3.1.49 \(\int (a+c x^2)^4 \, dx\) [49]

Optimal. Leaf size=51 \[ a^4 x+\frac {4}{3} a^3 c x^3+\frac {6}{5} a^2 c^2 x^5+\frac {4}{7} a c^3 x^7+\frac {c^4 x^9}{9} \]

[Out]

a^4*x+4/3*a^3*c*x^3+6/5*a^2*c^2*x^5+4/7*a*c^3*x^7+1/9*c^4*x^9

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Rubi [A]
time = 0.01, antiderivative size = 51, 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^4 x+\frac {4}{3} a^3 c x^3+\frac {6}{5} a^2 c^2 x^5+\frac {4}{7} a c^3 x^7+\frac {c^4 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^4,x]

[Out]

a^4*x + (4*a^3*c*x^3)/3 + (6*a^2*c^2*x^5)/5 + (4*a*c^3*x^7)/7 + (c^4*x^9)/9

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

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Mathematica [A]
time = 0.00, size = 51, normalized size = 1.00 \begin {gather*} a^4 x+\frac {4}{3} a^3 c x^3+\frac {6}{5} a^2 c^2 x^5+\frac {4}{7} a c^3 x^7+\frac {c^4 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^4,x]

[Out]

a^4*x + (4*a^3*c*x^3)/3 + (6*a^2*c^2*x^5)/5 + (4*a*c^3*x^7)/7 + (c^4*x^9)/9

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Maple [A]
time = 0.41, size = 44, normalized size = 0.86

method	result	size
gosper	\(a^{4} x +\frac {4}{3} c \,a^{3} x^{3}+\frac {6}{5} a^{2} c^{2} x^{5}+\frac {4}{7} a \,c^{3} x^{7}+\frac {1}{9} c^{4} x^{9}\)	\(44\)
default	\(a^{4} x +\frac {4}{3} c \,a^{3} x^{3}+\frac {6}{5} a^{2} c^{2} x^{5}+\frac {4}{7} a \,c^{3} x^{7}+\frac {1}{9} c^{4} x^{9}\)	\(44\)
norman	\(a^{4} x +\frac {4}{3} c \,a^{3} x^{3}+\frac {6}{5} a^{2} c^{2} x^{5}+\frac {4}{7} a \,c^{3} x^{7}+\frac {1}{9} c^{4} x^{9}\)	\(44\)
risch	\(a^{4} x +\frac {4}{3} c \,a^{3} x^{3}+\frac {6}{5} a^{2} c^{2} x^{5}+\frac {4}{7} a \,c^{3} x^{7}+\frac {1}{9} c^{4} x^{9}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^4,x,method=_RETURNVERBOSE)

[Out]

a^4*x+4/3*c*a^3*x^3+6/5*a^2*c^2*x^5+4/7*a*c^3*x^7+1/9*c^4*x^9

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Maxima [A]
time = 0.28, size = 43, normalized size = 0.84 \begin {gather*} \frac {1}{9} \, c^{4} x^{9} + \frac {4}{7} \, a c^{3} x^{7} + \frac {6}{5} \, a^{2} c^{2} x^{5} + \frac {4}{3} \, a^{3} c x^{3} + a^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4,x, algorithm="maxima")

[Out]

1/9*c^4*x^9 + 4/7*a*c^3*x^7 + 6/5*a^2*c^2*x^5 + 4/3*a^3*c*x^3 + a^4*x

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Fricas [A]
time = 1.70, size = 43, normalized size = 0.84 \begin {gather*} \frac {1}{9} \, c^{4} x^{9} + \frac {4}{7} \, a c^{3} x^{7} + \frac {6}{5} \, a^{2} c^{2} x^{5} + \frac {4}{3} \, a^{3} c x^{3} + a^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/9*c^4*x^9 + 4/7*a*c^3*x^7 + 6/5*a^2*c^2*x^5 + 4/3*a^3*c*x^3 + a^4*x

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Sympy [A]
time = 0.01, size = 49, normalized size = 0.96 \begin {gather*} a^{4} x + \frac {4 a^{3} c x^{3}}{3} + \frac {6 a^{2} c^{2} x^{5}}{5} + \frac {4 a c^{3} x^{7}}{7} + \frac {c^{4} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**4,x)

[Out]

a**4*x + 4*a**3*c*x**3/3 + 6*a**2*c**2*x**5/5 + 4*a*c**3*x**7/7 + c**4*x**9/9

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Giac [A]
time = 0.99, size = 43, normalized size = 0.84 \begin {gather*} \frac {1}{9} \, c^{4} x^{9} + \frac {4}{7} \, a c^{3} x^{7} + \frac {6}{5} \, a^{2} c^{2} x^{5} + \frac {4}{3} \, a^{3} c x^{3} + a^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4,x, algorithm="giac")

[Out]

1/9*c^4*x^9 + 4/7*a*c^3*x^7 + 6/5*a^2*c^2*x^5 + 4/3*a^3*c*x^3 + a^4*x

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Mupad [B]
time = 0.03, size = 43, normalized size = 0.84 \begin {gather*} a^4\,x+\frac {4\,a^3\,c\,x^3}{3}+\frac {6\,a^2\,c^2\,x^5}{5}+\frac {4\,a\,c^3\,x^7}{7}+\frac {c^4\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^4,x)

[Out]

a^4*x + (c^4*x^9)/9 + (4*a^3*c*x^3)/3 + (4*a*c^3*x^7)/7 + (6*a^2*c^2*x^5)/5

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