3.221 \(\int x^5 \sqrt{b x^2+c x^4} \, dx\)

Optimal. Leaf size=119 \[ \frac{5 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{128 c^3}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{128 c^{7/2}}-\frac{5 b \left (b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (b x^2+c x^4\right )^{3/2}}{8 c} \]

[Out]

(5*b^2*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(128*c^3) - (5*b*(b*x^2 + c*x^4)^(3/2))/(48*c^2) + (x^2*(b*x^2 + c*x
^4)^(3/2))/(8*c) - (5*b^4*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(128*c^(7/2))

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Rubi [A]  time = 0.130758, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2018, 670, 640, 612, 620, 206} \[ \frac{5 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{128 c^3}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{128 c^{7/2}}-\frac{5 b \left (b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (b x^2+c x^4\right )^{3/2}}{8 c} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[b*x^2 + c*x^4],x]

[Out]

(5*b^2*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(128*c^3) - (5*b*(b*x^2 + c*x^4)^(3/2))/(48*c^2) + (x^2*(b*x^2 + c*x
^4)^(3/2))/(8*c) - (5*b^4*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(128*c^(7/2))

Rule 2018

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^5 \sqrt{b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt{b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{x^2 \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac{(5 b) \operatorname{Subst}\left (\int x \sqrt{b x+c x^2} \, dx,x,x^2\right )}{16 c}\\ &=-\frac{5 b \left (b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (b x^2+c x^4\right )^{3/2}}{8 c}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \sqrt{b x+c x^2} \, dx,x,x^2\right )}{32 c^2}\\ &=\frac{5 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{128 c^3}-\frac{5 b \left (b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{256 c^3}\\ &=\frac{5 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{128 c^3}-\frac{5 b \left (b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{128 c^3}\\ &=\frac{5 b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{128 c^3}-\frac{5 b \left (b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac{x^2 \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{128 c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0757847, size = 114, normalized size = 0.96 \[ \frac{x \sqrt{b+c x^2} \left (\sqrt{c} x \sqrt{b+c x^2} \left (-10 b^2 c x^2+15 b^3+8 b c^2 x^4+48 c^3 x^6\right )-15 b^4 \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{384 c^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[b*x^2 + c*x^4],x]

[Out]

(x*Sqrt[b + c*x^2]*(Sqrt[c]*x*Sqrt[b + c*x^2]*(15*b^3 - 10*b^2*c*x^2 + 8*b*c^2*x^4 + 48*c^3*x^6) - 15*b^4*Log[
c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(384*c^(7/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.05, size = 124, normalized size = 1. \begin{align*}{\frac{1}{384\,x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 48\,{x}^{5} \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{5/2}-40\,{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{3}b+30\,\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}x{b}^{2}-15\,\sqrt{c}\sqrt{c{x}^{2}+b}x{b}^{3}-15\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{4} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{c}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(c*x^4+b*x^2)^(1/2),x)

[Out]

1/384*(c*x^4+b*x^2)^(1/2)*(48*x^5*(c*x^2+b)^(3/2)*c^(5/2)-40*c^(3/2)*(c*x^2+b)^(3/2)*x^3*b+30*c^(1/2)*(c*x^2+b
)^(3/2)*x*b^2-15*c^(1/2)*(c*x^2+b)^(1/2)*x*b^3-15*ln(x*c^(1/2)+(c*x^2+b)^(1/2))*b^4)/x/(c*x^2+b)^(1/2)/c^(7/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69643, size = 427, normalized size = 3.59 \begin{align*} \left [\frac{15 \, b^{4} \sqrt{c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) + 2 \,{\left (48 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} - 10 \, b^{2} c^{2} x^{2} + 15 \, b^{3} c\right )} \sqrt{c x^{4} + b x^{2}}}{768 \, c^{4}}, \frac{15 \, b^{4} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) +{\left (48 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} - 10 \, b^{2} c^{2} x^{2} + 15 \, b^{3} c\right )} \sqrt{c x^{4} + b x^{2}}}{384 \, c^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/768*(15*b^4*sqrt(c)*log(-2*c*x^2 - b + 2*sqrt(c*x^4 + b*x^2)*sqrt(c)) + 2*(48*c^4*x^6 + 8*b*c^3*x^4 - 10*b^
2*c^2*x^2 + 15*b^3*c)*sqrt(c*x^4 + b*x^2))/c^4, 1/384*(15*b^4*sqrt(-c)*arctan(sqrt(c*x^4 + b*x^2)*sqrt(-c)/(c*
x^2 + b)) + (48*c^4*x^6 + 8*b*c^3*x^4 - 10*b^2*c^2*x^2 + 15*b^3*c)*sqrt(c*x^4 + b*x^2))/c^4]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \sqrt{x^{2} \left (b + c x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(x**5*sqrt(x**2*(b + c*x**2)), x)

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Giac [A]  time = 1.32193, size = 136, normalized size = 1.14 \begin{align*} \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, x^{2} \mathrm{sgn}\left (x\right ) + \frac{b \mathrm{sgn}\left (x\right )}{c}\right )} x^{2} - \frac{5 \, b^{2} \mathrm{sgn}\left (x\right )}{c^{2}}\right )} x^{2} + \frac{15 \, b^{3} \mathrm{sgn}\left (x\right )}{c^{3}}\right )} \sqrt{c x^{2} + b} x + \frac{5 \, b^{4} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right ) \mathrm{sgn}\left (x\right )}{128 \, c^{\frac{7}{2}}} - \frac{5 \, b^{4} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{256 \, c^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*x^2*sgn(x) + b*sgn(x)/c)*x^2 - 5*b^2*sgn(x)/c^2)*x^2 + 15*b^3*sgn(x)/c^3)*sqrt(c*x^2 + b)*x + 5
/128*b^4*log(abs(-sqrt(c)*x + sqrt(c*x^2 + b)))*sgn(x)/c^(7/2) - 5/256*b^4*log(abs(b))*sgn(x)/c^(7/2)