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

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

[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]])

________________________________________________________________________________________

Rubi [A]  time = 5.63542, antiderivative size = 905, normalized size of antiderivative = 1., 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} \[ -\frac{(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}-\frac{\left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c^2 \left (c x^2+a\right )}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

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 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]

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 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 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 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]

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*}

Mathematica [A]  time = 1.09997, size = 343, normalized size = 0.38 \[ \frac{\frac{2 \sqrt [4]{c} \sqrt{d+e x} \left (-a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )-5 a^3 e^3+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )+6 c^3 d^3 x^3\right )}{a^2 \left (a+c x^2\right )^2}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (6 \sqrt{-a} c d^2 e+13 a \sqrt{c} d e^2+5 \sqrt{-a} a e^3+12 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{(-a)^{5/2}}+\frac{a \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (-6 \sqrt{-a} c d^2 e+13 a \sqrt{c} d e^2+5 (-a)^{3/2} e^3+12 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{(-a)^{7/2}}}{32 c^{9/4}} \]

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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Maple [B]  time = 0.247, size = 5915, normalized size = 6.5 \begin{align*} \text{output too large to display} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \end{align*}

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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Fricas [B]  time = 3.29809, size = 3903, normalized size = 4.31 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out