]])/(32*Sqrt[2]*a^2*c^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*
d^2 + a*e^2]]) - (e*(6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 8*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]]])/(32*Sqrt[2]*a^2*c^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^2*d^
4 + 11*a*c*d^2*e^2 + 5*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(3*c*d^2 + 4*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)])/(64*Sqrt[2]*a^2*c^
(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4
- 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(3*c*d^2 + 4*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)])/(64*Sqrt[2]*a^2*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 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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 )^3} \, dx &=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (6 c d^2+5 a e^2\right )+\frac {1}{2} c d e x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a+c x^2\right )}+\frac {\int \frac {\frac {1}{4} \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )+\frac {1}{2} c d e \left (3 c d^2+4 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} c d^2 e \left (3 c d^2+4 a e^2\right )+\frac {1}{4} e \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )+\frac {1}{2} c d e \left (3 c d^2+4 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 )}{4 a^2 c^2}\\ &=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{16 a^2 c^2 \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 d^2 e \left (3 c d^2+4 a e^2\right )+\frac {1}{4} e \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac {1}{2} c d^2 e \left (3 c d^2+4 a e^2\right )-\frac {1}{2} \sqrt {c} d e \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right )+\frac {1}{4} e \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 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 )}{8 \sqrt {2} a^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} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {1}{2} c d^2 e \left (3 c d^2+4 a e^2\right )+\frac {1}{4} e \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac {1}{2} c d^2 e \left (3 c d^2+4 a e^2\right )-\frac {1}{2} \sqrt {c} d e \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right )+\frac {1}{4} e \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 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 )}{8 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (\frac {1}{2} c d^2 e \left (3 c d^2+4 a e^2\right )+\frac {1}{2} \sqrt {c} d e \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right )-\frac {1}{4} e \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 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 )}{16 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (-\frac {1}{2} c d^2 e \left (3 c d^2+4 a e^2\right )-\frac {1}{2} \sqrt {c} d e \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right )+\frac {1}{4} e \left (3 c d^2+a e^2\right ) \left (4 c d^2+5 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 )}{16 \sqrt {2} a^2 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 (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{64 a^2 c^{5/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{64 a^2 c^{5/2} \sqrt {c d^2+a e^2}}\\ &=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a+c x^2\right )}-\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{64 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{64 \sqrt {2} a^2 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 (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{32 a^2 c^{5/2} \sqrt {c d^2+a e^2}}-\frac {\left (e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{32 a^2 c^{5/2} \sqrt {c d^2+a e^2}}\\ &=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a+c x^2\right )}+\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{64 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 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 )}{64 \sqrt {2} a^2 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 = 1.03, size = 343, normalized size = 0.38 \begin {gather*} \frac {\frac {2 \sqrt [4]{c} \sqrt {d+e x} \left (-5 a^3 e^3-a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )+6 c^3 d^3 x^3\right )}{a^2 \left (a+c x^2\right )^2}+\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (6 \sqrt {-a} c d^2 e+13 a \sqrt {c} d e^2+5 \sqrt {-a} a e^3+12 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 {a \sqrt {\sqrt {-a} e+\sqrt {c} d} \left (-6 \sqrt {-a} c d^2 e+13 a \sqrt {c} d e^2+5 (-a)^{3/2} e^3+12 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)^{7/2}}}{32 c^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(a + c*x^2)^3,x]

[Out]

((2*c^(1/4)*Sqrt[d + e*x]*(-5*a^3*e^3 + 6*c^3*d^3*x^3 + a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) - a^2*c*e*(11*d
^2 + 4*d*e*x + 9*e^2*x^2)))/(a^2*(a + c*x^2)^2) + (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(12*c^(3/2)*d^3 + 6*Sqrt[-a]*c
*d^2*e + 13*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)^(5/2) + (a*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(12*c^(3/2)*d^3 - 6*Sqrt[-a]*c*d^2*e + 13*a*Sqrt[c]*d*e^2 + 5*(-
a)^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(-a)^(7/2))/(32*c^(9/4))

