3.1.45 \(\int (a+b x^2)^3 \, dx\)

Optimal. Leaf size=35 \[ a^3 x+a^2 b x^3+\frac {3}{5} a b^2 x^5+\frac {b^3 x^7}{7} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {194} \begin {gather*} a^2 b x^3+a^3 x+\frac {3}{5} a b^2 x^5+\frac {b^3 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3,x]

[Out]

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

Rule 194

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 35, normalized size = 1.00 \begin {gather*} a^3 x+a^2 b x^3+\frac {3}{5} a b^2 x^5+\frac {b^3 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3,x]

[Out]

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

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^3, x]

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{7} x^{7} b^{3} + \frac {3}{5} x^{5} b^{2} a + x^{3} b a^{2} + x a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b^3 + 3/5*x^5*b^2*a + x^3*b*a^2 + x*a^3

________________________________________________________________________________________

giac [A]  time = 0.72, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {3}{5} \, a b^{2} x^{5} + a^{2} b x^{3} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^3*x^7 + 3/5*a*b^2*x^5 + a^2*b*x^3 + a^3*x

________________________________________________________________________________________

maple [A]  time = 0.00, size = 32, normalized size = 0.91 \begin {gather*} \frac {1}{7} b^{3} x^{7}+\frac {3}{5} a \,b^{2} x^{5}+a^{2} b \,x^{3}+a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x+a^2*b*x^3+3/5*a*b^2*x^5+1/7*b^3*x^7

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {3}{5} \, a b^{2} x^{5} + a^{2} b x^{3} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^3*x^7 + 3/5*a*b^2*x^5 + a^2*b*x^3 + a^3*x

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 31, normalized size = 0.89 \begin {gather*} a^3\,x+a^2\,b\,x^3+\frac {3\,a\,b^2\,x^5}{5}+\frac {b^3\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 32, normalized size = 0.91 \begin {gather*} a^{3} x + a^{2} b x^{3} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + a**2*b*x**3 + 3*a*b**2*x**5/5 + b**3*x**7/7

________________________________________________________________________________________