3.2.36 \(\int x^3 (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x)) \, dx\) [136]

Optimal. Leaf size=266 \[ \frac {2 b d^2 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {b d^2 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2} \]

[Out]

-1/7*(c^2*d*x^2+d)^(7/2)*(a+b*arcsinh(c*x))/c^4/d+1/9*(c^2*d*x^2+d)^(9/2)*(a+b*arcsinh(c*x))/c^4/d^2+2/63*b*d^
2*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-1/189*b*d^2*x^3*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/21*b*c
*d^2*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-19/441*b*c^3*d^2*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/81
*b*c^5*d^2*x^9*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {272, 45, 5804, 12, 380} \begin {gather*} \frac {\left (c^2 d x^2+d\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}-\frac {b c d^2 x^5 \sqrt {c^2 d x^2+d}}{21 \sqrt {c^2 x^2+1}}-\frac {b d^2 x^3 \sqrt {c^2 d x^2+d}}{189 c \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^9 \sqrt {c^2 d x^2+d}}{81 \sqrt {c^2 x^2+1}}+\frac {2 b d^2 x \sqrt {c^2 d x^2+d}}{63 c^3 \sqrt {c^2 x^2+1}}-\frac {19 b c^3 d^2 x^7 \sqrt {c^2 d x^2+d}}{441 \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(2*b*d^2*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) - (b*d^2*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c
^2*x^2]) - (b*c*d^2*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*x^7*Sqrt[d + c^2*d*x^2])/(
441*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(7/2)*(
a + b*ArcSinh[c*x]))/(7*c^4*d) + ((d + c^2*d*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4*d^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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 380

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

Rule 5804

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[
SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p -
1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps

\begin {align*} \int x^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx}{\sqrt {1+c^2 x^2}}+\left (a+b \sinh ^{-1}(c x)\right ) \int x^3 \left (d+c^2 d x^2\right )^{5/2} \, dx\\ &=-\frac {\left (b d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}}+\frac {1}{2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Subst}\left (\int x \left (d+c^2 d x\right )^{5/2} \, dx,x,x^2\right )\\ &=-\frac {\left (b d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}}+\frac {1}{2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Subst}\left (\int \left (-\frac {\left (d+c^2 d x\right )^{5/2}}{c^2}+\frac {\left (d+c^2 d x\right )^{7/2}}{c^2 d}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d^2 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {b d^2 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{9/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 140, normalized size = 0.53 \begin {gather*} \frac {d^2 \sqrt {d+c^2 d x^2} \left (63 a \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right )-b c x \sqrt {1+c^2 x^2} \left (-126+21 c^2 x^2+189 c^4 x^4+171 c^6 x^6+49 c^8 x^8\right )+63 b \left (1+c^2 x^2\right )^4 \left (-2+7 c^2 x^2\right ) \sinh ^{-1}(c x)\right )}{3969 c^4 \left (1+c^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(d^2*Sqrt[d + c^2*d*x^2]*(63*a*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) - b*c*x*Sqrt[1 + c^2*x^2]*(-126 + 21*c^2*x^2 +
189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8) + 63*b*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2)*ArcSinh[c*x]))/(3969*c^4*(1 +
c^2*x^2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(995\) vs. \(2(228)=456\).
time = 1.58, size = 996, normalized size = 3.74

method	result	size
default	\(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{9 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{63 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 x^{10} c^{10}+256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 x^{8} c^{8}+576 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+688 x^{6} c^{6}+432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}+120 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+41 c^{2} x^{2}+9 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (-1+9 \arcsinh \left (c x \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 x^{8} c^{8}+64 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+144 x^{6} c^{6}+112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}+56 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+25 c^{2} x^{2}+7 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (-1+7 \arcsinh \left (c x \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (-1+3 \arcsinh \left (c x \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (\arcsinh \left (c x \right )-1\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+\arcsinh \left (c x \right )\right ) d^{2}}{256 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+3 \arcsinh \left (c x \right )\right ) d^{2}}{576 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 x^{8} c^{8}-64 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+144 x^{6} c^{6}-112 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+104 c^{4} x^{4}-56 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+25 c^{2} x^{2}-7 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+7 \arcsinh \left (c x \right )\right ) d^{2}}{25088 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (256 x^{10} c^{10}-256 \sqrt {c^{2} x^{2}+1}\, x^{9} c^{9}+704 x^{8} c^{8}-576 \sqrt {c^{2} x^{2}+1}\, x^{7} c^{7}+688 x^{6} c^{6}-432 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+280 c^{4} x^{4}-120 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+41 c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+9 \arcsinh \left (c x \right )\right ) d^{2}}{41472 c^{4} \left (c^{2} x^{2}+1\right )}\right )\)	\(996\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

