3.3.31 \(\int x^2 (c (a+b x^2)^2)^{3/2} \, dx\) [231]

Optimal. Leaf size=143 \[ \frac {a^3 c x^3 \sqrt {c \left (a+b x^2\right )^2}}{3 \left (a+b x^2\right )}+\frac {3 a^2 b c x^5 \sqrt {c \left (a+b x^2\right )^2}}{5 \left (a+b x^2\right )}+\frac {3 a b^2 c x^7 \sqrt {c \left (a+b x^2\right )^2}}{7 \left (a+b x^2\right )}+\frac {b^3 c x^9 \sqrt {c \left (a+b x^2\right )^2}}{9 \left (a+b x^2\right )} \]

[Out]

1/3*a^3*c*x^3*(c*(b*x^2+a)^2)^(1/2)/(b*x^2+a)+3/5*a^2*b*c*x^5*(c*(b*x^2+a)^2)^(1/2)/(b*x^2+a)+3/7*a*b^2*c*x^7*
(c*(b*x^2+a)^2)^(1/2)/(b*x^2+a)+1/9*b^3*c*x^9*(c*(b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Rubi [A]
time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1973, 276} \begin {gather*} \frac {a^3 c x^3 \sqrt {c \left (a+b x^2\right )^2}}{3 \left (a+b x^2\right )}+\frac {3 a^2 b c x^5 \sqrt {c \left (a+b x^2\right )^2}}{5 \left (a+b x^2\right )}+\frac {b^3 c x^9 \sqrt {c \left (a+b x^2\right )^2}}{9 \left (a+b x^2\right )}+\frac {3 a b^2 c x^7 \sqrt {c \left (a+b x^2\right )^2}}{7 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(c*(a + b*x^2)^2)^(3/2),x]

[Out]

(a^3*c*x^3*Sqrt[c*(a + b*x^2)^2])/(3*(a + b*x^2)) + (3*a^2*b*c*x^5*Sqrt[c*(a + b*x^2)^2])/(5*(a + b*x^2)) + (3
*a*b^2*c*x^7*Sqrt[c*(a + b*x^2)^2])/(7*(a + b*x^2)) + (b^3*c*x^9*Sqrt[c*(a + b*x^2)^2])/(9*(a + b*x^2))

Rule 276

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

Rule 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]

Rubi steps

\begin {align*} \int x^2 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx &=\int x^2 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2} \, dx\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \int x^2 \left (a b c+b^2 c x^2\right )^3 \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \int \left (a^3 b^3 c^3 x^2+3 a^2 b^4 c^3 x^4+3 a b^5 c^3 x^6+b^6 c^3 x^8\right ) \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {a^3 c x^3 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{3 \left (a+b x^2\right )}+\frac {3 a^2 b c x^5 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{5 \left (a+b x^2\right )}+\frac {3 a b^2 c x^7 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{7 \left (a+b x^2\right )}+\frac {b^3 c x^9 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{9 \left (a+b x^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 63, normalized size = 0.44 \begin {gather*} \frac {\left (c \left (a+b x^2\right )^2\right )^{3/2} \left (105 a^3 x^3+189 a^2 b x^5+135 a b^2 x^7+35 b^3 x^9\right )}{315 \left (a+b x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*(a + b*x^2)^2)^(3/2)*(105*a^3*x^3 + 189*a^2*b*x^5 + 135*a*b^2*x^7 + 35*b^3*x^9))/(315*(a + b*x^2)^3)

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Maple [A]
time = 0.06, size = 60, normalized size = 0.42

method	result	size
gosper	\(\frac {x^{3} \left (35 b^{3} x^{6}+135 a \,b^{2} x^{4}+189 a^{2} b \,x^{2}+105 a^{3}\right ) \left (c \left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{315 \left (b \,x^{2}+a \right )^{3}}\)	\(60\)
default	\(\frac {x^{3} \left (35 b^{3} x^{6}+135 a \,b^{2} x^{4}+189 a^{2} b \,x^{2}+105 a^{3}\right ) \left (c \left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{315 \left (b \,x^{2}+a \right )^{3}}\)	\(60\)
trager	\(\frac {c \,x^{3} \left (35 b^{3} x^{6}+135 a \,b^{2} x^{4}+189 a^{2} b \,x^{2}+105 a^{3}\right ) \sqrt {b^{2} c \,x^{4}+2 a b c \,x^{2}+a^{2} c}}{315 b \,x^{2}+315 a}\)	\(72\)
risch	\(\frac {a^{3} c \,x^{3} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{3 b \,x^{2}+3 a}+\frac {3 a^{2} b c \,x^{5} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{5 \left (b \,x^{2}+a \right )}+\frac {3 a \,b^{2} c \,x^{7} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{7 \left (b \,x^{2}+a \right )}+\frac {b^{3} c \,x^{9} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{9 b \,x^{2}+9 a}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*x^3*(35*b^3*x^6+135*a*b^2*x^4+189*a^2*b*x^2+105*a^3)*(c*(b*x^2+a)^2)^(3/2)/(b*x^2+a)^3

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Maxima [A]
time = 0.28, size = 47, normalized size = 0.33 \begin {gather*} \frac {1}{9} \, b^{3} c^{\frac {3}{2}} x^{9} + \frac {3}{7} \, a b^{2} c^{\frac {3}{2}} x^{7} + \frac {3}{5} \, a^{2} b c^{\frac {3}{2}} x^{5} + \frac {1}{3} \, a^{3} c^{\frac {3}{2}} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^3*c^(3/2)*x^9 + 3/7*a*b^2*c^(3/2)*x^7 + 3/5*a^2*b*c^(3/2)*x^5 + 1/3*a^3*c^(3/2)*x^3

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Fricas [A]
time = 0.35, size = 74, normalized size = 0.52 \begin {gather*} \frac {{\left (35 \, b^{3} c x^{9} + 135 \, a b^{2} c x^{7} + 189 \, a^{2} b c x^{5} + 105 \, a^{3} c x^{3}\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{315 \, {\left (b x^{2} + a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(35*b^3*c*x^9 + 135*a*b^2*c*x^7 + 189*a^2*b*c*x^5 + 105*a^3*c*x^3)*sqrt(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)
/(b*x^2 + a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (c \left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 5.08, size = 72, normalized size = 0.50 \begin {gather*} \frac {1}{315} \, {\left (35 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + 135 \, a b^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 189 \, a^{2} b x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, a^{3} x^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} c^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(35*b^3*x^9*sgn(b*x^2 + a) + 135*a*b^2*x^7*sgn(b*x^2 + a) + 189*a^2*b*x^5*sgn(b*x^2 + a) + 105*a^3*x^3*s
gn(b*x^2 + a))*c^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (c\,{\left (b\,x^2+a\right )}^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*(a + b*x^2)^2)^(3/2),x)

[Out]

int(x^2*(c*(a + b*x^2)^2)^(3/2), x)

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