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

Optimal. Leaf size=181 \[ -\frac{b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (7 b B-10 A c)}{256 c^4}+\frac{b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^{9/2}}+\frac{b \left (b x^2+c x^4\right )^{3/2} (7 b B-10 A c)}{96 c^3}-\frac{x^2 \left (b x^2+c x^4\right )^{3/2} (7 b B-10 A c)}{80 c^2}+\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c} \]

[Out]

-(b^2*(7*b*B - 10*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(256*c^4) + (b*(7*b*B - 10*A*c)*(b*x^2 + c*x^4)^(3/2
))/(96*c^3) - ((7*b*B - 10*A*c)*x^2*(b*x^2 + c*x^4)^(3/2))/(80*c^2) + (B*x^4*(b*x^2 + c*x^4)^(3/2))/(10*c) + (
b^4*(7*b*B - 10*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(256*c^(9/2))

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Rubi [A]  time = 0.334355, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2034, 794, 670, 640, 612, 620, 206} \[ -\frac{b^2 \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4} (7 b B-10 A c)}{256 c^4}+\frac{b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^{9/2}}+\frac{b \left (b x^2+c x^4\right )^{3/2} (7 b B-10 A c)}{96 c^3}-\frac{x^2 \left (b x^2+c x^4\right )^{3/2} (7 b B-10 A c)}{80 c^2}+\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]

[Out]

-(b^2*(7*b*B - 10*A*c)*(b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4])/(256*c^4) + (b*(7*b*B - 10*A*c)*(b*x^2 + c*x^4)^(3/2
))/(96*c^3) - ((7*b*B - 10*A*c)*x^2*(b*x^2 + c*x^4)^(3/2))/(80*c^2) + (B*x^4*(b*x^2 + c*x^4)^(3/2))/(10*c) + (
b^4*(7*b*B - 10*A*c)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(256*c^(9/2))

Rule 2034

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

Rule 794

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

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 \left (A+B x^2\right ) \sqrt{b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (A+B x) \sqrt{b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\left (2 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int x^2 \sqrt{b x+c x^2} \, dx,x,x^2\right )}{10 c}\\ &=-\frac{(7 b B-10 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{80 c^2}+\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c}+\frac{(b (7 b B-10 A c)) \operatorname{Subst}\left (\int x \sqrt{b x+c x^2} \, dx,x,x^2\right )}{32 c^2}\\ &=\frac{b (7 b B-10 A c) \left (b x^2+c x^4\right )^{3/2}}{96 c^3}-\frac{(7 b B-10 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{80 c^2}+\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c}-\frac{\left (b^2 (7 b B-10 A c)\right ) \operatorname{Subst}\left (\int \sqrt{b x+c x^2} \, dx,x,x^2\right )}{64 c^3}\\ &=-\frac{b^2 (7 b B-10 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{256 c^4}+\frac{b (7 b B-10 A c) \left (b x^2+c x^4\right )^{3/2}}{96 c^3}-\frac{(7 b B-10 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{80 c^2}+\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\left (b^4 (7 b B-10 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{512 c^4}\\ &=-\frac{b^2 (7 b B-10 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{256 c^4}+\frac{b (7 b B-10 A c) \left (b x^2+c x^4\right )^{3/2}}{96 c^3}-\frac{(7 b B-10 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{80 c^2}+\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c}+\frac{\left (b^4 (7 b B-10 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^4}\\ &=-\frac{b^2 (7 b B-10 A c) \left (b+2 c x^2\right ) \sqrt{b x^2+c x^4}}{256 c^4}+\frac{b (7 b B-10 A c) \left (b x^2+c x^4\right )^{3/2}}{96 c^3}-\frac{(7 b B-10 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{80 c^2}+\frac{B x^4 \left (b x^2+c x^4\right )^{3/2}}{10 c}+\frac{b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{256 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.272944, size = 173, normalized size = 0.96 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{c} x \sqrt{\frac{c x^2}{b}+1} \left (-4 b^2 c^2 x^2 \left (25 A+14 B x^2\right )+10 b^3 c \left (15 A+7 B x^2\right )+16 b c^3 x^4 \left (5 A+3 B x^2\right )+96 c^4 x^6 \left (5 A+4 B x^2\right )-105 b^4 B\right )+15 b^{7/2} (7 b B-10 A c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{3840 c^{9/2} x \sqrt{\frac{c x^2}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/b]*(-105*b^4*B + 16*b*c^3*x^4*(5*A + 3*B*x^2) + 96*c^4*x^6*
(5*A + 4*B*x^2) + 10*b^3*c*(15*A + 7*B*x^2) - 4*b^2*c^2*x^2*(25*A + 14*B*x^2)) + 15*b^(7/2)*(7*b*B - 10*A*c)*A
rcSinh[(Sqrt[c]*x)/Sqrt[b]]))/(3840*c^(9/2)*x*Sqrt[1 + (c*x^2)/b])

