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

Optimal. Leaf size=254 \[ -\frac{21 a^6 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{21 a^5 c x \sqrt{c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

[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])/(10
24*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*Sqr
t[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))

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Rubi [A]  time = 0.471998, 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 \[ -\frac{21 a^6 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{3/2} \left (a+b x^2\right )^{3/2}}+\frac{21 a^5 c x \sqrt{c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac{21 a^4 c x^3 \sqrt{c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac{7}{128} a^3 c x^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]

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])/(10
24*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*Sqr
t[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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.202246, size = 135, normalized size = 0.53 \[ \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt{b} x \sqrt{a+b x^2} \left (315 a^5+4910 a^4 b x^2+11432 a^3 b^2 x^4+12144 a^2 b^3 x^6+6272 a b^4 x^8+1280 b^5 x^{10}\right )-315 a^6 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )\right )}{15360 b^{3/2} \left (a+b x^2\right )^{9/2}} \]

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[a + b*x^2]*(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^
6*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]))/(15360*b^(3/2)*(a + b*x^2)^(9/2))

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Maple [A]  time = 0.048, size = 236, normalized size = 0.9 \[{\frac{1}{15360\, \left ( b{x}^{2}+a \right ) ^{3}bc} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 1280\,{b}^{3}{x}^{7} \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}+3712\,{b}^{2}a{x}^{5} \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}+3440\,{a}^{2}{x}^{3} \left ( bc{x}^{2}+ac \right ) ^{5/2}b\sqrt{bc}+840\,{a}^{3}x \left ( bc{x}^{2}+ac \right ) ^{5/2}\sqrt{bc}-210\,{a}^{4}x \left ( bc{x}^{2}+ac \right ) ^{3/2}c\sqrt{bc}-315\,{a}^{5}{c}^{2}x\sqrt{bc{x}^{2}+ac}\sqrt{bc}-315\,{a}^{6}{c}^{3}\ln \left ({\frac{bcx+\sqrt{bc{x}^{2}+ac}\sqrt{bc}}{\sqrt{bc}}} \right ) \right ) \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.428264, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt{\frac{c}{b}} \log \left (-\frac{2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c - 2 \, \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt{\frac{c}{b}}}{b x^{2} + a}\right ) + 2 \,{\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{30720 \,{\left (b^{2} x^{2} + a b\right )}}, -\frac{315 \,{\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt{-\frac{c}{b}} \arctan \left (\frac{b c x^{3} + a c x}{\sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{-\frac{c}{b}}}\right ) -{\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{15360 \,{\left (b^{2} x^{2} + a b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^2 + a)^3*c)^(3/2)*x^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 + 12144*a^2*b^3*c*x^7 + 11
432*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((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))) - (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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.280046, size = 239, normalized size = 0.94 \[ \frac{1}{15360} \,{\left (\frac{315 \, a^{6} c{\rm ln}\left ({\left | -\sqrt{b c} x + \sqrt{b c x^{2} + a c} \right |}\right ){\rm sign}\left (b x^{2} + a\right )}{\sqrt{b c} b} +{\left (\frac{315 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{b} + 2 \,{\left (2455 \, a^{4}{\rm sign}\left (b x^{2} + a\right ) + 4 \,{\left (1429 \, a^{3} b{\rm sign}\left (b x^{2} + a\right ) + 2 \,{\left (759 \, a^{2} b^{2}{\rm sign}\left (b x^{2} + a\right ) + 8 \,{\left (10 \, b^{4} x^{2}{\rm sign}\left (b x^{2} + a\right ) + 49 \, a b^{3}{\rm sign}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b c x^{2} + a c} x\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360*(315*a^6*c*ln(abs(-sqrt(b*c)*x + sqrt(b*c*x^2 + a*c)))*sign(b*x^2 + a)/(
sqrt(b*c)*b) + (315*a^5*sign(b*x^2 + a)/b + 2*(2455*a^4*sign(b*x^2 + a) + 4*(142
9*a^3*b*sign(b*x^2 + a) + 2*(759*a^2*b^2*sign(b*x^2 + a) + 8*(10*b^4*x^2*sign(b*
x^2 + a) + 49*a*b^3*sign(b*x^2 + a))*x^2)*x^2)*x^2)*x^2)*sqrt(b*c*x^2 + a*c)*x)*
c