3.145 \(\int \frac {x^5 (a+b \sinh ^{-1}(c x))}{\sqrt {d+c^2 d x^2}} \, dx\)

Optimal. Leaf size=215 \[ \frac {x^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}+\frac {8 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^6 d}-\frac {4 x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^4 d}-\frac {b x^5 \sqrt {c^2 x^2+1}}{25 c \sqrt {c^2 d x^2+d}}-\frac {8 b x \sqrt {c^2 x^2+1}}{15 c^5 \sqrt {c^2 d x^2+d}}+\frac {4 b x^3 \sqrt {c^2 x^2+1}}{45 c^3 \sqrt {c^2 d x^2+d}} \]

[Out]

-8/15*b*x*(c^2*x^2+1)^(1/2)/c^5/(c^2*d*x^2+d)^(1/2)+4/45*b*x^3*(c^2*x^2+1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-1/25*
b*x^5*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+8/15*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^6/d-4/15*x^2*(a+b*
arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^4/d+1/5*x^4*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2/d

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Rubi [A]  time = 0.26, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5758, 5717, 8, 30} \[ \frac {x^4 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}-\frac {4 x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^4 d}+\frac {8 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^6 d}-\frac {b x^5 \sqrt {c^2 x^2+1}}{25 c \sqrt {c^2 d x^2+d}}+\frac {4 b x^3 \sqrt {c^2 x^2+1}}{45 c^3 \sqrt {c^2 d x^2+d}}-\frac {8 b x \sqrt {c^2 x^2+1}}{15 c^5 \sqrt {c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*x*Sqrt[1 + c^2*x^2])/(15*c^5*Sqrt[d + c^2*d*x^2]) + (4*b*x^3*Sqrt[1 + c^2*x^2])/(45*c^3*Sqrt[d + c^2*d*x
^2]) - (b*x^5*Sqrt[1 + c^2*x^2])/(25*c*Sqrt[d + c^2*d*x^2]) + (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(15
*c^6*d) - (4*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(15*c^4*d) + (x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSin
h[c*x]))/(5*c^2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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]

Rule 5758

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

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}-\frac {4 \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx}{5 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^4 \, dx}{5 c \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^5 \sqrt {1+c^2 x^2}}{25 c \sqrt {d+c^2 d x^2}}-\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}+\frac {8 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx}{15 c^4}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int x^2 \, dx}{15 c^3 \sqrt {d+c^2 d x^2}}\\ &=\frac {4 b x^3 \sqrt {1+c^2 x^2}}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^5 \sqrt {1+c^2 x^2}}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{15 c^5 \sqrt {d+c^2 d x^2}}\\ &=-\frac {8 b x \sqrt {1+c^2 x^2}}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {4 b x^3 \sqrt {1+c^2 x^2}}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^5 \sqrt {1+c^2 x^2}}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 119, normalized size = 0.55 \[ \frac {15 a \left (3 c^6 x^6-c^4 x^4+4 c^2 x^2+8\right )+b c x \sqrt {c^2 x^2+1} \left (-9 c^4 x^4+20 c^2 x^2-120\right )+15 b \left (3 c^6 x^6-c^4 x^4+4 c^2 x^2+8\right ) \sinh ^{-1}(c x)}{225 c^6 \sqrt {c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*x*Sqrt[1 + c^2*x^2]*(-120 + 20*c^2*x^2 - 9*c^4*x^4) + 15*a*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6) + 15*b*(
8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6)*ArcSinh[c*x])/(225*c^6*Sqrt[d + c^2*d*x^2])

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fricas [A]  time = 0.54, size = 161, normalized size = 0.75 \[ \frac {15 \, {\left (3 \, b c^{6} x^{6} - b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (45 \, a c^{6} x^{6} - 15 \, a c^{4} x^{4} + 60 \, a c^{2} x^{2} - {\left (9 \, b c^{5} x^{5} - 20 \, b c^{3} x^{3} + 120 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 120 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{225 \, {\left (c^{8} d x^{2} + c^{6} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/225*(15*(3*b*c^6*x^6 - b*c^4*x^4 + 4*b*c^2*x^2 + 8*b)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (45
*a*c^6*x^6 - 15*a*c^4*x^4 + 60*a*c^2*x^2 - (9*b*c^5*x^5 - 20*b*c^3*x^3 + 120*b*c*x)*sqrt(c^2*x^2 + 1) + 120*a)
*sqrt(c^2*d*x^2 + d))/(c^8*d*x^2 + c^6*d)

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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^5*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),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 [B]  time = 0.29, size = 625, normalized size = 2.91 \[ a \left (\frac {x^{4} \sqrt {c^{2} d \,x^{2}+d}}{5 c^{2} d}-\frac {4 \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )}{5 c^{2}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \arcsinh \left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}+1\right )}-\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \arcsinh \left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\arcsinh \left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+\arcsinh \left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}+1\right )}-\frac {5 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+3 \arcsinh \left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \arcsinh \left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}+1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x)

[Out]

a*(1/5*x^4/c^2/d*(c^2*d*x^2+d)^(1/2)-4/5/c^2*(1/3*x^2/c^2/d*(c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(c^2*d*x^2+d)^(1/2))
)+b*(1/800*(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))/c^6/d/(c^2*x^2+1)-5/288*(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))/c^6/d/(c^2*x^2+1
)+5/16*(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))/c^6/d/(c^2*x^2+1)+5/16*(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))/c^6/d/(c^2*x^2+1)-5/288*(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))/c^6/d/(c^2*x
^2+1)+1/800*(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))/c^6/d/(c^2*x^2+1))

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maxima [A]  time = 0.38, size = 174, normalized size = 0.81 \[ \frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} a - \frac {{\left (9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x\right )} b}{225 \, c^{5} \sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(c^2*d*x^2 + d)/(c^6*d))*b
*arcsinh(c*x) + 1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(c^2*d*x^2
+ d)/(c^6*d))*a - 1/225*(9*c^4*x^5 - 20*c^2*x^3 + 120*x)*b/(c^5*sqrt(d))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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