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Maple [A]  time = 0.008, size = 248, normalized size = 1.4 \begin{align*}{\frac{1}{3840\,x}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 384\,B{c}^{7/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{7}+480\,A{c}^{7/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{5}-336\,B{c}^{5/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{5}b-400\,A{c}^{5/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{3}b+280\,B{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{3}{b}^{2}+300\,A{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{3/2}x{b}^{2}-210\,B\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}x{b}^{3}-150\,A{c}^{3/2}\sqrt{c{x}^{2}+b}x{b}^{3}+105\,B\sqrt{c}\sqrt{c{x}^{2}+b}x{b}^{4}-150\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{4}c+105\,B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{5} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{c}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(c*x^4+b*x^2)^(1/2)*(384*B*c^(7/2)*(c*x^2+b)^(3/2)*x^7+480*A*c^(7/2)*(c*x^2+b)^(3/2)*x^5-336*B*c^(5/2)*
(c*x^2+b)^(3/2)*x^5*b-400*A*c^(5/2)*(c*x^2+b)^(3/2)*x^3*b+280*B*c^(3/2)*(c*x^2+b)^(3/2)*x^3*b^2+300*A*c^(3/2)*
(c*x^2+b)^(3/2)*x*b^2-210*B*c^(1/2)*(c*x^2+b)^(3/2)*x*b^3-150*A*c^(3/2)*(c*x^2+b)^(1/2)*x*b^3+105*B*c^(1/2)*(c
*x^2+b)^(1/2)*x*b^4-150*A*ln(x*c^(1/2)+(c*x^2+b)^(1/2))*b^4*c+105*B*ln(x*c^(1/2)+(c*x^2+b)^(1/2))*b^5)/x/(c*x^
2+b)^(1/2)/c^(9/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3207, size = 737, normalized size = 4.07 \begin{align*} \left [-\frac{15 \,{\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \,{\left (384 \, B c^{5} x^{8} + 48 \,{\left (B b c^{4} + 10 \, A c^{5}\right )} x^{6} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} - 8 \,{\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 10 \,{\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{7680 \, c^{5}}, -\frac{15 \,{\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) -{\left (384 \, B c^{5} x^{8} + 48 \,{\left (B b c^{4} + 10 \, A c^{5}\right )} x^{6} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} - 8 \,{\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 10 \,{\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3840 \, c^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(7*B*b^5 - 10*A*b^4*c)*sqrt(c)*log(-2*c*x^2 - b + 2*sqrt(c*x^4 + b*x^2)*sqrt(c)) - 2*(384*B*c^5*x
^8 + 48*(B*b*c^4 + 10*A*c^5)*x^6 - 105*B*b^4*c + 150*A*b^3*c^2 - 8*(7*B*b^2*c^3 - 10*A*b*c^4)*x^4 + 10*(7*B*b^
3*c^2 - 10*A*b^2*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^5, -1/3840*(15*(7*B*b^5 - 10*A*b^4*c)*sqrt(-c)*arctan(sqrt(c
*x^4 + b*x^2)*sqrt(-c)/(c*x^2 + b)) - (384*B*c^5*x^8 + 48*(B*b*c^4 + 10*A*c^5)*x^6 - 105*B*b^4*c + 150*A*b^3*c
^2 - 8*(7*B*b^2*c^3 - 10*A*b*c^4)*x^4 + 10*(7*B*b^3*c^2 - 10*A*b^2*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^5]

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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 )} \left (A + B x^{2}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15415, size = 285, normalized size = 1.57 \begin{align*} \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, B x^{2} \mathrm{sgn}\left (x\right ) + \frac{B b c^{7} \mathrm{sgn}\left (x\right ) + 10 \, A c^{8} \mathrm{sgn}\left (x\right )}{c^{8}}\right )} x^{2} - \frac{7 \, B b^{2} c^{6} \mathrm{sgn}\left (x\right ) - 10 \, A b c^{7} \mathrm{sgn}\left (x\right )}{c^{8}}\right )} x^{2} + \frac{5 \,{\left (7 \, B b^{3} c^{5} \mathrm{sgn}\left (x\right ) - 10 \, A b^{2} c^{6} \mathrm{sgn}\left (x\right )\right )}}{c^{8}}\right )} x^{2} - \frac{15 \,{\left (7 \, B b^{4} c^{4} \mathrm{sgn}\left (x\right ) - 10 \, A b^{3} c^{5} \mathrm{sgn}\left (x\right )\right )}}{c^{8}}\right )} \sqrt{c x^{2} + b} x - \frac{{\left (7 \, B b^{5} \mathrm{sgn}\left (x\right ) - 10 \, A b^{4} c \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + b} \right |}\right )}{256 \, c^{\frac{9}{2}}} + \frac{{\left (7 \, B b^{5} \log \left ({\left | b \right |}\right ) - 10 \, A b^{4} c \log \left ({\left | b \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{512 \, c^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(2*(4*(6*(8*B*x^2*sgn(x) + (B*b*c^7*sgn(x) + 10*A*c^8*sgn(x))/c^8)*x^2 - (7*B*b^2*c^6*sgn(x) - 10*A*b*c
^7*sgn(x))/c^8)*x^2 + 5*(7*B*b^3*c^5*sgn(x) - 10*A*b^2*c^6*sgn(x))/c^8)*x^2 - 15*(7*B*b^4*c^4*sgn(x) - 10*A*b^
3*c^5*sgn(x))/c^8)*sqrt(c*x^2 + b)*x - 1/256*(7*B*b^5*sgn(x) - 10*A*b^4*c*sgn(x))*log(abs(-sqrt(c)*x + sqrt(c*
x^2 + b)))/c^(9/2) + 1/512*(7*B*b^5*log(abs(b)) - 10*A*b^4*c*log(abs(b)))*sgn(x)/c^(9/2)