a*(1/9*x^2*(c^2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(c^2*d*x^2+d)^(7/2))+b*(1/41472*(d*(c^2*x^2+1))^(1/2)*(256*x^1
0*c^10+256*(c^2*x^2+1)^(1/2)*x^9*c^9+704*x^8*c^8+576*(c^2*x^2+1)^(1/2)*x^7*c^7+688*x^6*c^6+432*(c^2*x^2+1)^(1/
2)*x^5*c^5+280*c^4*x^4+120*(c^2*x^2+1)^(1/2)*x^3*c^3+41*c^2*x^2+9*(c^2*x^2+1)^(1/2)*c*x+1)*(-1+9*arcsinh(c*x))
*d^2/c^4/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(1/2)*(64*x^8*c^8+64*(c^2*x^2+1)^(1/2)*x^7*c^7+144*x^6*c^6+112*(c
^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4+56*(c^2*x^2+1)^(1/2)*x^3*c^3+25*c^2*x^2+7*(c^2*x^2+1)^(1/2)*c*x+1)*(-1+7*a
rcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-1/576*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2+
3*(c^2*x^2+1)^(1/2)*c*x+1)*(-1+3*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x
^2+1)^(1/2)*c*x+1)*(arcsinh(c*x)-1)*d^2/c^4/(c^2*x^2+1)-3/256*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)
*c*x+1)*(1+arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)-1/576*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*(c^2*x^2+1)^(1/2)*x^3*c^
3+5*c^2*x^2-3*(c^2*x^2+1)^(1/2)*c*x+1)*(1+3*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+3/25088*(d*(c^2*x^2+1))^(1/2)*(6
4*x^8*c^8-64*(c^2*x^2+1)^(1/2)*x^7*c^7+144*x^6*c^6-112*(c^2*x^2+1)^(1/2)*x^5*c^5+104*c^4*x^4-56*(c^2*x^2+1)^(1
/2)*x^3*c^3+25*c^2*x^2-7*(c^2*x^2+1)^(1/2)*c*x+1)*(1+7*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1)+1/41472*(d*(c^2*x^2+1
))^(1/2)*(256*x^10*c^10-256*(c^2*x^2+1)^(1/2)*x^9*c^9+704*x^8*c^8-576*(c^2*x^2+1)^(1/2)*x^7*c^7+688*x^6*c^6-43
2*(c^2*x^2+1)^(1/2)*x^5*c^5+280*c^4*x^4-120*(c^2*x^2+1)^(1/2)*x^3*c^3+41*c^2*x^2-9*(c^2*x^2+1)^(1/2)*c*x+1)*(1
+9*arcsinh(c*x))*d^2/c^4/(c^2*x^2+1))

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Maxima [A]
time = 0.30, size = 156, normalized size = 0.59 \begin {gather*} \frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{63} \, {\left (\frac {7 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} + 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} + 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} b}{3969 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

1/63*(7*(c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(7/2)/(c^4*d))*b*arcsinh(c*x) + 1/63*(7*(c^2*d*x
^2 + d)^(7/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(7/2)/(c^4*d))*a - 1/3969*(49*c^8*d^(5/2)*x^9 + 171*c^6*d^(5/2)*
x^7 + 189*c^4*d^(5/2)*x^5 + 21*c^2*d^(5/2)*x^3 - 126*d^(5/2)*x)*b/c^3

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Fricas [A]
time = 0.36, size = 263, normalized size = 0.99 \begin {gather*} \frac {63 \, {\left (7 \, b c^{10} d^{2} x^{10} + 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} + 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} - 2 \, b d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (441 \, a c^{10} d^{2} x^{10} + 1638 \, a c^{8} d^{2} x^{8} + 2142 \, a c^{6} d^{2} x^{6} + 1008 \, a c^{4} d^{2} x^{4} - 63 \, a c^{2} d^{2} x^{2} - 126 \, a d^{2} - {\left (49 \, b c^{9} d^{2} x^{9} + 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} + 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} + c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

1/3969*(63*(7*b*c^10*d^2*x^10 + 26*b*c^8*d^2*x^8 + 34*b*c^6*d^2*x^6 + 16*b*c^4*d^2*x^4 - b*c^2*d^2*x^2 - 2*b*d
^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (441*a*c^10*d^2*x^10 + 1638*a*c^8*d^2*x^8 + 2142*a*c^6*
d^2*x^6 + 1008*a*c^4*d^2*x^4 - 63*a*c^2*d^2*x^2 - 126*a*d^2 - (49*b*c^9*d^2*x^9 + 171*b*c^7*d^2*x^7 + 189*b*c^
5*d^2*x^5 + 21*b*c^3*d^2*x^3 - 126*b*c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3060 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2),x)

[Out]

int(x^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2), x)

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