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IntegrateAlgebraic [C]  time = 3.67, size = 518, normalized size = 0.57 \begin {gather*} \frac {\left (18 \sqrt {a} \sqrt {c} d e+5 i a e^2-12 i c d^2\right ) \left (\sqrt {c} d-i \sqrt {a} e\right )^2 \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 )}{32 a^{5/2} c^2 \sqrt {i \sqrt {c} \left (\sqrt {a} e+i \sqrt {c} d\right )}}+\frac {\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \left (18 \sqrt {a} \sqrt {c} d e-5 i a e^2+12 i c d^2\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 )}{32 a^{5/2} c^2 \sqrt {-i \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\right )}}-\frac {e \sqrt {d+e x} \left (5 a^3 e^6+16 a^2 c d^2 e^4-14 a^2 c d e^4 (d+e x)+9 a^2 c e^4 (d+e x)^2+17 a c^2 d^4 e^2-32 a c^2 d^3 e^2 (d+e x)+23 a c^2 d^2 e^2 (d+e x)^2-8 a c^2 d e^2 (d+e x)^3+6 c^3 d^6-18 c^3 d^5 (d+e x)+18 c^3 d^4 (d+e x)^2-6 c^3 d^3 (d+e x)^3\right )}{16 a^2 c^2 \left (a e^2+c d^2-2 c d (d+e x)+c (d+e x)^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(7/2)/(a + c*x^2)^3,x]

[Out]

-1/16*(e*Sqrt[d + e*x]*(6*c^3*d^6 + 17*a*c^2*d^4*e^2 + 16*a^2*c*d^2*e^4 + 5*a^3*e^6 - 18*c^3*d^5*(d + e*x) - 3
2*a*c^2*d^3*e^2*(d + e*x) - 14*a^2*c*d*e^4*(d + e*x) + 18*c^3*d^4*(d + e*x)^2 + 23*a*c^2*d^2*e^2*(d + e*x)^2 +
9*a^2*c*e^4*(d + e*x)^2 - 6*c^3*d^3*(d + e*x)^3 - 8*a*c^2*d*e^2*(d + e*x)^3))/(a^2*c^2*(c*d^2 + a*e^2 - 2*c*d
*(d + e*x) + c*(d + e*x)^2)^2) + ((Sqrt[c]*d + I*Sqrt[a]*e)^2*((12*I)*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e - (5*I)*a
*e^2)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(32*a^(5/2)*c^2*Sq
rt[(-I)*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)]) + ((Sqrt[c]*d - I*Sqrt[a]*e)^2*((-12*I)*c*d^2 + 18*Sqrt[a]*Sqrt
[c]*d*e + (5*I)*a*e^2)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(
32*a^(5/2)*c^2*Sqrt[I*Sqrt[c]*(I*Sqrt[c]*d + Sqrt[a]*e)])

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fricas [B]  time = 0.53, size = 1751, normalized size = 1.93

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)^3,x, algorithm="fricas")

[Out]

1/64*((a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105
*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((3
024*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(e*x +
d) + (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 + 13*a^6*c^7*d*e^2)*sqrt
(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 3
85*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^
9)))/(a^5*c^4))) - (a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*
d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5
*c^4))*log((3024*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^3*c*d^2*e^11 + 625*a^4*e^1
3)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 + 13*a^6*c^
7*d*e^2)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt(-(144*c^3*d^7 + 420*a*c^
2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*
e^14)/(a^5*c^9)))/(a^5*c^4))) + (a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2
+ 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^
5*c^9)))/(a^5*c^4))*log((3024*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^3*c*d^2*e^11
+ 625*a^4*e^13)*sqrt(e*x + d) + (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^
3 + 13*a^6*c^7*d*e^2)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt(-(144*c^3*d
^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^
12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))) - (a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*
a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*
a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((3024*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^
3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (
12*a^5*c^8*d^3 + 13*a^6*c^7*d*e^2)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqr
t(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 10
50*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))) - 4*(11*a^2*c*d^2*e + 5*a^3*e^3 - 2*(3*c^3*d^3 + 4*a*c
^2*d*e^2)*x^3 - (a*c^2*d^2*e - 9*a^2*c*e^3)*x^2 - 2*(5*a*c^2*d^3 - 2*a^2*c*d*e^2)*x)*sqrt(e*x + d))/(a^2*c^4*x
^4 + 2*a^3*c^3*x^2 + a^4*c^2)

