3.165 \(\int (f x)^m (d+e x^2)^3 (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=585 \[ \frac{3 d^2 e (f x)^{m+3} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{d^3 (f x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{f (m+1)}+\frac{3 d e^2 (f x)^{m+5} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (m+7)}+\frac{b x \sqrt{1-c^2 x^2} (f x)^{m+1} \left (\frac{e (m+1) \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{(m+3) (m+5) (m+7)}+\frac{c^6 d^3 (m+2) (m+4) (m+6)}{m+1}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right )}{c^5 f (m+1) (m+2) (m+4) (m+6) \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}+\frac{b e x \sqrt{c^2 x^2-1} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}+\frac{b e^2 x \sqrt{c^2 x^2-1} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}+\frac{b e^3 x \sqrt{c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt{c^2 x^2}} \]

[Out]

(b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m
^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^5*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2
*x^2]) + (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f^3*(4 + m)
*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 + c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sq
rt[c^2*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCsc[c*x]
))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCsc[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCs
c[c*x]))/(f^7*(7 + m)) + (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 +
3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/((3 + m)*(5 + m)*(7
+ m)))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^5*f*(1 + m
)*(2 + m)*(4 + m)*(6 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.40489, antiderivative size = 566, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {270, 5239, 1809, 1267, 459, 365, 364} \[ \frac{3 d^2 e (f x)^{m+3} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{d^3 (f x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{f (m+1)}+\frac{3 d e^2 (f x)^{m+5} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (m+7)}+\frac{b c x \sqrt{1-c^2 x^2} (f x)^{m+1} \left (\frac{e \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac{d^3}{(m+1)^2}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right )}{f \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}+\frac{b e x \sqrt{c^2 x^2-1} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}+\frac{b e^2 x \sqrt{c^2 x^2-1} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}+\frac{b e^3 x \sqrt{c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsc[c*x]),x]

[Out]

(b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m
^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^5*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2
*x^2]) + (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f^3*(4 + m)
*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 + c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sq
rt[c^2*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCsc[c*x]
))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCsc[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCs
c[c*x]))/(f^7*(7 + m)) + (b*c*(d^3/(1 + m)^2 + (e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m
^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/(c^6*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m))
)*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*Sqrt[c^2*x^2]*Sq
rt[-1 + c^2*x^2])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 5239

