3.237 \(\int x^2 (c (a+b x^2)^3)^{3/2} \, dx\)

Optimal. Leaf size=253 \[ -\frac{21 a^{9/2} c \sqrt{c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{1024 b^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/2}}+\frac{21 a^5 c x \sqrt{c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

[Out]

(7*a^3*c*x^3*Sqrt[c*(a + b*x^2)^3])/128 + (21*a^5*c*x*Sqrt[c*(a + b*x^2)^3])/(1024*b*(a + b*x^2)) + (21*a^4*c*
x^3*Sqrt[c*(a + b*x^2)^3])/(512*(a + b*x^2)) + (21*a^2*c*x^3*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/320 + (3*a*c*x
^3*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/40 + (c*x^3*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3])/12 - (21*a^(9/2)*c*Sq
rt[c*(a + b*x^2)^3]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(1024*b^(3/2)*(1 + (b*x^2)/a)^(3/2))

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Rubi [A]  time = 0.246433, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6720, 279, 321, 217, 206} \[ -\frac{21 a^6 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{21 a^5 c x \sqrt{c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c*(a + b*x^2)^3)^(3/2),x]

[Out]

(7*a^3*c*x^3*Sqrt[c*(a + b*x^2)^3])/128 + (21*a^5*c*x*Sqrt[c*(a + b*x^2)^3])/(1024*b*(a + b*x^2)) + (21*a^4*c*
x^3*Sqrt[c*(a + b*x^2)^3])/(512*(a + b*x^2)) + (21*a^2*c*x^3*(a + b*x^2)*Sqrt[c*(a + b*x^2)^3])/320 + (3*a*c*x
^3*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/40 + (c*x^3*(a + b*x^2)^3*Sqrt[c*(a + b*x^2)^3])/12 - (21*a^6*c*Sqrt[c
*(a + b*x^2)^3]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(3/2)*(a + b*x^2)^(3/2))

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 279

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

Rule 321

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{9/2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (3 a c \sqrt{c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{7/2} \, dx}{4 \left (a+b x^2\right )^{3/2}}\\ &=\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (21 a^2 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{5/2} \, dx}{40 \left (a+b x^2\right )^{3/2}}\\ &=\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (21 a^3 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{3/2} \, dx}{64 \left (a+b x^2\right )^{3/2}}\\ &=\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (21 a^4 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int x^2 \sqrt{a+b x^2} \, dx}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (21 a^5 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{512 \left (a+b x^2\right )^{3/2}}\\ &=\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21 a^5 c x \sqrt{c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}-\frac{\left (21 a^6 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{1024 b \left (a+b x^2\right )^{3/2}}\\ &=\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21 a^5 c x \sqrt{c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}-\frac{\left (21 a^6 c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{1024 b \left (a+b x^2\right )^{3/2}}\\ &=\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21 a^5 c x \sqrt{c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}-\frac{21 a^6 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{3/2} \left (a+b x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.150343, size = 143, normalized size = 0.57 \[ \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt{b} x \sqrt{\frac{b x^2}{a}+1} \left (12144 a^2 b^3 x^6+11432 a^3 b^2 x^4+4910 a^4 b x^2+315 a^5+6272 a b^4 x^8+1280 b^5 x^{10}\right )-315 a^{11/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{15360 b^{3/2} \left (a+b x^2\right )^4 \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(c*(a + b*x^2)^3)^(3/2),x]

[Out]

((c*(a + b*x^2)^3)^(3/2)*(Sqrt[b]*x*Sqrt[1 + (b*x^2)/a]*(315*a^5 + 4910*a^4*b*x^2 + 11432*a^3*b^2*x^4 + 12144*
a^2*b^3*x^6 + 6272*a*b^4*x^8 + 1280*b^5*x^10) - 315*a^(11/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]]))/(15360*b^(3/2)*(a
+ b*x^2)^4*Sqrt[1 + (b*x^2)/a])

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Maple [A]  time = 0.03, size = 236, normalized size = 0.9 \begin{align*}{\frac{1}{15360\,b \left ( b{x}^{2}+a \right ) ^{3}c} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 1280\,{x}^{7} \left ( bc{x}^{2}+ac \right ) ^{5/2}{b}^{3}\sqrt{bc}+3712\,\sqrt{bc} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{5}a{b}^{2}+3440\,\sqrt{bc} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{3}{a}^{2}b+840\,\sqrt{bc} \left ( bc{x}^{2}+ac \right ) ^{5/2}x{a}^{3}-210\,\sqrt{bc} \left ( bc{x}^{2}+ac \right ) ^{3/2}x{a}^{4}c-315\,\sqrt{bc}\sqrt{bc{x}^{2}+ac}x{a}^{5}{c}^{2}-315\,\ln \left ({\frac{bcx+\sqrt{bc{x}^{2}+ac}\sqrt{bc}}{\sqrt{bc}}} \right ){a}^{6}{c}^{3} \right ) \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*(b*x^2+a)^3)^(3/2),x)