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giac [A]  time = 0.56, size = 550, normalized size = 0.61 \begin {gather*} -\frac {{\left (12 \, c^{2} d^{3} - 6 \, \sqrt {-a c} c d^{2} e + 13 \, a c d e^{2} - 5 \, \sqrt {-a c} a e^{3}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, \sqrt {-a c} a^{2} c^{4}} + \frac {{\left (12 \, c^{2} d^{3} + 6 \, \sqrt {-a c} c d^{2} e + 13 \, a c d e^{2} + 5 \, \sqrt {-a c} a e^{3}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, \sqrt {-a c} a^{2} c^{4}} + \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {x e + d} c^{3} d^{6} e + 8 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} - 23 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} + 32 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} - 17 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} - 9 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 14 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 16 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} - 5 \, \sqrt {x e + d} a^{3} e^{7}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}^{2} a^{2} 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)^3,x, algorithm="giac")

[Out]

-1/32*(12*c^2*d^3 - 6*sqrt(-a*c)*c*d^2*e + 13*a*c*d*e^2 - 5*sqrt(-a*c)*a*e^3)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*ab
s(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d + sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 + a^3*c^2*e^2)*a^2*c^3))/(a^2*c^3
)))/(sqrt(-a*c)*a^2*c^4) + 1/32*(12*c^2*d^3 + 6*sqrt(-a*c)*c*d^2*e + 13*a*c*d*e^2 + 5*sqrt(-a*c)*a*e^3)*sqrt(-
c^2*d + sqrt(-a*c)*c*e)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d - sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 + a^3*c
^2*e^2)*a^2*c^3))/(a^2*c^3)))/(sqrt(-a*c)*a^2*c^4) + 1/16*(6*(x*e + d)^(7/2)*c^3*d^3*e - 18*(x*e + d)^(5/2)*c^
3*d^4*e + 18*(x*e + d)^(3/2)*c^3*d^5*e - 6*sqrt(x*e + d)*c^3*d^6*e + 8*(x*e + d)^(7/2)*a*c^2*d*e^3 - 23*(x*e +
d)^(5/2)*a*c^2*d^2*e^3 + 32*(x*e + d)^(3/2)*a*c^2*d^3*e^3 - 17*sqrt(x*e + d)*a*c^2*d^4*e^3 - 9*(x*e + d)^(5/2
)*a^2*c*e^5 + 14*(x*e + d)^(3/2)*a^2*c*d*e^5 - 16*sqrt(x*e + d)*a^2*c*d^2*e^5 - 5*sqrt(x*e + d)*a^3*e^7)/(((x*
e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + a*e^2)^2*a^2*c^2)

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maple [B]  time = 0.22, size = 5915, normalized size = 6.54 \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)^3,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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + a)^3, x)

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mupad [B]  time = 1.07, size = 2569, normalized size = 2.84

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)^3,x)

[Out]

