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

Optimal. Leaf size=596 \[ \frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}-\frac {b e^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^4 f^3 (m+4) (m+5) (m+6) (m+7)}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (f x)^{m+1} \left (\frac {c^6 d^3 (m+2) (m+4) (m+6)}{m+1}+\frac {e (m+1) \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{(m+3) (m+5) (m+7)}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{c^6 f (m+1) (m+2) (m+4) (m+6)} \]

[Out]

d^3*(f*x)^(1+m)*(a+b*arcsech(c*x))/f/(1+m)+3*d^2*e*(f*x)^(3+m)*(a+b*arcsech(c*x))/f^3/(3+m)+3*d*e^2*(f*x)^(5+m
)*(a+b*arcsech(c*x))/f^5/(5+m)+e^3*(f*x)^(7+m)*(a+b*arcsech(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))*(f*x)^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],c^2*x^2)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^
6/f/(1+m)/(2+m)/(4+m)/(6+m)-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))*(f*x)^(1+m)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^6/f/(6+m)/(m^2+6*m+8)/(m^3+15
*m^2+71*m+105)-b*e^2*(e*(5+m)^2+3*c^2*d*(m^2+13*m+42))*(f*x)^(3+m)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1
)^(1/2)/c^4/f^3/(4+m)/(5+m)/(6+m)/(7+m)-b*e^3*(f*x)^(5+m)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c
^2/f^5/(6+m)/(7+m)

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Rubi [A]  time = 2.55, antiderivative size = 576, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {270, 6301, 1809, 1267, 459, 364} \[ \frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} (f x)^{m+1} \left (\frac {e \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac {d^3}{(m+1)^2}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right )}{f}-\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^6 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}-\frac {b e^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^4 f^3 (m+4) (m+5) (m+6) (m+7)}-\frac {b e^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSech[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))*(f*x)^(1 + m)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^6*f*(2 + m)*(3 + m)*(4 + m
)*(5 + m)*(6 + m)*(7 + m))) - (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*(f*x)^(3 + m)*Sqrt[(1 + c*x)^(-
1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^4*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)) - (b*e^3*(f*x)^(5 + m)*Sqrt[(1
+ c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^2*f^5*(6 + m)*(7 + m)) + (d^3*(f*x)^(1 + m)*(a + b*ArcSech[c*
x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcSech[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*
ArcSech[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSech[c*x]))/(f^7*(7 + m)) + (b*(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)))*(f*x)^(1 + m)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x
]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/f

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 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])

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 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 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 6301

Int[((a_.) + ArcSech[(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*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)],
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), 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])) || (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps

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

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Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \operatorname {arsech}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a e^{3} f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a d e^{2} f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a d^{2} e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d^{3}}{f {\left (m + 1\right )}} + \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b e^{3} f^{m} x^{7} x^{m} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b d e^{2} f^{m} x^{5} x^{m} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b d^{2} e f^{m} x^{3} x^{m} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b d^{3} f^{m} x x^{m}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right ) - {\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b e^{3} f^{m} x^{7} x^{m} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b d e^{2} f^{m} x^{5} x^{m} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b d^{2} e f^{m} x^{3} x^{m} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b d^{3} f^{m} x x^{m}\right )} \log \relax (x)}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} - \int \frac {{\left (b c^{2} e^{3} f^{m} {\left (m + 7\right )} x^{2} \log \relax (c) - {\left (e^{3} f^{m} {\left (m + 7\right )} \log \relax (c) - e^{3} f^{m}\right )} b\right )} x^{6} x^{m}}{c^{2} {\left (m + 7\right )} x^{2} - m - 7}\,{d x} - \int \frac {3 \, {\left (b c^{2} d e^{2} f^{m} {\left (m + 5\right )} x^{2} \log \relax (c) - {\left (d e^{2} f^{m} {\left (m + 5\right )} \log \relax (c) - d e^{2} f^{m}\right )} b\right )} x^{4} x^{m}}{c^{2} {\left (m + 5\right )} x^{2} - m - 5}\,{d x} - \int \frac {3 \, {\left (b c^{2} d^{2} e f^{m} {\left (m + 3\right )} x^{2} \log \relax (c) - {\left (d^{2} e f^{m} {\left (m + 3\right )} \log \relax (c) - d^{2} e f^{m}\right )} b\right )} x^{2} x^{m}}{c^{2} {\left (m + 3\right )} x^{2} - m - 3}\,{d x} - \int \frac {{\left (b c^{2} d^{3} f^{m} {\left (m + 1\right )} x^{2} \log \relax (c) - {\left (d^{3} f^{m} {\left (m + 1\right )} \log \relax (c) - d^{3} f^{m}\right )} b\right )} x^{m}}{c^{2} {\left (m + 1\right )} x^{2} - m - 1}\,{d x} + \int \frac {{\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b c^{2} e^{3} f^{m} x^{8} x^{m} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} b c^{2} d e^{2} f^{m} x^{6} x^{m} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} b c^{2} d^{2} e f^{m} x^{4} x^{m} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b c^{2} d^{3} f^{m} x^{2} x^{m}}{{\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{2} x^{2} - m^{4} - 16 \, m^{3} + {\left ({\left (m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105\right )} c^{2} x^{2} - m^{4} - 16 \, m^{3} - 86 \, m^{2} - 176 \, m - 105\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - 86 \, m^{2} - 176 \, m - 105}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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