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

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

[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 = 0.10, antiderivative size = 187, normalized size of antiderivative = 1.31, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1989, 1112, 270} \[ \frac {b^3 c x^9 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{9 \left (a+b x^2\right )}+\frac {3 a b^2 c x^7 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{7 \left (a+b x^2\right )}+\frac {3 a^2 b c x^5 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{5 \left (a+b x^2\right )}+\frac {a^3 c x^3 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c*(a + b*x^2)^2)^(3/2),x]

[Out]

(a^3*c*x^3*Sqrt[a^2*c + 2*a*b*c*x^2 + b^2*c*x^4])/(3*(a + b*x^2)) + (3*a^2*b*c*x^5*Sqrt[a^2*c + 2*a*b*c*x^2 +
b^2*c*x^4])/(5*(a + b*x^2)) + (3*a*b^2*c*x^7*Sqrt[a^2*c + 2*a*b*c*x^2 + b^2*c*x^4])/(7*(a + b*x^2)) + (b^3*c*x
^9*Sqrt[a^2*c + 2*a*b*c*x^2 + b^2*c*x^4])/(9*(a + b*x^2))

Rule 270

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 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 1989

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] &&
TrinomialQ[u, x] &&  !TrinomialMatchQ[u, x]

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 = 0.02, size = 63, normalized size = 0.44 \[ \frac {\left (105 a^3 x^3+189 a^2 b x^5+135 a b^2 x^7+35 b^3 x^9\right ) \left (c \left (a+b x^2\right )^2\right )^{3/2}}{315 \left (a+b x^2\right )^3} \]

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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fricas [A]  time = 0.71, size = 74, normalized size = 0.52 \[ \frac {{\left (35 \, b^{3} c x^{9} + 135 \, a b^{2} c x^{7} + 189 \, a^{2} b c x^{5} + 105 \, a^{3} c x^{3}\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{315 \, {\left (b x^{2} + a\right )}} \]

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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giac [A]  time = 0.27, size = 72, normalized size = 0.50 \[ \frac {1}{315} \, {\left (35 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + 135 \, a b^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 189 \, a^{2} b x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, a^{3} x^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} c^{\frac {3}{2}} \]

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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maple [A]  time = 0.01, size = 60, normalized size = 0.42 \[ \frac {\left (35 b^{3} x^{6}+135 a \,b^{2} x^{4}+189 a^{2} b \,x^{2}+105 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2} c \right )^{\frac {3}{2}} x^{3}}{315 \left (b \,x^{2}+a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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maxima [A]  time = 1.20, size = 47, normalized size = 0.33 \[ \frac {1}{9} \, b^{3} c^{\frac {3}{2}} x^{9} + \frac {3}{7} \, a b^{2} c^{\frac {3}{2}} x^{7} + \frac {3}{5} \, a^{2} b c^{\frac {3}{2}} x^{5} + \frac {1}{3} \, a^{3} c^{\frac {3}{2}} x^{3} \]

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

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

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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