\(\int (f x)^m (d+e x^2)^3 (a+b \csc ^{-1}(c x)) \, dx\) [165]

Optimal result

Integrand size = 23, antiderivative size = 585 \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) x (f x)^{1+m} \sqrt {-1+c^2 x^2}}{c^5 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) x (f x)^{3+m} \sqrt {-1+c^2 x^2}}{c^3 f^3 (4+m) (5+m) (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {b e^3 x (f x)^{5+m} \sqrt {-1+c^2 x^2}}{c f^5 (6+m) (7+m) \sqrt {c^2 x^2}}+\frac {d^3 (f x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (7+m)}+\frac {b \left (\frac {c^6 d^3 (2+m) (4+m) (6+m)}{1+m}+\frac {e (1+m) \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{(3+m) (5+m) (7+m)}\right ) x (f x)^{1+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{c^5 f (1+m) (2+m) (4+m) (6+m) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}} \]

[Out]

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*arccsc(c*x))/f^7/(7+m)+b*(c^6*d^3*(2+m)*(4+m)*(6+m)/(1+m)+e*(
1+m)*(e^2*(m^2+8*m+15)^2+3*c^2*d*e*(3+m)^2*(m^2+13*m+42)+3*c^4*d^2*(m^4+22*m^3+179*m^2+638*m+840))/(m^3+15*m^2
+71*m+105))*x*(f*x)^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],c^2*x^2)*(-c^2*x^2+1)^(1/2)/c^5/f/(1+m)/(2+m)
/(4+m)/(6+m)/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)+b*e*(e^2*(m^2+8*m+15)^2+3*c^2*d*e*(3+m)^2*(m^2+13*m+42)+3*c^4*d
^2*(m^4+22*m^3+179*m^2+638*m+840))*x*(f*x)^(1+m)*(c^2*x^2-1)^(1/2)/c^5/f/(6+m)/(m^2+6*m+8)/(m^3+15*m^2+71*m+10
5)/(c^2*x^2)^(1/2)+b*e^2*(e*(5+m)^2+3*c^2*d*(m^2+13*m+42))*x*(f*x)^(3+m)*(c^2*x^2-1)^(1/2)/c^3/f^3/(4+m)/(5+m)
/(6+m)/(7+m)/(c^2*x^2)^(1/2)+b*e^3*x*(f*x)^(5+m)*(c^2*x^2-1)^(1/2)/c/f^5/(6+m)/(7+m)/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 1.57 (sec) , antiderivative size = 566, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {276, 5347, 1823, 1281, 470, 372, 371} \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {d^3 (f x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \csc ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \csc ^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \csc ^{-1}(c x)\right )}{f^7 (m+7)}+\frac {b e^3 x \sqrt {c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt {c^2 x^2}}+\frac {b e^2 x \sqrt {c^2 x^2-1} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}}+\frac {b c x \sqrt {1-c^2 x^2} (f x)^{m+1} \left (\frac {e \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac {d^3}{(m+1)^2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right )}{f \sqrt {c^2 x^2} \sqrt {c^2 x^2-1}}+\frac {b e x \sqrt {c^2 x^2-1} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt {c^2 x^2}} \]

[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 276

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 470

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 1281

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 1823

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 5347

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]))

Rubi steps \begin{align*} \text {integral}& = \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} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.88 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.69 \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=x (f x)^m \left (\frac {a d^3}{1+m}+\frac {3 a d^2 e x^2}{3+m}+\frac {3 a d e^2 x^4}{5+m}+\frac {a e^3 x^6}{7+m}+\frac {b d^3 \csc ^{-1}(c x)}{1+m}+\frac {3 b d^2 e x^2 \csc ^{-1}(c x)}{3+m}+\frac {3 b d e^2 x^4 \csc ^{-1}(c x)}{5+m}+\frac {b e^3 x^6 \csc ^{-1}(c x)}{7+m}-\frac {b c d^3 \sqrt {1-\frac {1}{c^2 x^2}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{(1+m)^2 \sqrt {1-c^2 x^2}}-\frac {3 b c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(3+m)^2 \sqrt {1-c^2 x^2}}-\frac {3 b c d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{2},\frac {7+m}{2},c^2 x^2\right )}{(5+m)^2 \sqrt {1-c^2 x^2}}-\frac {b c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^7 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7+m}{2},\frac {9+m}{2},c^2 x^2\right )}{(7+m)^2 \sqrt {1-c^2 x^2}}\right ) \]

[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]))

Maple [F]

\[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )d x\]

[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)

Fricas [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[In]

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

[Out]

a*e^3*f^m*x^7*x^m/(m + 7) + 3*a*d*e^2*f^m*x^5*x^m/(m + 5) + 3*a*d^2*e*f^m*x^3*x^m/(m + 3) + (f*x)^(m + 1)*a*d^
3/(f*(m + 1)) + (((b*e^3*f^m*m^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 9*b*e^3*f^m*m^2*arctan2(1, sqrt(c*x
+ 1)*sqrt(c*x - 1)) + 23*b*e^3*f^m*m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 15*b*e^3*f^m*arctan2(1, sqrt(c
*x + 1)*sqrt(c*x - 1)))*x^7 + 3*(b*d*e^2*f^m*m^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 11*b*d*e^2*f^m*m^2*
arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 31*b*d*e^2*f^m*m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 21*b*d*e^
2*f^m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x^5 + 3*(b*d^2*e*f^m*m^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)
) + 13*b*d^2*e*f^m*m^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 47*b*d^2*e*f^m*m*arctan2(1, sqrt(c*x + 1)*sqr
t(c*x - 1)) + 35*b*d^2*e*f^m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x^3 + (b*d^3*f^m*m^3*arctan2(1, sqrt(c*x
+ 1)*sqrt(c*x - 1)) + 15*b*d^3*f^m*m^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 71*b*d^3*f^m*m*arctan2(1, sq
rt(c*x + 1)*sqrt(c*x - 1)) + 105*b*d^3*f^m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x)*x^m + (m^4 + 16*m^3 + 8
6*m^2 + 176*m + 105)*integrate(-(b*d^3*f^m*m^3 + 15*b*d^3*f^m*m^2 + (b*e^3*f^m*m^3 + 9*b*e^3*f^m*m^2 + 23*b*e^
3*f^m*m + 15*b*e^3*f^m)*x^6 + 71*b*d^3*f^m*m + 105*b*d^3*f^m + 3*(b*d*e^2*f^m*m^3 + 11*b*d*e^2*f^m*m^2 + 31*b*
d*e^2*f^m*m + 21*b*d*e^2*f^m)*x^4 + 3*(b*d^2*e*f^m*m^3 + 13*b*d^2*e*f^m*m^2 + 47*b*d^2*e*f^m*m + 35*b*d^2*e*f^
m)*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1)*x^m/(m^4 + 16*m^3 - (c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 + 176*c^2*m + 105*c
^2)*x^2 + 86*m^2 + 176*m + 105), x))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

Giac [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

int((f*x)^m*(d + e*x^2)^3*(a + b*asin(1/(c*x))),x)

[Out]

int((f*x)^m*(d + e*x^2)^3*(a + b*asin(1/(c*x))), x)