3.74 \(\int (a+b x^2)^5 \, dx\)

Optimal. Leaf size=62 \[ \frac{10}{7} a^2 b^3 x^7+2 a^3 b^2 x^5+\frac{5}{3} a^4 b x^3+a^5 x+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{11}}{11} \]

[Out]

a^5*x + (5*a^4*b*x^3)/3 + 2*a^3*b^2*x^5 + (10*a^2*b^3*x^7)/7 + (5*a*b^4*x^9)/9 + (b^5*x^11)/11

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Rubi [A]  time = 0.019159, antiderivative size = 62, normalized size of antiderivative = 1., 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} \[ \frac{10}{7} a^2 b^3 x^7+2 a^3 b^2 x^5+\frac{5}{3} a^4 b x^3+a^5 x+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5,x]

[Out]

a^5*x + (5*a^4*b*x^3)/3 + 2*a^3*b^2*x^5 + (10*a^2*b^3*x^7)/7 + (5*a*b^4*x^9)/9 + (b^5*x^11)/11

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 )^5 \, dx &=\int \left (a^5+5 a^4 b x^2+10 a^3 b^2 x^4+10 a^2 b^3 x^6+5 a b^4 x^8+b^5 x^{10}\right ) \, dx\\ &=a^5 x+\frac{5}{3} a^4 b x^3+2 a^3 b^2 x^5+\frac{10}{7} a^2 b^3 x^7+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0009757, size = 62, normalized size = 1. \[ \frac{10}{7} a^2 b^3 x^7+2 a^3 b^2 x^5+\frac{5}{3} a^4 b x^3+a^5 x+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5,x]

[Out]

a^5*x + (5*a^4*b*x^3)/3 + 2*a^3*b^2*x^5 + (10*a^2*b^3*x^7)/7 + (5*a*b^4*x^9)/9 + (b^5*x^11)/11

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Maple [A]  time = 0.001, size = 55, normalized size = 0.9 \begin{align*}{a}^{5}x+{\frac{5\,{a}^{4}b{x}^{3}}{3}}+2\,{a}^{3}{b}^{2}{x}^{5}+{\frac{10\,{a}^{2}{b}^{3}{x}^{7}}{7}}+{\frac{5\,a{b}^{4}{x}^{9}}{9}}+{\frac{{b}^{5}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5,x)

[Out]

a^5*x+5/3*a^4*b*x^3+2*a^3*b^2*x^5+10/7*a^2*b^3*x^7+5/9*a*b^4*x^9+1/11*b^5*x^11

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Maxima [A]  time = 2.31468, size = 73, normalized size = 1.18 \begin{align*} \frac{1}{11} \, b^{5} x^{11} + \frac{5}{9} \, a b^{4} x^{9} + \frac{10}{7} \, a^{2} b^{3} x^{7} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/11*b^5*x^11 + 5/9*a*b^4*x^9 + 10/7*a^2*b^3*x^7 + 2*a^3*b^2*x^5 + 5/3*a^4*b*x^3 + a^5*x

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Fricas [A]  time = 1.1136, size = 122, normalized size = 1.97 \begin{align*} \frac{1}{11} x^{11} b^{5} + \frac{5}{9} x^{9} b^{4} a + \frac{10}{7} x^{7} b^{3} a^{2} + 2 x^{5} b^{2} a^{3} + \frac{5}{3} x^{3} b a^{4} + x a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/11*x^11*b^5 + 5/9*x^9*b^4*a + 10/7*x^7*b^3*a^2 + 2*x^5*b^2*a^3 + 5/3*x^3*b*a^4 + x*a^5

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Sympy [A]  time = 0.067331, size = 61, normalized size = 0.98 \begin{align*} a^{5} x + \frac{5 a^{4} b x^{3}}{3} + 2 a^{3} b^{2} x^{5} + \frac{10 a^{2} b^{3} x^{7}}{7} + \frac{5 a b^{4} x^{9}}{9} + \frac{b^{5} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5,x)

[Out]

a**5*x + 5*a**4*b*x**3/3 + 2*a**3*b**2*x**5 + 10*a**2*b**3*x**7/7 + 5*a*b**4*x**9/9 + b**5*x**11/11

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Giac [A]  time = 2.29265, size = 73, normalized size = 1.18 \begin{align*} \frac{1}{11} \, b^{5} x^{11} + \frac{5}{9} \, a b^{4} x^{9} + \frac{10}{7} \, a^{2} b^{3} x^{7} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5,x, algorithm="giac")

[Out]

1/11*b^5*x^11 + 5/9*a*b^4*x^9 + 10/7*a^2*b^3*x^7 + 2*a^3*b^2*x^5 + 5/3*a^4*b*x^3 + a^5*x