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

Optimal. Leaf size=208 \[ \frac{63 a^5 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 \sqrt{b} \left (a+b x^2\right )^{3/2}}+\frac{63 a^4 c x \sqrt{c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

[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*Sqrt[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]])/(25
6*Sqrt[b]*(a + b*x^2)^(3/2))

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Rubi [A]  time = 0.156294, 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 \[ \frac{63 a^5 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 \sqrt{b} \left (a+b x^2\right )^{3/2}}+\frac{63 a^4 c x \sqrt{c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac{21}{128} a^3 c x \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{160} a^2 c x \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{9}{80} a c x \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{10} c x \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

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*Sqrt[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]])/(25
6*Sqrt[b]*(a + b*x^2)^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.111739, size = 124, normalized size = 0.6 \[ \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (315 a^5 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} x \sqrt{a+b x^2} \left (965 a^4+1490 a^3 b x^2+1368 a^2 b^2 x^4+656 a b^3 x^6+128 b^4 x^8\right )\right )}{1280 \sqrt{b} \left (a+b x^2\right )^{9/2}} \]

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[a + b*x^2]*(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^5*Log[b*x + Sqrt[b]*Sqrt
[a + b*x^2]]))/(1280*Sqrt[b]*(a + b*x^2)^(9/2))

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Maple [A]  time = 0.01, size = 205, normalized size = 1. \[{\frac{1}{1280\,c \left ( b{x}^{2}+a \right ) ^{3}} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 128\,{b}^{2}{x}^{5} \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}+400\,ba{x}^{3} \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}+315\,{a}^{5}{c}^{3}\ln \left ({\frac{bcx+\sqrt{bc{x}^{2}+ac}\sqrt{bc}}{\sqrt{bc}}} \right ) +440\,{a}^{2}x \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}+210\,{a}^{3}x \left ( bc{x}^{2}+ac \right ) ^{3/2}\sqrt{bc}c+315\,{a}^{4}{c}^{2}x\sqrt{bc{x}^{2}+ac}\sqrt{bc} \right ) \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bc}}}} \]

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*b^2*x^5*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)+400*b*
a*x^3*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)+315*a^5*c^3*ln((b*c*x+(b*c*x^2+a*c)^(1/2)*
(b*c)^(1/2))/(b*c)^(1/2))+440*a^2*x*(b*c*x^2+a*c)^(5/2)*(b*c)^(1/2)+210*a^3*x*(b
*c*x^2+a*c)^(3/2)*(b*c)^(1/2)*c+315*a^4*c^2*x*(b*c*x^2+a*c)^(1/2)*(b*c)^(1/2))/(
b*x^2+a)^3/(c*(b*x^2+a))^(3/2)/(b*c)^(1/2)/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.358238, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (a^{5} b c x^{2} + a^{6} c\right )} \sqrt{\frac{c}{b}} \log \left (-\frac{2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c + 2 \, \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt{\frac{c}{b}}}{b x^{2} + a}\right ) + 2 \,{\left (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} c x\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{2560 \,{\left (b x^{2} + a\right )}}, \frac{315 \,{\left (a^{5} b c x^{2} + a^{6} c\right )} \sqrt{-\frac{c}{b}} \arctan \left (\frac{b c x^{3} + a c x}{\sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{-\frac{c}{b}}}\right ) +{\left (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} c x\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{1280 \,{\left (b x^{2} + a\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^2 + a)^3*c)^(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((b*c*x^3 + a*
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)*sqrt(-c/b))) + (12
8*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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280902, size = 207, normalized size = 1. \[ -\frac{1}{1280} \,{\left (\frac{315 \, a^{5} c{\rm ln}\left ({\left | -\sqrt{b c} x + \sqrt{b c x^{2} + a c} \right |}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{b c}} -{\left (965 \, a^{4}{\rm sign}\left (b x^{2} + a\right ) + 2 \,{\left (745 \, a^{3} b{\rm sign}\left (b x^{2} + a\right ) + 4 \,{\left (171 \, a^{2} b^{2}{\rm sign}\left (b x^{2} + a\right ) + 2 \,{\left (8 \, b^{4} x^{2}{\rm sign}\left (b x^{2} + a\right ) + 41 \, a b^{3}{\rm sign}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b c x^{2} + a c} x\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1280*(315*a^5*c*ln(abs(-sqrt(b*c)*x + sqrt(b*c*x^2 + a*c)))*sign(b*x^2 + a)/s
qrt(b*c) - (965*a^4*sign(b*x^2 + a) + 2*(745*a^3*b*sign(b*x^2 + a) + 4*(171*a^2*
b^2*sign(b*x^2 + a) + 2*(8*b^4*x^2*sign(b*x^2 + a) + 41*a*b^3*sign(b*x^2 + a))*x
^2)*x^2)*x^2)*sqrt(b*c*x^2 + a*c)*x)*c