((e*(3*c*d^3 + 4*a*d*e^2)*(d + e*x)^(7/2))/(8*a^2) + ((d + e*x)^(3/2)*(7*a^2*d*e^5 + 9*c^2*d^5*e + 16*a*c*d^3*
e^3))/(8*a^2*c) - ((d + e*x)^(1/2)*(5*a^3*e^7 + 6*c^3*d^6*e + 17*a*c^2*d^4*e^3 + 16*a^2*c*d^2*e^5))/(16*a^2*c^
2) - (e*(d + e*x)^(5/2)*(9*a^2*e^4 + 18*c^2*d^4 + 23*a*c*d^2*e^2))/(16*a^2*c))/(c^2*(d + e*x)^4 + a^2*e^4 + c^
2*d^4 + (6*c^2*d^2 + 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 + 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 + 2*a*
c*d^2*e^2) - 2*atanh((25*e^10*(d + e*x)^(1/2)*(- (9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) - (385*d^3*e
^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(-a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(-
a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((825*d^5*e^9)/(2048*a^3) + (325*d*e^13)/(2048*a*c^2) + (63*c*d^7
*e^7)/(512*a^4) + (449*d^3*e^11)/(1024*a^2*c) + (125*e^14*(-a^15*c^9)^(1/2))/(2048*a^8*c^7) + (95*d^2*e^12*(-a
^15*c^9)^(1/2))/(512*a^9*c^6) + (381*d^4*e^10*(-a^15*c^9)^(1/2))/(2048*a^10*c^5) + (63*d^6*e^8*(-a^15*c^9)^(1/
2))/(1024*a^11*c^4))) + (21*d^2*e^8*(d + e*x)^(1/2)*(- (9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) - (385
*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(-a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*
e^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((325*d*e^13)/(2048*c^3) + (63*d^7*e^7)/(512*a^3) + (449*d^
3*e^11)/(1024*a*c^2) + (825*d^5*e^9)/(2048*a^2*c) + (125*e^14*(-a^15*c^9)^(1/2))/(2048*a^7*c^8) + (95*d^2*e^12
*(-a^15*c^9)^(1/2))/(512*a^8*c^7) + (381*d^4*e^10*(-a^15*c^9)^(1/2))/(2048*a^9*c^6) + (63*d^6*e^8*(-a^15*c^9)^
(1/2))/(1024*a^10*c^5))) - (25*d*e^9*(-a^15*c^9)^(1/2)*(d + e*x)^(1/2)*(- (9*d^7)/(256*a^5*c) - (105*d*e^6)/(4
096*a^2*c^4) - (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(-a^15*c^9)^(1/2))/(4096*
a^9*c^9) - (21*d^2*e^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((125*e^14*(-a^15*c^9)^(1/2))/(2048*c^3)
+ (325*a^7*c^2*d*e^13)/2048 + (63*a^4*c^5*d^7*e^7)/512 + (825*a^5*c^4*d^5*e^9)/2048 + (449*a^6*c^3*d^3*e^11)/
1024 + (63*d^6*e^8*(-a^15*c^9)^(1/2))/(1024*a^3) + (95*d^2*e^12*(-a^15*c^9)^(1/2))/(512*a*c^2) + (381*d^4*e^10
*(-a^15*c^9)^(1/2))/(2048*a^2*c))) - (21*d^3*e^7*(-a^15*c^9)^(1/2)*(d + e*x)^(1/2)*(- (9*d^7)/(256*a^5*c) - (1
05*d*e^6)/(4096*a^2*c^4) - (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(-a^15*c^9)^(
1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((63*a^5*c^4*d^7*e^7)/512 +
(825*a^6*c^3*d^5*e^9)/2048 + (449*a^7*c^2*d^3*e^11)/1024 + (125*a*e^14*(-a^15*c^9)^(1/2))/(2048*c^4) + (325*a^
8*c*d*e^13)/2048 + (95*d^2*e^12*(-a^15*c^9)^(1/2))/(512*c^3) + (381*d^4*e^10*(-a^15*c^9)^(1/2))/(2048*a*c^2) +
(63*d^6*e^8*(-a^15*c^9)^(1/2))/(1024*a^2*c))))*(-(144*a^5*c^8*d^7 + 25*a*e^7*(-a^15*c^9)^(1/2) + 105*a^8*c^5*
