\(\int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx\) [146]

Optimal result

Integrand size = 26, antiderivative size = 192 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {3 b x^2 \sqrt {1+c^2 x^2}}{16 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{4 c^2 d}+\frac {3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^5 \sqrt {d+c^2 d x^2}} \]

[Out]

3/16*b*x^2*(c^2*x^2+1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-1/16*b*x^4*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+3/16*(
a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/b/c^5/(c^2*d*x^2+d)^(1/2)-3/8*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c
^4/d+1/4*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5812, 5783, 30} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{4 c^2 d}+\frac {3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{16 b c^5 \sqrt {c^2 d x^2+d}}-\frac {3 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{8 c^4 d}-\frac {b x^4 \sqrt {c^2 x^2+1}}{16 c \sqrt {c^2 d x^2+d}}+\frac {3 b x^2 \sqrt {c^2 x^2+1}}{16 c^3 \sqrt {c^2 d x^2+d}} \]

[In]

Int[(x^4*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]

[Out]

(3*b*x^2*Sqrt[1 + c^2*x^2])/(16*c^3*Sqrt[d + c^2*d*x^2]) - (b*x^4*Sqrt[1 + c^2*x^2])/(16*c*Sqrt[d + c^2*d*x^2]
) - (3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c^4*d) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/
(4*c^2*d) + (3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c^5*Sqrt[d + c^2*d*x^2])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5783

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

Rule 5812

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{4 c^2 d}-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx}{4 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^3 \, dx}{4 c \sqrt {d+c^2 d x^2}} \\ & = -\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{4 c^2 d}+\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx}{8 c^4}+\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{8 c^3 \sqrt {d+c^2 d x^2}} \\ & = \frac {3 b x^2 \sqrt {1+c^2 x^2}}{16 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2}}{16 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{4 c^2 d}+\frac {3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^5 \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {\frac {16 a c x \left (-3+2 c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{d}+\frac {48 a \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{\sqrt {d}}+\frac {b \sqrt {1+c^2 x^2} (16 \cosh (2 \text {arcsinh}(c x))-\cosh (4 \text {arcsinh}(c x))+4 \text {arcsinh}(c x) (6 \text {arcsinh}(c x)-8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))))}{\sqrt {d+c^2 d x^2}}}{128 c^5} \]

[In]

Integrate[(x^4*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]

[Out]

((16*a*c*x*(-3 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2])/d + (48*a*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/Sqrt[d] +
(b*Sqrt[1 + c^2*x^2]*(16*Cosh[2*ArcSinh[c*x]] - Cosh[4*ArcSinh[c*x]] + 4*ArcSinh[c*x]*(6*ArcSinh[c*x] - 8*Sin
h[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/Sqrt[d + c^2*d*x^2])/(128*c^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(518\) vs. \(2(166)=332\).

Time = 0.20 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.70

method	result	size
default	\(\frac {a \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\)	\(519\)
parts	\(\frac {a \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\)	\(519\)

[In]

int(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*a*x^3/c^2/d*(c^2*d*x^2+d)^(1/2)-3/8*a/c^4*x/d*(c^2*d*x^2+d)^(1/2)+3/8*a/c^4*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*
d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(3/16*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d*arcsinh(c*x)^2+1/256*(d*(c
^2*x^2+1))^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+
1)^(1/2))*(-1+4*arcsinh(c*x))/c^5/d/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1
/2)+2*c*x+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(c*x))/c^5/d/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2
*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(c*x))/c^5/d/(c^2*x^2+1)+1/256*(d*(c^2*x^2+1))^(1/
2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*(1+4
*arcsinh(c*x))/c^5/d/(c^2*x^2+1))

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral((b*x^4*arcsinh(c*x) + a*x^4)/sqrt(c^2*d*x^2 + d), x)

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(x**4*(a+b*asinh(c*x))/(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**4*(a + b*asinh(c*x))/sqrt(d*(c**2*x**2 + 1)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)*x^4/sqrt(c^2*d*x^2 + d), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]

[In]

int((x^4*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(1/2),x)

[Out]

int((x^4*(a + b*asinh(c*x)))/(d + c^2*d*x^2)^(1/2), x)