3.185 \(\int x^m (d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=313 \[ \frac {c^6 d^3 x^{m+7} \left (a+b \sinh ^{-1}(c x)\right )}{m+7}+\frac {3 c^4 d^3 x^{m+5} \left (a+b \sinh ^{-1}(c x)\right )}{m+5}+\frac {3 c^2 d^3 x^{m+3} \left (a+b \sinh ^{-1}(c x)\right )}{m+3}+\frac {d^3 x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{m+1}-\frac {3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};-c^2 x^2\right )}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}-\frac {b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt {c^2 x^2+1} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}-\frac {b c^5 d^3 \sqrt {c^2 x^2+1} x^{m+6}}{(m+7)^2}-\frac {b c^3 d^3 (m+9) (2 m+13) \sqrt {c^2 x^2+1} x^{m+4}}{(m+5)^2 (m+7)^2} \]

[Out]

d^3*x^(1+m)*(a+b*arcsinh(c*x))/(1+m)+3*c^2*d^3*x^(3+m)*(a+b*arcsinh(c*x))/(3+m)+3*c^4*d^3*x^(5+m)*(a+b*arcsinh
(c*x))/(5+m)+c^6*d^3*x^(7+m)*(a+b*arcsinh(c*x))/(7+m)-3*b*c*d^3*(35*m^3+455*m^2+1813*m+2161)*x^(2+m)*hypergeom
([1/2, 1+1/2*m],[2+1/2*m],-c^2*x^2)/(m^2+3*m+2)/(m^3+15*m^2+71*m+105)^2-b*c*d^3*(m^4+27*m^3+284*m^2+1329*m+227
1)*x^(2+m)*(c^2*x^2+1)^(1/2)/(7+m)^2/(m^2+8*m+15)^2-b*c^3*d^3*(9+m)*(13+2*m)*x^(4+m)*(c^2*x^2+1)^(1/2)/(5+m)^2
/(7+m)^2-b*c^5*d^3*x^(6+m)*(c^2*x^2+1)^(1/2)/(7+m)^2

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Rubi [A]  time = 2.17, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {270, 5730, 12, 1809, 1267, 459, 364} \[ \frac {3 c^2 d^3 x^{m+3} \left (a+b \sinh ^{-1}(c x)\right )}{m+3}+\frac {3 c^4 d^3 x^{m+5} \left (a+b \sinh ^{-1}(c x)\right )}{m+5}+\frac {c^6 d^3 x^{m+7} \left (a+b \sinh ^{-1}(c x)\right )}{m+7}+\frac {d^3 x^{m+1} \left (a+b \sinh ^{-1}(c x)\right )}{m+1}-\frac {3 b c d^3 \left (35 m^3+455 m^2+1813 m+2161\right ) x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};-c^2 x^2\right )}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}-\frac {b c d^3 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt {c^2 x^2+1} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}-\frac {b c^3 d^3 (m+9) (2 m+13) \sqrt {c^2 x^2+1} x^{m+4}}{(m+5)^2 (m+7)^2}-\frac {b c^5 d^3 \sqrt {c^2 x^2+1} x^{m+6}}{(m+7)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*x^(2 + m)*Sqrt[1 + c^2*x^2])/((3 + m)^2*(5 + m)^2*(7 + m)^
2)) - (b*c^3*d^3*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt[1 + c^2*x^2])/((5 + m)^2*(7 + m)^2) - (b*c^5*d^3*x^(6 + m)*
Sqrt[1 + c^2*x^2])/(7 + m)^2 + (d^3*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (3*c^2*d^3*x^(3 + m)*(a + b*ArcS
inh[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcSinh[c*x]))/(5 + m) + (c^6*d^3*x^(7 + m)*(a + b*ArcSinh[c*
x]))/(7 + m) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 +
m)/2, -(c^2*x^2)])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 5730

