3.1013 \(\int \frac {x^3}{\sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=36 \[ \frac {\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac {a \sqrt {a+b x^2}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {5, 266, 43} \[ \frac {\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac {a \sqrt {a+b x^2}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]

[Out]

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

Rule 5

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + b*x^n)^p, x] /; FreeQ[{
a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[c, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt {a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {x^3}{\sqrt {a+b x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {a+b x^2}}{b^2}+\frac {\left (a+b x^2\right )^{3/2}}{3 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.75 \[ \frac {\left (b x^2-2 a\right ) \sqrt {a+b x^2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4],x]

[Out]

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

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fricas [A]  time = 0.76, size = 23, normalized size = 0.64 \[ \frac {\sqrt {b x^{2} + a} {\left (b x^{2} - 2 \, a\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(b*x^2 + a)*(b*x^2 - 2*a)/b^2

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giac [A]  time = 0.15, size = 30, normalized size = 0.83 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {\sqrt {b x^{2} + a} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/3*(b*x^2 + a)^(3/2)/b^2 - sqrt(b*x^2 + a)*a/b^2

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maple [A]  time = 0.00, size = 25, normalized size = 0.69 \[ -\frac {\sqrt {b \,x^{2}+a}\, \left (-b \,x^{2}+2 a \right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2+a)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.02, size = 33, normalized size = 0.92 \[ \frac {\sqrt {b x^{2} + a} x^{2}}{3 \, b} - \frac {2 \, \sqrt {b x^{2} + a} a}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(b*x^2 + a)*x^2/b - 2/3*sqrt(b*x^2 + a)*a/b^2

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mupad [B]  time = 4.60, size = 24, normalized size = 0.67 \[ -\frac {\sqrt {b\,x^2+a}\,\left (2\,a-b\,x^2\right )}{3\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.55, size = 44, normalized size = 1.22 \[ \begin {cases} - \frac {2 a \sqrt {a + b x^{2}}}{3 b^{2}} + \frac {x^{2} \sqrt {a + b x^{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2+a)**(1/2),x)

[Out]

Piecewise((-2*a*sqrt(a + b*x**2)/(3*b**2) + x**2*sqrt(a + b*x**2)/(3*b), Ne(b, 0)), (x**4/(4*sqrt(a)), True))

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