d*e^6 + 420*a^6*c^7*d^5*e^2 + 385*a^7*c^6*d^3*e^4 + 21*c*d^2*e^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^9))^(1/2) - 2
*atanh((25*e^10*(d + e*x)^(1/2)*((25*e^7*(-a^15*c^9)^(1/2))/(4096*a^9*c^9) - (105*d*e^6)/(4096*a^2*c^4) - (385
*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (9*d^7)/(256*a^5*c) + (21*d^2*e^5*(-a^15*c^9)^(1/2))
/(4096*a^10*c^8))^(1/2))/(32*((825*d^5*e^9)/(2048*a^3) + (325*d*e^13)/(2048*a*c^2) + (63*c*d^7*e^7)/(512*a^4)
+ (449*d^3*e^11)/(1024*a^2*c) - (125*e^14*(-a^15*c^9)^(1/2))/(2048*a^8*c^7) - (95*d^2*e^12*(-a^15*c^9)^(1/2))/
(512*a^9*c^6) - (381*d^4*e^10*(-a^15*c^9)^(1/2))/(2048*a^10*c^5) - (63*d^6*e^8*(-a^15*c^9)^(1/2))/(1024*a^11*c
^4))) + (21*d^2*e^8*(d + e*x)^(1/2)*((25*e^7*(-a^15*c^9)^(1/2))/(4096*a^9*c^9) - (105*d*e^6)/(4096*a^2*c^4) -
(385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (9*d^7)/(256*a^5*c) + (21*d^2*e^5*(-a^15*c^9)^(1
/2))/(4096*a^10*c^8))^(1/2))/(32*((325*d*e^13)/(2048*c^3) + (63*d^7*e^7)/(512*a^3) + (449*d^3*e^11)/(1024*a*c^
2) + (825*d^5*e^9)/(2048*a^2*c) - (125*e^14*(-a^15*c^9)^(1/2))/(2048*a^7*c^8) - (95*d^2*e^12*(-a^15*c^9)^(1/2)
)/(512*a^8*c^7) - (381*d^4*e^10*(-a^15*c^9)^(1/2))/(2048*a^9*c^6) - (63*d^6*e^8*(-a^15*c^9)^(1/2))/(1024*a^10*
c^5))) - (25*d*e^9*(-a^15*c^9)^(1/2)*(d + e*x)^(1/2)*((25*e^7*(-a^15*c^9)^(1/2))/(4096*a^9*c^9) - (105*d*e^6)/
(4096*a^2*c^4) - (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (9*d^7)/(256*a^5*c) + (21*d^2*e
^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((125*e^14*(-a^15*c^9)^(1/2))/(2048*c^3) - (325*a^7*c^2*d*e^
13)/2048 - (63*a^4*c^5*d^7*e^7)/512 - (825*a^5*c^4*d^5*e^9)/2048 - (449*a^6*c^3*d^3*e^11)/1024 + (63*d^6*e^8*(
-a^15*c^9)^(1/2))/(1024*a^3) + (95*d^2*e^12*(-a^15*c^9)^(1/2))/(512*a*c^2) + (381*d^4*e^10*(-a^15*c^9)^(1/2))/
(2048*a^2*c))) + (21*d^3*e^7*(-a^15*c^9)^(1/2)*(d + e*x)^(1/2)*((25*e^7*(-a^15*c^9)^(1/2))/(4096*a^9*c^9) - (1
05*d*e^6)/(4096*a^2*c^4) - (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (9*d^7)/(256*a^5*c) +
(21*d^2*e^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((63*a^5*c^4*d^7*e^7)/512 + (825*a^6*c^3*d^5*e^9)/
2048 + (449*a^7*c^2*d^3*e^11)/1024 - (125*a*e^14*(-a^15*c^9)^(1/2))/(2048*c^4) + (325*a^8*c*d*e^13)/2048 - (95
*d^2*e^12*(-a^15*c^9)^(1/2))/(512*c^3) - (381*d^4*e^10*(-a^15*c^9)^(1/2))/(2048*a*c^2) - (63*d^6*e^8*(-a^15*c^
9)^(1/2))/(1024*a^2*c))))*(-(144*a^5*c^8*d^7 - 25*a*e^7*(-a^15*c^9)^(1/2) + 105*a^8*c^5*d*e^6 + 420*a^6*c^7*d^
5*e^2 + 385*a^7*c^6*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^9))^(1/2)

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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)**3,x)

[Out]

Timed out

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