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 459

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac{(b c x) \int \frac{(f x)^m \left (\frac{d^3}{1+m}+\frac{3 d^2 e x^2}{3+m}+\frac{3 d e^2 x^4}{5+m}+\frac{e^3 x^6}{7+m}\right )}{\sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{b e^3 x (f x)^{5+m} \sqrt{-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac{(b x) \int \frac{(f x)^m \left (\frac{c^2 d^3 (6+m)}{1+m}+\frac{3 c^2 d^2 e (6+m) x^2}{3+m}+\frac{e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x^4}{(5+m) (7+m)}\right )}{\sqrt{-1+c^2 x^2}} \, dx}{c (6+m) \sqrt{c^2 x^2}}\\ &=\frac{b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt{-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{b e^3 x (f x)^{5+m} \sqrt{-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac{(b x) \int \frac{(f x)^m \left (\frac{c^4 d^3 (4+m) (6+m)}{1+m}+\frac{e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt{-1+c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt{c^2 x^2}}\\ &=\frac{b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt{-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt{-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{b e^3 x (f x)^{5+m} \sqrt{-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac{\left (b \left (\frac{c^4 d^3 (4+m) (6+m)}{1+m}+\frac{e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{c^2 (2+m) (3+m) (5+m) (7+m)}\right ) x\right ) \int \frac{(f x)^m}{\sqrt{-1+c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt{c^2 x^2}}\\ &=\frac{b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt{-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt{-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{b e^3 x (f x)^{5+m} \sqrt{-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac{\left (b \left (\frac{c^4 d^3 (4+m) (6+m)}{1+m}+\frac{e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{c^2 (2+m) (3+m) (5+m) (7+m)}\right ) x \sqrt{1-c^2 x^2}\right ) \int \frac{(f x)^m}{\sqrt{1-c^2 x^2}} \, dx}{c^3 (4+m) (6+m) \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}\\ &=\frac{b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt{-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt{-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{b e^3 x (f x)^{5+m} \sqrt{-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt{c^2 x^2}}+\frac{d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac{3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac{b \left (\frac{c^6 d^3}{(1+m)^2}+\frac{e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{(2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}\right ) x (f x)^{1+m} \sqrt{1-c^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};c^2 x^2\right )}{c^5 f \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.67206, size = 402, normalized size = 0.69 \[ x (f x)^m \left (\frac{3 a d^2 e x^2}{m+3}+\frac{a d^3}{m+1}+\frac{3 a d e^2 x^4}{m+5}+\frac{a e^3 x^6}{m+7}-\frac{3 b c d^2 e x^3 \sqrt{1-\frac{1}{c^2 x^2}} \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right )}{(m+3)^2 \sqrt{1-c^2 x^2}}-\frac{b c d^3 x \sqrt{1-\frac{1}{c^2 x^2}} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right )}{(m+1)^2 \sqrt{1-c^2 x^2}}-\frac{3 b c d e^2 x^5 \sqrt{1-\frac{1}{c^2 x^2}} \, _2F_1\left (\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};c^2 x^2\right )}{(m+5)^2 \sqrt{1-c^2 x^2}}-\frac{b c e^3 x^7 \sqrt{1-\frac{1}{c^2 x^2}} \, _2F_1\left (\frac{1}{2},\frac{m+7}{2};\frac{m+9}{2};c^2 x^2\right )}{(m+7)^2 \sqrt{1-c^2 x^2}}+\frac{3 b d^2 e x^2 \csc ^{-1}(c x)}{m+3}+\frac{b d^3 \csc ^{-1}(c x)}{m+1}+\frac{3 b d e^2 x^4 \csc ^{-1}(c x)}{m+5}+\frac{b e^3 x^6 \csc ^{-1}(c x)}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsc[c*x]),x]

[Out]

x*(f*x)^m*((a*d^3)/(1 + m) + (3*a*d^2*e*x^2)/(3 + m) + (3*a*d*e^2*x^4)/(5 + m) + (a*e^3*x^6)/(7 + m) + (b*d^3*
ArcCsc[c*x])/(1 + m) + (3*b*d^2*e*x^2*ArcCsc[c*x])/(3 + m) + (3*b*d*e^2*x^4*ArcCsc[c*x])/(5 + m) + (b*e^3*x^6*
ArcCsc[c*x])/(7 + m) - (b*c*d^3*Sqrt[1 - 1/(c^2*x^2)]*x*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])
/((1 + m)^2*Sqrt[1 - c^2*x^2]) - (3*b*c*d^2*e*Sqrt[1 - 1/(c^2*x^2)]*x^3*Hypergeometric2F1[1/2, (3 + m)/2, (5 +
m)/2, c^2*x^2])/((3 + m)^2*Sqrt[1 - c^2*x^2]) - (3*b*c*d*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^5*Hypergeometric2F1[1/2,
(5 + m)/2, (7 + m)/2, c^2*x^2])/((5 + m)^2*Sqrt[1 - c^2*x^2]) - (b*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x^7*Hypergeome
tric2F1[1/2, (7 + m)/2, (9 + m)/2, c^2*x^2])/((7 + m)^2*Sqrt[1 - c^2*x^2]))

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Maple [F]  time = 8.362, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) ^{3} \left ( a+b{\rm arccsc} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(e*x^2+d)^3*(a+b*arccsc(c*x)),x)

[Out]

int((f*x)^m*(e*x^2+d)^3*(a+b*arccsc(c*x)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} +{\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \operatorname{arccsc}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

integral((a*e^3*x^6 + 3*a*d*e^2*x^4 + 3*a*d^2*e*x^2 + a*d^3 + (b*e^3*x^6 + 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d
^3)*arccsc(c*x))*(f*x)^m, x)

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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((f*x)**m*(e*x**2+d)**3*(a+b*acsc(c*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{3}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^3*(b*arccsc(c*x) + a)*(f*x)^m, x)