3.1.74 \(\int (a+b x^2)^5 \, dx\) [74]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 62, 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 = {200} \begin {gather*} 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 {gather*}

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 200

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*}

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Mathematica [A]
time = 0.00, size = 62, normalized size = 1.00 \begin {gather*} 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 {gather*}

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 = 0.02, size = 55, normalized size = 0.89

method	result	size
gosper	\(a^{5} x +\frac {5}{3} a^{4} b \,x^{3}+2 a^{3} b^{2} x^{5}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{11} b^{5} x^{11}\)	\(55\)
default	\(a^{5} x +\frac {5}{3} a^{4} b \,x^{3}+2 a^{3} b^{2} x^{5}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{11} b^{5} x^{11}\)	\(55\)
norman	\(a^{5} x +\frac {5}{3} a^{4} b \,x^{3}+2 a^{3} b^{2} x^{5}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{11} b^{5} x^{11}\)	\(55\)
risch	\(a^{5} x +\frac {5}{3} a^{4} b \,x^{3}+2 a^{3} b^{2} x^{5}+\frac {10}{7} a^{2} b^{3} x^{7}+\frac {5}{9} a \,b^{4} x^{9}+\frac {1}{11} b^{5} x^{11}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 54, normalized size = 0.87 \begin {gather*} \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 {gather*}

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 = 1.03, size = 54, normalized size = 0.87 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Sympy [A]
time = 0.01, size = 61, normalized size = 0.98 \begin {gather*} 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 {gather*}

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 = 1.60, size = 54, normalized size = 0.87 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.02, size = 54, normalized size = 0.87 \begin {gather*} 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5,x)

[Out]

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

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