\(\int x (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x)) \, dx\) [138]

Optimal result

Integrand size = 24, antiderivative size = 193 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d^2 x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d} \]

[Out]

1/7*(c^2*d*x^2+d)^(7/2)*(a+b*arcsinh(c*x))/c^2/d-1/7*b*d^2*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/7*b*c*d
^2*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-3/35*b*c^3*d^2*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/49*b*c
^5*d^2*x^7*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5798, 200} \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d}-\frac {b d^2 x \sqrt {c^2 d x^2+d}}{7 c \sqrt {c^2 x^2+1}}-\frac {b c d^2 x^3 \sqrt {c^2 d x^2+d}}{7 \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^7 \sqrt {c^2 d x^2+d}}{49 \sqrt {c^2 x^2+1}}-\frac {3 b c^3 d^2 x^5 \sqrt {c^2 d x^2+d}}{35 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

-1/7*(b*d^2*x*Sqrt[d + c^2*d*x^2])/(c*Sqrt[1 + c^2*x^2]) - (b*c*d^2*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x
^2]) - (3*b*c^3*d^2*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^7*Sqrt[d + c^2*d*x^2])/(49*
Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^2*d)

Rule 200

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

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d}-\frac {\left (b d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {1+c^2 x^2}} \\ & = \frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d}-\frac {\left (b d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {1+c^2 x^2}} \\ & = -\frac {b d^2 x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d^2 x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.58 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \sqrt {d+c^2 d x^2} \left (35 a \left (1+c^2 x^2\right )^4-b c x \sqrt {1+c^2 x^2} \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )+35 b \left (1+c^2 x^2\right )^4 \text {arcsinh}(c x)\right )}{245 c^2 \left (1+c^2 x^2\right )} \]

[In]

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

[Out]

(d^2*Sqrt[d + c^2*d*x^2]*(35*a*(1 + c^2*x^2)^4 - b*c*x*Sqrt[1 + c^2*x^2]*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6
*x^6) + 35*b*(1 + c^2*x^2)^4*ArcSinh[c*x]))/(245*c^2*(1 + c^2*x^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(862\) vs. \(2(165)=330\).

Time = 0.22 (sec) , antiderivative size = 863, normalized size of antiderivative = 4.47

method	result	size
default	\(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}+56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}+7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}-56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}-7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(863\)
parts	\(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}+56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}+7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right ) d^{2}}{128 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{640 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}-56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}-7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d^{2}}{6272 c^{2} \left (c^{2} x^{2}+1\right )}\right )\)	\(863\)

[In]

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

[Out]

1/7*a*(c^2*d*x^2+d)^(7/2)/c^2/d+b*(1/6272*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c
^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1/
2)+1)*(-1+7*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+1/640*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(
1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+5*arcsinh(c*x))*d^2/c^2
/(c^2*x^2+1)+1/128*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1
/2)+1)*(-1+3*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+5/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(
-1+arcsinh(c*x))*d^2/c^2/(c^2*x^2+1)+5/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*
x)+1)*d^2/c^2/(c^2*x^2+1)+1/128*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(
c^2*x^2+1)^(1/2)+1)*(3*arcsinh(c*x)+1)*d^2/c^2/(c^2*x^2+1)+1/640*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*
(c^2*x^2+1)^(1/2)+28*c^4*x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+1)*(1+5*arcsinh(c
*x))*d^2/c^2/(c^2*x^2+1)+1/6272*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8-64*c^7*x^7*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112
*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4-56*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^(1/2)+1)*(1+7
*arcsinh(c*x))*d^2/c^2/(c^2*x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.17 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {35 \, {\left (b c^{8} d^{2} x^{8} + 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (35 \, a c^{8} d^{2} x^{8} + 140 \, a c^{6} d^{2} x^{6} + 210 \, a c^{4} d^{2} x^{4} + 140 \, a c^{2} d^{2} x^{2} + 35 \, a d^{2} - {\left (5 \, b c^{7} d^{2} x^{7} + 21 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3} + 35 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{245 \, {\left (c^{4} x^{2} + c^{2}\right )}} \]

[In]

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

[Out]

1/245*(35*(b*c^8*d^2*x^8 + 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*sqrt(c^2*d*x^2 + d)*lo
g(c*x + sqrt(c^2*x^2 + 1)) + (35*a*c^8*d^2*x^8 + 140*a*c^6*d^2*x^6 + 210*a*c^4*d^2*x^4 + 140*a*c^2*d^2*x^2 + 3
5*a*d^2 - (5*b*c^7*d^2*x^7 + 21*b*c^5*d^2*x^5 + 35*b*c^3*d^2*x^3 + 35*b*c*d^2*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d
*x^2 + d))/(c^4*x^2 + c^2)

Sympy [F]

\[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]

[In]

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

[Out]

Integral(x*(d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.50 \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b \operatorname {arsinh}\left (c x\right )}{7 \, c^{2} d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a}{7 \, c^{2} d} - \frac {{\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} + 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} + 35 \, d^{\frac {7}{2}} x\right )} b}{245 \, c d} \]

[In]

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

[Out]

1/7*(c^2*d*x^2 + d)^(7/2)*b*arcsinh(c*x)/(c^2*d) + 1/7*(c^2*d*x^2 + d)^(7/2)*a/(c^2*d) - 1/245*(5*c^6*d^(7/2)*
x^7 + 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7/2)*x^3 + 35*d^(7/2)*x)*b/(c*d)

Giac [F(-2)]

Exception generated. \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*(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

Mupad [F(-1)]

Timed out. \[ \int x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]

[In]

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

[Out]

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