Int[((a_.) + ArcSinh[(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*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^m \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac {3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac {3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac {c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-(b c) \int \frac {d^3 x^{1+m} \left (\frac {1}{1+m}+\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}+\frac {c^6 x^6}{7+m}\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac {3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac {3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac {c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\left (b c d^3\right ) \int \frac {x^{1+m} \left (\frac {1}{1+m}+\frac {3 c^2 x^2}{3+m}+\frac {3 c^4 x^4}{5+m}+\frac {c^6 x^6}{7+m}\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {b c^5 d^3 x^{6+m} \sqrt {1+c^2 x^2}}{(7+m)^2}+\frac {d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac {3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac {3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac {c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac {\left (b d^3\right ) \int \frac {x^{1+m} \left (\frac {c^2 (7+m)}{1+m}+\frac {3 c^4 (7+m) x^2}{3+m}+\frac {c^6 (9+m) (13+2 m) x^4}{(5+m) (7+m)}\right )}{\sqrt {1+c^2 x^2}} \, dx}{c (7+m)}\\ &=-\frac {b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt {1+c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac {b c^5 d^3 x^{6+m} \sqrt {1+c^2 x^2}}{(7+m)^2}+\frac {d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac {3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac {3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac {c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac {\left (b d^3\right ) \int \frac {x^{1+m} \left (\frac {c^4 (5+m) (7+m)}{1+m}+\frac {c^6 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^2}{(3+m) (5+m) (7+m)}\right )}{\sqrt {1+c^2 x^2}} \, dx}{c^3 (5+m) (7+m)}\\ &=-\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^{2+m} \sqrt {1+c^2 x^2}}{(3+m)^2 (5+m)^2 (7+m)^2}-\frac {b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt {1+c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac {b c^5 d^3 x^{6+m} \sqrt {1+c^2 x^2}}{(7+m)^2}+\frac {d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac {3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac {3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac {c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac {\left (3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right )\right ) \int \frac {x^{1+m}}{\sqrt {1+c^2 x^2}} \, dx}{(1+m) (3+m)^2 (5+m)^2 (7+m)^2}\\ &=-\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^{2+m} \sqrt {1+c^2 x^2}}{(3+m)^2 (5+m)^2 (7+m)^2}-\frac {b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt {1+c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac {b c^5 d^3 x^{6+m} \sqrt {1+c^2 x^2}}{(7+m)^2}+\frac {d^3 x^{1+m} \left (a+b \sinh ^{-1}(c x)\right )}{1+m}+\frac {3 c^2 d^3 x^{3+m} \left (a+b \sinh ^{-1}(c x)\right )}{3+m}+\frac {3 c^4 d^3 x^{5+m} \left (a+b \sinh ^{-1}(c x)\right )}{5+m}+\frac {c^6 d^3 x^{7+m} \left (a+b \sinh ^{-1}(c x)\right )}{7+m}-\frac {3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) x^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};-c^2 x^2\right )}{(1+m) (2+m) (3+m)^2 (5+m)^2 (7+m)^2}\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 257, normalized size = 0.82 \[ \frac {x^{m+1} \left (\frac {6 d \left (\frac {4 d^2 \left ((m+2) \left (c^2 m x^2+c^2 x^2+m+3\right ) \left (a+b \sinh ^{-1}(c x)\right )-b c (m+1) x \, _2F_1\left (-\frac {1}{2},\frac {m}{2}+1;\frac {m}{2}+2;-c^2 x^2\right )-2 b c x \, _2F_1\left (\frac {1}{2},\frac {m}{2}+1;\frac {m}{2}+2;-c^2 x^2\right )\right )}{(m+1) (m+2) (m+3)}+\left (c^2 d x^2+d\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 x \, _2F_1\left (-\frac {3}{2},\frac {m}{2}+1;\frac {m}{2}+2;-c^2 x^2\right )}{m+2}\right )}{m+5}+\left (c^2 d x^2+d\right )^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^3 x \, _2F_1\left (-\frac {5}{2},\frac {m}{2}+1;\frac {m}{2}+2;-c^2 x^2\right )}{m+2}\right )}{m+7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]) - (b*c*d^3*x*Hypergeometric2F1[-5/2, 1 + m/2, 2 + m/2, -(c^
2*x^2)])/(2 + m) + (6*d*((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]) - (b*c*d^2*x*Hypergeometric2F1[-3/2, 1 + m/2,
2 + m/2, -(c^2*x^2)])/(2 + m) + (4*d^2*((2 + m)*(3 + m + c^2*x^2 + c^2*m*x^2)*(a + b*ArcSinh[c*x]) - b*c*(1 +
m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)] - 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2,
-(c^2*x^2)]))/((1 + m)*(2 + m)*(3 + m))))/(5 + m)))/(7 + m)

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{6} d^{3} x^{6} + 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} + a d^{3} + {\left (b c^{6} d^{3} x^{6} + 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} + b d^{3}\right )} \operatorname {arsinh}\left (c x\right )\right )} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x)

[Out]

int(x^m*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3,x)

[Out]

int(x^m*(a + b*asinh(c*x))*(d + c^2*d*x^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ d^{3} \left (\int a x^{m}\, dx + \int b x^{m} \operatorname {asinh}{\left (c x \right )}\, dx + \int 3 a c^{2} x^{2} x^{m}\, dx + \int 3 a c^{4} x^{4} x^{m}\, dx + \int a c^{6} x^{6} x^{m}\, dx + \int 3 b c^{2} x^{2} x^{m} \operatorname {asinh}{\left (c x \right )}\, dx + \int 3 b c^{4} x^{4} x^{m} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{6} x^{6} x^{m} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(c**2*d*x**2+d)**3*(a+b*asinh(c*x)),x)

[Out]

d**3*(Integral(a*x**m, x) + Integral(b*x**m*asinh(c*x), x) + Integral(3*a*c**2*x**2*x**m, x) + Integral(3*a*c*
*4*x**4*x**m, x) + Integral(a*c**6*x**6*x**m, x) + Integral(3*b*c**2*x**2*x**m*asinh(c*x), x) + Integral(3*b*c
**4*x**4*x**m*asinh(c*x), x) + Integral(b*c**6*x**6*x**m*asinh(c*x), x))

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