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

Optimal. Leaf size=207 \[ \frac{63 a^4 c x \sqrt{c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{63 a^{7/2} c \sqrt{c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/2}}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

[Out]

(21*a^3*c*x*Sqrt[c*(a + b*x^2)^3])/128 + (63*a^4*c*x*Sqrt[c*(a + b*x^2)^3])/(256*(a + b*x^2)) + (21*a^2*c*x*(a
+ b*x^2)*Sqrt[c*(a + b*x^2)^3])/160 + (9*a*c*x*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/80 + (c*x*(a + b*x^2)^3*S
qrt[c*(a + b*x^2)^3])/10 + (63*a^(7/2)*c*Sqrt[c*(a + b*x^2)^3]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(256*Sqrt[b]*(1 +
(b*x^2)/a)^(3/2))

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Rubi [A]  time = 0.0733558, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6720, 195, 217, 206} \[ \frac{63 a^4 c x \sqrt{c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{63 a^5 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 \sqrt{b} \left (a+b x^2\right )^{3/2}}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x^2)^3)^(3/2),x]

[Out]

(21*a^3*c*x*Sqrt[c*(a + b*x^2)^3])/128 + (63*a^4*c*x*Sqrt[c*(a + b*x^2)^3])/(256*(a + b*x^2)) + (21*a^2*c*x*(a
+ b*x^2)*Sqrt[c*(a + b*x^2)^3])/160 + (9*a*c*x*(a + b*x^2)^2*Sqrt[c*(a + b*x^2)^3])/80 + (c*x*(a + b*x^2)^3*S
qrt[c*(a + b*x^2)^3])/10 + (63*a^5*c*Sqrt[c*(a + b*x^2)^3]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*Sqrt[b]*
(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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 \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 \left (a+b x^2\right )^{9/2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (9 a c \sqrt{c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{7/2} \, dx}{10 \left (a+b x^2\right )^{3/2}}\\ &=\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (63 a^2 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{5/2} \, dx}{80 \left (a+b x^2\right )^{3/2}}\\ &=\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \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 \left (a+b x^2\right )^{3/2} \, dx}{32 \left (a+b x^2\right )^{3/2}}\\ &=\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (63 a^4 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int \sqrt{a+b x^2} \, dx}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{63 a^4 c x \sqrt{c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (63 a^5 c \sqrt{c \left (a+b x^2\right )^3}\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{256 \left (a+b x^2\right )^{3/2}}\\ &=\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{63 a^4 c x \sqrt{c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (63 a^5 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 )}{256 \left (a+b x^2\right )^{3/2}}\\ &=\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{63 a^4 c x \sqrt{c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{63 a^5 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 \sqrt{b} \left (a+b x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.1129, size = 132, normalized size = 0.64 \[ \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt{b} x \sqrt{\frac{b x^2}{a}+1} \left (1368 a^2 b^2 x^4+1490 a^3 b x^2+965 a^4+656 a b^3 x^6+128 b^4 x^8\right )+315 a^{9/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{1280 \sqrt{b} \left (a+b x^2\right )^4 \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]*(965*a^4 + 1490*a^3*b*x^2 + 1368*a^2*b^2*x^4 + 656*a*b
^3*x^6 + 128*b^4*x^8) + 315*a^(9/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]]))/(1280*Sqrt[b]*(a + b*x^2)^4*Sqrt[1 + (b*x^2
)/a])

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Maple [A]  time = 0.006, size = 205, normalized size = 1. \begin{align*}{\frac{1}{1280\,c \left ( b{x}^{2}+a \right ) ^{3}} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 128\,{x}^{5} \left ( bc{x}^{2}+ac \right ) ^{5/2}{b}^{2}\sqrt{bc}+400\, \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}{x}^{3}ab+440\, \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}x{a}^{2}+210\, \left ( bc{x}^{2}+ac \right ) ^{3/2}\sqrt{bc}x{a}^{3}c+315\,\sqrt{bc{x}^{2}+ac}\sqrt{bc}x{a}^{4}{c}^{2}+315\,\ln \left ({\frac{bcx+\sqrt{bc{x}^{2}+ac}\sqrt{bc}}{\sqrt{bc}}} \right ){a}^{5}{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((c*(b*x^2+a)^3)^(3/2),x)

[Out]

1/1280*(c*(b*x^2+a)^3)^(3/2)*(128*x^5*(b*c*x^2+a*c)^(5/2)*b^2*(b*c)^(1/2)+400*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)*
x^3*a*b+440*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)*x*a^2+210*(b*c*x^2+a*c)^(3/2)*(b*c)^(1/2)*x*a^3*c+315*(b*c*x^2+a*c
)^(1/2)*(b*c)^(1/2)*x*a^4*c^2+315*ln((b*c*x+(b*c*x^2+a*c)^(1/2)*(b*c)^(1/2))/(b*c)^(1/2))*a^5*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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((b*x^2 + a)^3*c)^(3/2), x)

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Fricas [A]  time = 1.7172, size = 887, normalized size = 4.29 \begin{align*} \left [\frac{315 \,{\left (a^{5} b c x^{2} + a^{6} 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 (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} 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}}{2560 \,{\left (b x^{2} + a\right )}}, -\frac{315 \,{\left (a^{5} b c x^{2} + a^{6} 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 (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} 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}}{1280 \,{\left (b x^{2} + a\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2560*(315*(a^5*b*c*x^2 + a^6*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*(128*b^4*c*x^9 + 656*a*b^3*c*x^7 + 1368*a^2
*b^2*c*x^5 + 1490*a^3*b*c*x^3 + 965*a^4*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*x^2 +
a), -1/1280*(315*(a^5*b*c*x^2 + a^6*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)) - (128*b^4*c*x^9 + 656*a*b^3*c*x^7 + 1368*a^2*b^2*c*x^5
+ 1490*a^3*b*c*x^3 + 965*a^4*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*x^2 + a)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23069, size = 207, normalized size = 1. \begin{align*} -\frac{1}{1280} \,{\left (\frac{315 \, a^{5} 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}} -{\left (965 \, a^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 2 \,{\left (745 \, a^{3} b \mathrm{sgn}\left (b x^{2} + a\right ) + 4 \,{\left (171 \, a^{2} b^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 2 \,{\left (8 \, b^{4} x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 41 \, a b^{3} \mathrm{sgn}\left (b x^{2} + a\right )\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((c*(b*x^2+a)^3)^(3/2),x, algorithm="giac")

[Out]

-1/1280*(315*a^5*c*log(abs(-sqrt(b*c)*x + sqrt(b*c*x^2 + a*c)))*sgn(b*x^2 + a)/sqrt(b*c) - (965*a^4*sgn(b*x^2
+ a) + 2*(745*a^3*b*sgn(b*x^2 + a) + 4*(171*a^2*b^2*sgn(b*x^2 + a) + 2*(8*b^4*x^2*sgn(b*x^2 + a) + 41*a*b^3*sg
n(b*x^2 + a))*x^2)*x^2)*x^2)*sqrt(b*c*x^2 + a*c)*x)*c