[Out]

1/15360*(c*(b*x^2+a)^3)^(3/2)/b*(1280*x^7*(b*c*x^2+a*c)^(5/2)*b^3*(b*c)^(1/2)+3712*(b*c)^(1/2)*(b*c*x^2+a*c)^(
5/2)*x^5*a*b^2+3440*(b*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*x^3*a^2*b+840*(b*c)^(1/2)*(b*c*x^2+a*c)^(5/2)*x*a^3-210*(b
*c)^(1/2)*(b*c*x^2+a*c)^(3/2)*x*a^4*c-315*(b*c)^(1/2)*(b*c*x^2+a*c)^(1/2)*x*a^5*c^2-315*ln((b*c*x+(b*c*x^2+a*c
)^(1/2)*(b*c)^(1/2))/(b*c)^(1/2))*a^6*c^3)/(b*x^2+a)^3/(c*(b*x^2+a))^(3/2)/c/(b*c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac{3}{2}} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*(b*x^2+a)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate(((b*x^2 + a)^3*c)^(3/2)*x^2, x)

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Fricas [A]  time = 1.98179, size = 969, normalized size = 3.83 \begin{align*} \left [\frac{315 \,{\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt{\frac{c}{b}} \log \left (-\frac{2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c - 2 \, \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt{\frac{c}{b}}}{b x^{2} + a}\right ) + 2 \,{\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{30720 \,{\left (b^{2} x^{2} + a b\right )}}, \frac{315 \,{\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt{-\frac{c}{b}} \arctan \left (\frac{\sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt{-\frac{c}{b}}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) +{\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{15360 \,{\left (b^{2} x^{2} + a b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*(b*x^2+a)^3)^(3/2),x, algorithm="fricas")

[Out]

[1/30720*(315*(a^6*b*c*x^2 + a^7*c)*sqrt(c/b)*log(-(2*b^2*c*x^4 + 3*a*b*c*x^2 + a^2*c - 2*sqrt(b^3*c*x^6 + 3*a
*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*b*x*sqrt(c/b))/(b*x^2 + a)) + 2*(1280*b^5*c*x^11 + 6272*a*b^4*c*x^9 + 1214
4*a^2*b^3*c*x^7 + 11432*a^3*b^2*c*x^5 + 4910*a^4*b*c*x^3 + 315*a^5*c*x)*sqrt(b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2
*b*c*x^2 + a^3*c))/(b^2*x^2 + a*b), 1/15360*(315*(a^6*b*c*x^2 + a^7*c)*sqrt(-c/b)*arctan(sqrt(b^3*c*x^6 + 3*a*
b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)*b*x*sqrt(-c/b)/(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)) + (1280*b^5*c*x^11 + 6272
*a*b^4*c*x^9 + 12144*a^2*b^3*c*x^7 + 11432*a^3*b^2*c*x^5 + 4910*a^4*b*c*x^3 + 315*a^5*c*x)*sqrt(b^3*c*x^6 + 3*
a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c))/(b^2*x^2 + a*b)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*(b*x**2+a)**3)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.30981, size = 239, normalized size = 0.94 \begin{align*} \frac{1}{15360} \,{\left (\frac{315 \, a^{6} c \log \left ({\left | -\sqrt{b c} x + \sqrt{b c x^{2} + a c} \right |}\right ) \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{b c} b} +{\left (\frac{315 \, a^{5} \mathrm{sgn}\left (b x^{2} + a\right )}{b} + 2 \,{\left (2455 \, a^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 4 \,{\left (1429 \, a^{3} b \mathrm{sgn}\left (b x^{2} + a\right ) + 2 \,{\left (759 \, a^{2} b^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 8 \,{\left (10 \, b^{4} x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 49 \, a b^{3} \mathrm{sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b c x^{2} + a c} x\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*(b*x^2+a)^3)^(3/2),x, algorithm="giac")

[Out]

1/15360*(315*a^6*c*log(abs(-sqrt(b*c)*x + sqrt(b*c*x^2 + a*c)))*sgn(b*x^2 + a)/(sqrt(b*c)*b) + (315*a^5*sgn(b*
x^2 + a)/b + 2*(2455*a^4*sgn(b*x^2 + a) + 4*(1429*a^3*b*sgn(b*x^2 + a) + 2*(759*a^2*b^2*sgn(b*x^2 + a) + 8*(10
*b^4*x^2*sgn(b*x^2 + a) + 49*a*b^3*sgn(b*x^2 + a))*x^2)*x^2)*x^2)*x^2)*sqrt(b*c*x^2 + a*c)*x)*c