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

Optimal. Leaf size=32 \[ \frac{c \left (a+b x^2\right )^4 \sqrt{c \left (a+b x^2\right )^3}}{11 b} \]

[Out]

(c*(a + b*x^2)^4*Sqrt[c*(a + b*x^2)^3])/(11*b)

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Rubi [A]  time = 0.0314822, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{c \left (a+b x^2\right )^4 \sqrt{c \left (a+b x^2\right )^3}}{11 b} \]

Antiderivative was successfully verified.

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

[Out]

(c*(a + b*x^2)^4*Sqrt[c*(a + b*x^2)^3])/(11*b)

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Rubi in Sympy [A]  time = 2.68851, size = 26, normalized size = 0.81 \[ \frac{c \sqrt{c \left (a + b x^{2}\right )^{3}} \left (a + b x^{2}\right )^{4}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*sqrt(c*(a + b*x**2)**3)*(a + b*x**2)**4/(11*b)

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Mathematica [A]  time = 0.0272466, size = 29, normalized size = 0.91 \[ \frac{\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^3\right )^{3/2}}{11 b} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)*(c*(a + b*x^2)^3)^(3/2))/(11*b)

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Maple [A]  time = 0.005, size = 26, normalized size = 0.8 \[{\frac{b{x}^{2}+a}{11\,b} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*(b*x^2+a)^3)^(3/2),x)

[Out]

1/11*(b*x^2+a)/b*(c*(b*x^2+a)^3)^(3/2)

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Maxima [A]  time = 0.706534, size = 95, normalized size = 2.97 \[ \frac{{\left (b^{4} c^{\frac{3}{2}} x^{8} + 4 \, a b^{3} c^{\frac{3}{2}} x^{6} + 6 \, a^{2} b^{2} c^{\frac{3}{2}} x^{4} + 4 \, a^{3} b c^{\frac{3}{2}} x^{2} + a^{4} c^{\frac{3}{2}}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*(b^4*c^(3/2)*x^8 + 4*a*b^3*c^(3/2)*x^6 + 6*a^2*b^2*c^(3/2)*x^4 + 4*a^3*b*c^
(3/2)*x^2 + a^4*c^(3/2))*(b*x^2 + a)^(3/2)/b

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Fricas [A]  time = 0.291354, size = 117, normalized size = 3.66 \[ \frac{{\left (b^{4} c x^{8} + 4 \, a b^{3} c x^{6} + 6 \, a^{2} b^{2} c x^{4} + 4 \, a^{3} b c x^{2} + a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*(b^4*c*x^8 + 4*a*b^3*c*x^6 + 6*a^2*b^2*c*x^4 + 4*a^3*b*c*x^2 + a^4*c)*sqrt(
b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)/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*(c*(b*x**2+a)**3)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27018, size = 450, normalized size = 14.06 \[ \frac{1155 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{4}{\rm sign}\left (b x^{2} + a\right ) - \frac{924 \,{\left (5 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}}\right )} a^{3}{\rm sign}\left (b x^{2} + a\right )}{c} + \frac{198 \,{\left (35 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{2} c^{2} - 42 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a c + 15 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}}\right )} a^{2}{\rm sign}\left (b x^{2} + a\right )}{c^{2}} - \frac{44 \,{\left (105 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{3} c^{3} - 189 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a^{2} c^{2} + 135 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}} a c - 35 \,{\left (b c x^{2} + a c\right )}^{\frac{9}{2}}\right )} a{\rm sign}\left (b x^{2} + a\right )}{c^{3}} + \frac{{\left (1155 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{4} c^{4} - 2772 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a^{3} c^{3} + 2970 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}} a^{2} c^{2} - 1540 \,{\left (b c x^{2} + a c\right )}^{\frac{9}{2}} a c + 315 \,{\left (b c x^{2} + a c\right )}^{\frac{11}{2}}\right )}{\rm sign}\left (b x^{2} + a\right )}{c^{4}}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3465*(1155*(b*c*x^2 + a*c)^(3/2)*a^4*sign(b*x^2 + a) - 924*(5*(b*c*x^2 + a*c)^
(3/2)*a*c - 3*(b*c*x^2 + a*c)^(5/2))*a^3*sign(b*x^2 + a)/c + 198*(35*(b*c*x^2 +
a*c)^(3/2)*a^2*c^2 - 42*(b*c*x^2 + a*c)^(5/2)*a*c + 15*(b*c*x^2 + a*c)^(7/2))*a^
2*sign(b*x^2 + a)/c^2 - 44*(105*(b*c*x^2 + a*c)^(3/2)*a^3*c^3 - 189*(b*c*x^2 + a
*c)^(5/2)*a^2*c^2 + 135*(b*c*x^2 + a*c)^(7/2)*a*c - 35*(b*c*x^2 + a*c)^(9/2))*a*
sign(b*x^2 + a)/c^3 + (1155*(b*c*x^2 + a*c)^(3/2)*a^4*c^4 - 2772*(b*c*x^2 + a*c)
^(5/2)*a^3*c^3 + 2970*(b*c*x^2 + a*c)^(7/2)*a^2*c^2 - 1540*(b*c*x^2 + a*c)^(9/2)
*a*c + 315*(b*c*x^2 + a*c)^(11/2))*sign(b*x^2 + a)/c^4)/b